Volume
LIPIcs, Volume 374
53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)
Event
Editors
- Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Publication Details
- published at: 2026-07-01
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-428-4
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Complete Volume
LIPIcs, Volume 374, ICALP 2026, Complete Volume
Abstract
LIPIcs, Volume 374, ICALP 2026, Complete Volume
Cite as
53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 1-4010, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{bhattacharya_et_al:LIPIcs.ICALP.2026,
title = {{LIPIcs, Volume 374, ICALP 2026, Complete Volume}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {1--4010},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026},
URN = {urn:nbn:de:0030-drops-268877},
doi = {10.4230/LIPIcs.ICALP.2026},
annote = {Keywords: LIPIcs, Volume 374, ICALP 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 0:i-0:xl, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhattacharya_et_al:LIPIcs.ICALP.2026.0,
author = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {0:i--0:xl},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.0},
URN = {urn:nbn:de:0030-drops-268827},
doi = {10.4230/LIPIcs.ICALP.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Algebraic Proof Systems: An Algebraic Approach to Analysing Proofs (Invited Talk)
Abstract
Proof complexity studies the power of formal proof systems. More specifically, it investigates the resources required to certify the truth of mathematical statements. The area plays a central role in computational complexity and has close connections to circuit complexity, automated reasoning, and mathematical logic.
A key direction within the field is algebraic proof complexity, which studies proof systems based on algebraic representations of logical formulas. These systems provide insight into the power and limitations of algebraic methods for reasoning. Among them, the Ideal Proof System (IPS), introduced by Grochow and Pitassi, is the focus of this talk. IPS represents proofs as algebraic circuits certifying the unsatisfiability of systems of polynomial equations.
In this talk, we will explore the IPS proof system, its connections to algebraic complexity, and recent developments in the area. We will discuss lower bounds for IPS, structural properties of algebraic proofs, and their implications for classical questions in proof complexity. We will also discuss recent developments involving symmetry and lifting, barriers to current lower-bound techniques, and recent progress toward proving lower bounds for CNF instances. Finally, we will highlight several open problems and directions for future research.
Cite as
Nutan Limaye. Algebraic Proof Systems: An Algebraic Approach to Analysing Proofs (Invited Talk). In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{limaye:LIPIcs.ICALP.2026.1,
author = {Limaye, Nutan},
title = {{Algebraic Proof Systems: An Algebraic Approach to Analysing Proofs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {1:1--1:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.1},
URN = {urn:nbn:de:0030-drops-263903},
doi = {10.4230/LIPIcs.ICALP.2026.1},
annote = {Keywords: Proof Complexity, Ideal Proof System, Lower Bounds, Refuting CNFs}
}
Document
Invited Talk
Decidability and Complexity Borders of Reachability Problems (Invited Talk)
Abstract
Reachability problems are arguably one of the most fundamental type of decision problems in the area of infinite-state system: Essentially every non-trivial decision problem involves solving reachability problems of one kind or another.
Because of this, reachability has continuously received attention since the very early days of automata theory. It therefore seems worthwhile to characterize the decidability and complexity borders of reachability problems. By this we mean results that consider a family of decision problems and describe precisely where, within this family, a decidability or complexity border lies.
The talk will focus on two such settings: One is about decidability, where we aim to describe the state spaces for which reachability is decidable. The other is about complexity, where we aim to describe which kinds of target sets permit polynomial-time algorithms.
Cite as
Georg Zetzsche. Decidability and Complexity Borders of Reachability Problems (Invited Talk). In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 2:1-2:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{zetzsche:LIPIcs.ICALP.2026.2,
author = {Zetzsche, Georg},
title = {{Decidability and Complexity Borders of Reachability Problems}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {2:1--2:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.2},
URN = {urn:nbn:de:0030-drops-263919},
doi = {10.4230/LIPIcs.ICALP.2026.2},
annote = {Keywords: infinite-state systems, pushdown, vector addition systems, reachability, decidability, complexity}
}
Document
Track A: Algorithms, Complexity and Games
Pseudo-Deterministic Quantum Algorithms
Abstract
We initiate a systematic study of pseudo-deterministic quantum algorithms. These are quantum algorithms that, for any input, output a canonical solution with high probability. Focusing on the query complexity model, our main contributions include the following complexity separations, which require new lower bound techniques specifically tailored to pseudo-determinism:
- We exhibit a problem, Avoid One Encrypted String (AOES), whose classical randomized query complexity is O(1) but is maximally hard for pseudo-deterministic quantum algorithms (Ω(N) query complexity).
- We exhibit a problem, Quantum-Locked Estimation (QL-Estimation), for which pseudo-deterministic quantum algorithms admit an exponential speed-up over classical pseudo-deterministic algorithms (O(log(N)) vs. Θ(√N)), while the randomized query complexity is O(1).
Complementing these separations, we show that for any total problem R, pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms, i.e., 𝖣(R) = Õ(psQ(R)⁵). On the algorithmic side, we identify a class of quantum search problems that can be made pseudo-deterministic with small overhead, including Grover search, element distinctness, triangle finding, k-sum, and graph collision.
Cite as
Hugo Aaronson, Tom Gur, and Jiawei Li. Pseudo-Deterministic Quantum Algorithms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aaronson_et_al:LIPIcs.ICALP.2026.3,
author = {Aaronson, Hugo and Gur, Tom and Li, Jiawei},
title = {{Pseudo-Deterministic Quantum Algorithms}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {3:1--3:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.3},
URN = {urn:nbn:de:0030-drops-263928},
doi = {10.4230/LIPIcs.ICALP.2026.3},
annote = {Keywords: Pseudo-determinism, Quantum Computing, Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Tight Algorithm and Hardness for Submodular Linear Ordering
Abstract
We consider the Minimum Linear Ordering Problem: given a ground set N of cardinality n and a non-negative set function f: 2^N → ℝ_{≥0}, the goal is to find an ordering π of N that minimizes the sum of the values of f over all prefixes of π. This problem has been studied for various classes of set functions, and the case of a submodular f is of special interest, as it captures classic problems including Minimum Linear Arrangement and Minimum Containing Interval Graph. In this work, we resolve the approximability of the Minimum Linear Ordering Problem for a general submodular f by establishing matching upper and lower bounds and present: (1) a polynomial-time algorithm achieving an O(√{n/ln n})-approximation; and (2) a matching information-theoretic hardness result, showing that no algorithm evaluating f a polynomial number of times can achieve an o(√{n/ln n})-approximation. Previously, the best known hardness of approximation was 2, and an O(√{n/ln n})-approximation was known only for the special case where f is both submodular and symmetric.
Cite as
Evan Abboud and Roy Schwartz. Tight Algorithm and Hardness for Submodular Linear Ordering. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{abboud_et_al:LIPIcs.ICALP.2026.4,
author = {Abboud, Evan and Schwartz, Roy},
title = {{Tight Algorithm and Hardness for Submodular Linear Ordering}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {4:1--4:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.4},
URN = {urn:nbn:de:0030-drops-263932},
doi = {10.4230/LIPIcs.ICALP.2026.4},
annote = {Keywords: Submodular optimization, approximation algorithms, hardness of approximation, linear ordering, combinatorial optimization}
}
Document
Track A: Algorithms, Complexity and Games
The Impossibility of Simultaneous Time and I/O Optimality for the Planar Maxima and Convex Hull Problems
Abstract
We prove that no deterministic output-sensitive algorithm for the planar convex hull and maxima problems can obtain both optimal time and I/O complexity, where the optimality is defined with respect to both the input and output sizes. This explains why the best previous algorithms achieved an optimal I/O bound at the cost of sub-optimal running time (Goodrich et al. [FOCS, 1993]). To the best of our knowledge, the impossibility of simultaneous optimality was only shown previously for the permutation problem by Brodal and Fagerberg [STOC, 2003]. Our results imply that no optimal deterministic output-sensitive cache-oblivious algorithm exists for either problem. In addition, we present simple deterministic algorithms that match our lower bounds and that provide a trade-off between time and I/Os. On the other hand, a simple modification of our deterministic algorithm results in a randomized algorithm that simultaneously achieves optimal (worst-case) time and optimal expected I/O bounds.
Cite as
Peyman Afshani, Gerth Stølting Brodal, and Nodari Sitchinava. The Impossibility of Simultaneous Time and I/O Optimality for the Planar Maxima and Convex Hull Problems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{afshani_et_al:LIPIcs.ICALP.2026.5,
author = {Afshani, Peyman and Brodal, Gerth St{\o}lting and Sitchinava, Nodari},
title = {{The Impossibility of Simultaneous Time and I/O Optimality for the Planar Maxima and Convex Hull Problems}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {5:1--5:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.5},
URN = {urn:nbn:de:0030-drops-263943},
doi = {10.4230/LIPIcs.ICALP.2026.5},
annote = {Keywords: External Memory model, cache-oblivious algorithms, lower bounds}
}
Document
Track A: Algorithms, Complexity and Games
Quantum Advantage in Proof Systems Without Entanglement
Abstract
The study of interactive proofs in the quantum setting has yielded profound insights in complexity theory and quantum information. A curious feature of these results is that the advantage, in terms of computational power, of quantum models over their classical counterparts is usually due to entanglement phenomena rather than quantum communication with the verifier. For example, it is known that QIP = IP = PSPACE, and QMIP with unentangled provers is equal to NEXP = MIP; on the other hand, MIP* = RE.
In this work we initiate the general study of (quantum) positional multi-prover interactive proofs ((Q)PMIP), in which provers and verifiers positioned in space communicate freely save for the constraints imposed by the speed of light. We investigate how the class of languages decidable by (Q)PMIPs depends on the arrangement of the verifiers and (honest) provers. In the case of classical PMIPs, we show a dichotomy: if the arrangement satisfies what we call the "min-ball" condition, then the class is NEXP, otherwise it is PSPACE. We then exhibit an arrangement that does not satisfy the min-ball condition for which there is a quantum PMIP for EXP in the no pre-shared entanglement model.
Our construction is based on positional cryptography and MIPs with no-signaling soundness. We introduce a new positional primitive, the positional hardcore bit, which allows a pair of spatially separated players to transmit a random bit to a particular location while guaranteeing that it remains strongly unguessable elsewhere.
Cite as
Krishna Agaram, Nicholas Spooner, and Yuxi Zheng. Quantum Advantage in Proof Systems Without Entanglement. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{agaram_et_al:LIPIcs.ICALP.2026.6,
author = {Agaram, Krishna and Spooner, Nicholas and Zheng, Yuxi},
title = {{Quantum Advantage in Proof Systems Without Entanglement}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {6:1--6:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.6},
URN = {urn:nbn:de:0030-drops-263957},
doi = {10.4230/LIPIcs.ICALP.2026.6},
annote = {Keywords: Quantum interactive proofs, positional cryptography, multi-prover interactive proofs, no-signaling soundness}
}
Document
Track A: Algorithms, Complexity and Games
Improved Tree Sparsifiers in Near-Linear Time
Abstract
A tree cut-sparsifier T of quality α of a graph G is a single tree that preserves the capacities of all cuts in the graph up to a factor of α. A tree flow-sparsifier T of quality α guarantees that every demand that can be routed in T can also be routed in G with congestion at most α.
We present a near-linear time algorithm that, for any undirected capacitated graph G = (V,E,c), constructs a tree cut-sparsifier T of quality O(log² n log log n), where n = |V|. This nearly matches the quality of the best known polynomial construction of a tree cut-sparsifier, of quality O(log^{1.5} n log log n) [Räcke and Shah, ESA 2014]. By the flow-cut gap, our result yields a tree flow-sparsifier (and congestion-approximator) of quality O(log³ n log log n). This improves on the celebrated result of [Räcke, Shah, and Täubig, SODA 2014] (RST) that gave a near-linear time construction of a tree flow-sparsifier of quality O(log⁴ n).
Our algorithm builds on a recent expander decomposition algorithm by [Agassy, Dorfman, and Kaplan, ICALP 2023], which we use as a black box to obtain a clean and modular foundation for tree cut-sparsifiers. This yields an improved and simplified version of the RST construction for cut-sparsifiers with quality O(log³ n). We then introduce a near-linear time refinement phase that controls the load accumulated on boundary edges of the sub-clusters across the levels of the tree. Combining the improved framework with this refinement phase leads to our final O(log² n log log n) tree cut-sparsifier.
Cite as
Daniel Agassy, Dani Dorfman, and Haim Kaplan. Improved Tree Sparsifiers in Near-Linear Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{agassy_et_al:LIPIcs.ICALP.2026.7,
author = {Agassy, Daniel and Dorfman, Dani and Kaplan, Haim},
title = {{Improved Tree Sparsifiers in Near-Linear Time}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {7:1--7:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.7},
URN = {urn:nbn:de:0030-drops-263967},
doi = {10.4230/LIPIcs.ICALP.2026.7},
annote = {Keywords: Tree sparsifiers, cut sparsifiers, flow sparsifiers, congestion approximators, expander decomposition, near-linear time algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Mind the Gap? Not for SVP Hardness Under ETH!
Abstract
We prove new hardness results for fundamental lattice problems under the Exponential Time Hypothesis (ETH). Building on a recent breakthrough by Bitansky et al. [BHIRW24], who gave a polynomial-time reduction from 3SAT to the (gap) MAXLIN problem - a class of CSPs with linear equations over finite fields - we derive ETH hardness for several lattice problems.
First, we show that for any p ∈ [1, ∞), there exists an explicit constant γ > 1 such that CVP _{p,γ} (the 𝓁_p-norm approximate Closest Vector Problem) does not admit a 2^o(n)-time algorithm unless ETH is false. Our reduction is deterministic and proceeds via a direct reduction from (gap) MAXLIN to CVP _{p,γ}.
Our main contribution is a randomized ETH hardness result for SVP _{p,γ} (the 𝓁_p-norm approximate Shortest Vector Problem) for all p ∈ (2, ∞). This result relies on a novel geometric property of the integer lattice ℤⁿ in the 𝓁_p norm, which says that for any p ∈ (2, ∞), the number of lattice vectors close to 1/2 1_n (in the 𝓁_p norm) is exponentially larger than the number of short vectors (namely those close to the origin). We establish this property via a new inequality for the Theta function, which we use to get a randomized reduction from CVP _{p,γ} to SVP _{p,γ'}.
Finally, we also use our ideas to give some minor improvements over prior reductions from 3SAT to BDD _{p, α} (the Bounded Distance Decoding Problem), yielding better ETH hardness results for BDD _{p, α} for any p ∈ [1, ∞) and α > α_p^{‡}, where α_p^{‡} is an explicit threshold depending on p.
Cite as
Divesh Aggarwal, Rishav Gupta, Aditya Morolia, and Chuanqi Zhang. Mind the Gap? Not for SVP Hardness Under ETH!. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 8:1-8:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aggarwal_et_al:LIPIcs.ICALP.2026.8,
author = {Aggarwal, Divesh and Gupta, Rishav and Morolia, Aditya and Zhang, Chuanqi},
title = {{Mind the Gap? Not for SVP Hardness Under ETH!}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {8:1--8:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.8},
URN = {urn:nbn:de:0030-drops-263979},
doi = {10.4230/LIPIcs.ICALP.2026.8},
annote = {Keywords: Lattices, Fine-Grained Complexity, Exponential Time Hypothesis, Post-Quantum Cryptography}
}
Document
Track A: Algorithms, Complexity and Games
Symmetric Parameterised Holants on Hypergraphs: Towards a Classification for Parameterised VCSPs
Abstract
We study the complexity of the parameterised counting constraint satisfaction problem: given a set of constraints over a set of variables and a positive integer k, how many ways are there to assign k variables to 1 (and the others to 0) such that all constraints are satisfied. While this problem, and its decision version, received significant attention during the last two decades, existing work has so far exclusively focused on restricted settings such as finding and counting homomorphisms between relational structures due to Grohe (JACM 2007) and Dalmau and Jonsson (TCS 2004), or the case of finite constraint languages due to Creignou and Vollmer (SAT 2012), and Bulatov and Marx (SICOMP 2014).
In this work, we tackle a more general setting of parameterised (counting) valued constraint satisfaction problems (VCSPs) with infinite constraint languages: we allow our constraints to be chosen from an infinite set of permitted constraints and we allow our constraints to map an assignment of its variables not only to True or False, but to arbitrary values. In this setting we are able to model and classify significantly more general problems such as (weighted) parameterised factor problems on hypergraphs and counting weight-k solutions of systems of linear equations, none of which are captured by existing complexity classifications of parameterised constraint satisfaction problems.
On a formal level, we express parameterised VCSPs as parameterised holant problems on uniform hypergraphs, and we establish complete and explicit complexity dichotomy theorems for this family of problems both w.r.t. classical complexity theory (P vs. #P) and parameterised complexity (FPT vs. #W[1]). For resolving the P vs. #P question, we mainly rely on the use of hypergraph gadgets, the existence of which we prove using properties of degree sequences necessary for realisability in uniform hypergraphs. As a technical highlight, we also employ Curticapean’s "CFI Filters" (SODA 2024) - named after the Cai-Fürer-Immermann construction for bounding the expressiveness of the Weisfeiler-Leman heuristic - to establish polynomial-time algorithms for isolating vectors in the homomorphism basis of some of our holant problems. For the FPT vs. #W[1] question, we build upon the recently established combinatorial toolkit for parameterised holants on the special case of graphs by Aivasiliotis et al. (ICALP 2025) and also rely on an extension of the framework of the homomorphism basis due to Curticapean, Dell and Marx (STOC 17) to uniform hypergraphs.
Cite as
Panagiotis Aivasiliotis, Andreas Göbel, and Marc Roth. Symmetric Parameterised Holants on Hypergraphs: Towards a Classification for Parameterised VCSPs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aivasiliotis_et_al:LIPIcs.ICALP.2026.9,
author = {Aivasiliotis, Panagiotis and G\"{o}bel, Andreas and Roth, Marc},
title = {{Symmetric Parameterised Holants on Hypergraphs: Towards a Classification for Parameterised VCSPs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {9:1--9:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.9},
URN = {urn:nbn:de:0030-drops-263986},
doi = {10.4230/LIPIcs.ICALP.2026.9},
annote = {Keywords: Parameterised Complexity, Counting Problems, Constraint Satisfaction Problems, Holant Problems}
}
Document
Track A: Algorithms, Complexity and Games
Node-Weighted Triangles: Faster and Simpler
Abstract
Weighted variants of triangle detection are an important object of study because of their prominence in fine-grained complexity. We revisit the Node-Weighted Triangle problem, where the goal is to decide if a vertex-weighted graph contains a triangle whose node weights sum to zero. This problem has been the focus of a celebrated line of work, beginning with a subcubic-time algorithm [Vassilevska, Williams; STOC '06], and culminating in algorithms running almost in matrix multiplication time, O(MM(n) + n²⋅2^O(√{log n})) [Czumaj, Lingas; SODA '07], [Vassilevska W., Williams; STOC '09]. This runtime is almost-optimal, since even detecting an unweighted triangle is conjectured to require matrix multiplication time MM(n). However, the superpolylogarithmic 2^Ω(√{log n}) overhead persists in a world where near-optimal matrix multiplication is possible (i.e., MM(n) ≤ n²poly(log n)).
In this paper, we present a new algorithm solving Node-Weighted Triangle in O(MM(n)) time, closing the gap to unweighted triangle detection completely. Remarkably, our algorithm is much simpler than previous approaches, which use involved recursion schemes and communication protocols.
Cite as
Shyan Akmal and Nick Fischer. Node-Weighted Triangles: Faster and Simpler. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 10:1-10:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{akmal_et_al:LIPIcs.ICALP.2026.10,
author = {Akmal, Shyan and Fischer, Nick},
title = {{Node-Weighted Triangles: Faster and Simpler}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {10:1--10:7},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.10},
URN = {urn:nbn:de:0030-drops-263998},
doi = {10.4230/LIPIcs.ICALP.2026.10},
annote = {Keywords: fine-grained complexity, triangle detection, node-weighted triangle}
}
Document
Track A: Algorithms, Complexity and Games
Pinning on Tight Cuts: Improved Algorithm and Bounds for Unsplittable Multicommodity Flows in Outerplanar Graphs
Abstract
The multicommodity flow problem in an undirected capacitated graph G is specified by a set of source-sink pairs with nonnegative demands. A flow is feasible if it routes all demands without exceeding the edge capacities, and it is unsplittable if it routes each demand along a single path.
Let α be the smallest value such that the existence of a feasible flow implies the existence of an unsplittable flow that exceeds the edge capacities by at most + α d_max. Schrijver, Seymour, and Winkler showed that α ∈ [1.01, 1.5] if G is a cycle. These bounds were ultimately improved to α ∈ [1.1, 1.3] by Skutella and Däubel. Recently, Alemán Espinosa and Kumar extended this constant upper bound to the broader class of outerplanar graphs, and showed that if G is outerplanar then α ≤ 3.6.
We show that α ∈ [4/3,2] if G is outerplanar. We introduce a novel technique that considers the global parameters of the instance, and that may be useful in other (more general) settings where the cut-condition is sufficient, or nearly sufficient, for the existence of a feasible flow.
Cite as
David Alemán Espinosa and Niklas Schlomberg. Pinning on Tight Cuts: Improved Algorithm and Bounds for Unsplittable Multicommodity Flows in Outerplanar Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alemanespinosa_et_al:LIPIcs.ICALP.2026.11,
author = {Alem\'{a}n Espinosa, David and Schlomberg, Niklas},
title = {{Pinning on Tight Cuts: Improved Algorithm and Bounds for Unsplittable Multicommodity Flows in Outerplanar Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {11:1--11:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.11},
URN = {urn:nbn:de:0030-drops-264008},
doi = {10.4230/LIPIcs.ICALP.2026.11},
annote = {Keywords: Unsplittable Flows, Multicommodity Flows, Planar Graphs}
}
Document
Track A: Algorithms, Complexity and Games
Faster Triangulation Mixing via Transport Flows
Abstract
We prove an Õ(n²) bound for the relaxation time and the log-Sobolev time (inverse log-Sobolev constant) of the classical triangulation flip chain on a convex (n+2)-gon, implying a mixing time of Õ(n²). The previous state of the art for the mixing time of this chain due to Eppstein and Frishberg [Eppstein and Frishberg, 2023] was Õ(n³), while the best known lower bound on the mixing time due to Molloy, Reed and Steiger [Molloy et al., 1997] is Ω(n^{3/2}). Our relaxation time bound makes significant progress towards Aldous' [Aldous, 2003] conjectured bound of Θ(n^{3/2}) for the relaxation time.
We improve upon the analysis of [Eppstein and Frishberg, 2023] by further developing the framework of transport flows introduced in the work [Xiaoyu Chen et al., 2025] of Chen et al. In this light, our results can be seen as a more efficient way of using combinatorial decompositions to obtain functional inequalities for Markov chains. We hope our ideas will find other applications in the future.
Cite as
Vedat Levi Alev, Daniel Frishberg, Michail Sarantis, and Prasad Tetali. Faster Triangulation Mixing via Transport Flows. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alev_et_al:LIPIcs.ICALP.2026.12,
author = {Alev, Vedat Levi and Frishberg, Daniel and Sarantis, Michail and Tetali, Prasad},
title = {{Faster Triangulation Mixing via Transport Flows}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {12:1--12:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.12},
URN = {urn:nbn:de:0030-drops-264011},
doi = {10.4230/LIPIcs.ICALP.2026.12},
annote = {Keywords: triangulations, mixing time, log-Sobolev inequality, spectral gap, Markov chain, random walk, MCMC, transport flow, multicommodity flow}
}
Document
Track A: Algorithms, Complexity and Games
Multiplicative Assignment with Upgrades
Abstract
We study a problem related to submodular function optimization and the exact matching problem for which we show a rather peculiar status: its natural LP-relaxation can have fractional optimal vertices, but there is always also an optimal integral vertex, which we can also compute in polynomial time.
More specifically, we consider the multiplicative assignment problem with upgrades in which we are given a set of customers and suppliers and we seek to assign each customer to a different supplier. Each customer has a demand and each supplier has a regular and an upgraded cost for each unit demand provided to the respective assigned client. Our goal is to upgrade at most k suppliers and to compute an assignment in order to minimize the total resulting cost. This can be cast as the problem to compute an optimal matching in a bipartite graph with the additional constraint that we must select k edges from a certain group of edges, similar to selecting k red edges in the exact matching problem. Also, selecting the suppliers to be upgraded corresponds to maximizing a submodular set function under a cardinality constraint.
Our result yields an efficient LP-based algorithm to solve our problem optimally. In addition, we also provide a purely strongly polynomial-time algorithm for it. As an application, we obtain exact algorithms for the upgrading variant of the problem to schedule jobs on identical or uniformly related machines in order to minimize their sum of completion times, i.e., where we may upgrade up to k jobs to reduce their respective processing times.
Cite as
Alexander Armbruster, Lars Rohwedder, Stefan Weltge, Andreas Wiese, and Ruilong Zhang. Multiplicative Assignment with Upgrades. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{armbruster_et_al:LIPIcs.ICALP.2026.13,
author = {Armbruster, Alexander and Rohwedder, Lars and Weltge, Stefan and Wiese, Andreas and Zhang, Ruilong},
title = {{Multiplicative Assignment with Upgrades}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {13:1--13:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.13},
URN = {urn:nbn:de:0030-drops-264022},
doi = {10.4230/LIPIcs.ICALP.2026.13},
annote = {Keywords: Scheduling, Bipartite Matching, LP-based Algorithm, Combinatorial Optimization, Resource Allocation}
}
Document
Track A: Algorithms, Complexity and Games
Testing Sparse Functions over the Reals
Abstract
Over the last three decades, function testing has been extensively studied over Boolean, finite fields, and discrete settings. However, to encode the real-world applications more succinctly, function testing over the reals (where the domain and range, both are reals) is of prime importance. Recently, there have been some works in the direction of testing for algebraic representations of such functions: the work by Fleming and Yoshida (ITCS 20), Arora, Kelman, and Meir (SOSA 25) on linearity testing and the work of Arora, Bhattacharyya, Fleming, Kelman, and Yoshida (SODA 23) for testing low-degree polynomials. Our work follows the same avenue, wherein we study three well-studied sparse representations of functions, over the reals, namely (i) k-linearity, (ii) k-sparse, low-degree polynomials, and (iii) k-juntas.
In this setting, given approximate query access to some f:ℝⁿ → ℝ, we want to decide if the function satisfies some property of interest, or if it is far from all functions that satisfy the property. Here, the distance is measured in the 𝓁₁-metric, under the assumption that we are drawing samples from the Standard Gaussian distribution. We present efficient testers and Ω(k) lower bounds for testing each of these three properties.
Cite as
Vipul Arora, Arnab Bhattacharyya, Philips George John, and Sayantan Sen. Testing Sparse Functions over the Reals. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 14:1-14:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{arora_et_al:LIPIcs.ICALP.2026.14,
author = {Arora, Vipul and Bhattacharyya, Arnab and George John, Philips and Sen, Sayantan},
title = {{Testing Sparse Functions over the Reals}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {14:1--14:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.14},
URN = {urn:nbn:de:0030-drops-264038},
doi = {10.4230/LIPIcs.ICALP.2026.14},
annote = {Keywords: Property testing, sparsity, linearity, low-degree polynomials, juntas, computation over reals}
}
Document
Track A: Algorithms, Complexity and Games
Parallel Reachability and Shortest Paths on Non-Sparse Digraphs: Near-Linear Work and Sub-Square-Root Depth
Abstract
We present parallel algorithms for computing single-source reachability and shortest paths on directed n-vertex m-edge graphs using near-linear Õ(m) work and o(√n) depth whenever m ≥ n^{1+o(1)}. At the extreme of m = Ω(n²), our reachability and shortest path algorithms have depth only n^0.136 and n^{0.25+o(1)}, respectively. The state-of-the-art parallel algorithms with near-linear work for both problems [Jambulapati et al., 2019; Cao et al., 2020; Rozhoň et al., 2023; Cao and Fineman, 2023; Brand et al., 2025] require Ω(√n) depth in all density regimes.
Cite as
Vikrant Ashvinkumar, Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Parallel Reachability and Shortest Paths on Non-Sparse Digraphs: Near-Linear Work and Sub-Square-Root Depth. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ashvinkumar_et_al:LIPIcs.ICALP.2026.15,
author = {Ashvinkumar, Vikrant and Bernstein, Aaron and Gutenberg, Maximilian Probst and Saranurak, Thatchaphol},
title = {{Parallel Reachability and Shortest Paths on Non-Sparse Digraphs: Near-Linear Work and Sub-Square-Root Depth}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {15:1--15:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.15},
URN = {urn:nbn:de:0030-drops-264045},
doi = {10.4230/LIPIcs.ICALP.2026.15},
annote = {Keywords: shortcut set, parallel reachability, hopset, parallel SSSP}
}
Document
Track A: Algorithms, Complexity and Games
Fully Dynamic Algorithms for Coloring Triangle-Free Graphs
Abstract
A celebrated result of Johansson in graph theory states that every triangle-free graph of maximum degree Δ can be properly colored with O(Δ/lnΔ) colors, improving upon the "greedy bound" of Δ+1 coloring in general graphs. This coloring can also be found in polynomial time.
We present an algorithm for maintaining an O(Δ/lnΔ) coloring of a dynamically changing triangle-free graph that undergoes edge insertions and deletions. The algorithm is randomized and on n-vertex graphs has amortized update time of Δ^o(1) log(n) per update with high probability, even against an adaptive adversary.
A key to the analysis of our algorithm is an application of the entropy compression method that to our knowledge is new in the context of dynamic algorithms. This technique appears general and is likely to find other applications in dynamic problems and thus can be of its own independent interest.
Cite as
Sepehr Assadi and Helia Yazdanyar. Fully Dynamic Algorithms for Coloring Triangle-Free Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 16:1-16:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{assadi_et_al:LIPIcs.ICALP.2026.16,
author = {Assadi, Sepehr and Yazdanyar, Helia},
title = {{Fully Dynamic Algorithms for Coloring Triangle-Free Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {16:1--16:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.16},
URN = {urn:nbn:de:0030-drops-264053},
doi = {10.4230/LIPIcs.ICALP.2026.16},
annote = {Keywords: Dynamic graphs - Graph coloring - Dynamic entropy compression}
}
Document
Track A: Algorithms, Complexity and Games
Competitive Bundle Trading
Abstract
Allocating a set of resources to an online sequence of customers is a fundamental problem in online algorithms with an extensive history. However, the natural extension where the algorithm is also allowed to purchase inventory from suppliers, who also arrive online, is essentially unexplored. We study this general trading problem under the objective of profit maximization, which is the difference between revenue from sales and cost of purchases. Maximizing the difference between two competing quantities is significantly more challenging than the sell-only case.
We show a logarithmic competitive ratio relative to the optimal offline solution. Our algorithm is an exponential-weight–update dynamic pricing scheme, and our analysis dual-fits the algorithm’s profit with respect to a linear programming relaxation that upper bounds the optimal offline profit; we also prove (nearly) matching lower bounds. Finally, we extend our results by designing an incentive-compatible mechanism for the setting in which customers are strategic and may misreport their true valuations.
Cite as
Yossi Azar, Niv Buchbinder, Roie Levin, and Or Vardi. Competitive Bundle Trading. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{azar_et_al:LIPIcs.ICALP.2026.17,
author = {Azar, Yossi and Buchbinder, Niv and Levin, Roie and Vardi, Or},
title = {{Competitive Bundle Trading}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {17:1--17:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.17},
URN = {urn:nbn:de:0030-drops-264066},
doi = {10.4230/LIPIcs.ICALP.2026.17},
annote = {Keywords: Online algorithms, competitive analysis, algorithmic game theory, mechanism design, dynamic pricing, resource allocation}
}
Document
Track A: Algorithms, Complexity and Games
Online Metric TSP: Beyond the √n Barrier
Abstract
We study an online variant of the Traveling Salesperson Problem (TSP) in which n points arrive sequentially and must be inserted into an evolving tour. In the classical setting where arbitrary insertions are allowed, an O(log n)-competitive algorithm has been known since the 1970s (Rosenkrantz, Stearns and Lewis 1977, Imase and Waxman 1991). Recently, Abrahamsen, Bercea, Beretta, Klausen, and Kozma [ESA 2024] introduced online metric TSP, a stricter model in which each arriving point must be assigned to a distinct cell of an array of size m ≥ n, with the final tour order induced by the non-empty cells; the parameter m captures the space usage of the algorithm.
When m = 2ⁿ, this model recovers arbitrary insertions and therefore admits an O(log n)-competitive algorithm. In contrast, when m = n, i.e., when each point’s position is fixed on arrival, Bertram [Christian Bertram, 2025] recently showed that the competitive ratio is Θ(√n). We investigate the tradeoff between space usage and competitiveness between these extremes. We note that this tradeoff was previously explored by the authors in [Yossi Azar et al., 2026] for the online sorting problem, which is the special case of online metric TSP on a line metric.
Our main result is a deterministic online metric TSP algorithm using m = (1+ε) n space that achieves a competitive ratio of O(log³ n/ε), for any ε ≤ 1. In particular, increasing the space from n to 2n improves the competitive ratio from Θ(√n) to O(log³ n). We complement this with a lower bound showing that for m = n^{1+ε}, any deterministic algorithm has a competitive ratio Ω(1/ε), for all ε ≥ Ω(log log n / log n). Consequently, even with m = O(n ⋅ polylog(n)), deterministic algorithms cannot achieve a constant competitive ratio.
Cite as
Yossi Azar, Debmalya Panigrahi, and Or Vardi. Online Metric TSP: Beyond the √n Barrier. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{azar_et_al:LIPIcs.ICALP.2026.18,
author = {Azar, Yossi and Panigrahi, Debmalya and Vardi, Or},
title = {{Online Metric TSP: Beyond the √n Barrier}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {18:1--18:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.18},
URN = {urn:nbn:de:0030-drops-264071},
doi = {10.4230/LIPIcs.ICALP.2026.18},
annote = {Keywords: Online algorithms, competitive analysis, metric TSP, space-competitiveness tradeoff, routing problems}
}
Document
Track A: Algorithms, Complexity and Games
Clustering Permutations Under the Ulam Metric: A Parameterized Complexity Study
Abstract
Rank aggregation seeks a representative permutation for a collection of rankings and plays a central role in areas such as social choice, information retrieval, and computational biology. Two fundamental aggregation tasks are the center and median problems, which minimize the maximum and the total distance to the input permutations, respectively. While these problems are well understood under Kendall’s tau and related distances, their parameterized complexity under the Ulam metric, an edit-distance-based metric on permutations, has remained largely unexplored.
In this work, we initiate a systematic study of the parameterized complexity of rank aggregation under the Ulam metric. We consider both the center and median problems, as well as their generalizations to the k-center and k-median clustering settings, parameterized by the number of centers k and the distance budget d (corresponding to the maximum distance for center variants and the total distance for median variants). Both problems are known to be NP-hard already for k = 1.
We show that the Ulam k-center problem remains NP-hard when d = 1, but is fixed-parameter tractable when parameterized by k + d. Our algorithm is based on a novel local-search framework tailored to the non-local nature of Ulam distances. We complement this by proving that no polynomial kernel exists for the k+d parameterization unless NP ⊆ coNP/poly. For the Ulam k-median problem parameterized by the total distance d, we establish W[1]-hardness and provide an XP algorithm. We also provide a polynomial kernel for the parameter k + d, which in turn yields a fixed-parameter tractable algorithm.
Cite as
Tian Bai, Fedor V. Fomin, Petr A. Golovach, Yash Hiren More, and Simon Wietheger. Clustering Permutations Under the Ulam Metric: A Parameterized Complexity Study. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 19:1-19:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bai_et_al:LIPIcs.ICALP.2026.19,
author = {Bai, Tian and Fomin, Fedor V. and Golovach, Petr A. and More, Yash Hiren and Wietheger, Simon},
title = {{Clustering Permutations Under the Ulam Metric: A Parameterized Complexity Study}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {19:1--19:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.19},
URN = {urn:nbn:de:0030-drops-264080},
doi = {10.4230/LIPIcs.ICALP.2026.19},
annote = {Keywords: parameterized complexity, Ulam distance, rank aggregation, clustering}
}
Document
Track A: Algorithms, Complexity and Games
On the Average-Case Performance of Greedy for Maximum Coverage
Abstract
For the classical maximum coverage problem, the greedy algorithm achieves a worst-case 1-1/e approximation, which is optimal unless P = NP. The notion of coverage appears in a wide range of optimization tasks, where empirical evaluations indicate approximation ratios close to 1 for the greedy algorithm on real data. Random models have provided average-case justifications for the empirical performance of many well-known algorithms, but little is known about the average-case performance of greedy for maximum coverage.
We analyze the expected approximation ratio of the greedy algorithm in a random model, which we call the left-regular random model. We first show that, for all parameter settings of this model, the expected approximation ratio of the greedy algorithm improves by a constant over its worst-case 1-1/e guarantee. We then identify two simple conditions, either of which ensures that the expected approximation ratio is close to 1 for sufficiently large graphs. Finally, we show that there is a regime where greedy does not achieve an expected approximation better than 0.94. To obtain these results, we develop analytical tools, including a novel application of the differential equation method and a connection to maximum matching in Erdős-Rényi graphs, which may be of independent interest for other random models.
Cite as
Eric Balkanski, Jason Chatzitheodorou, and Flore Sentenac. On the Average-Case Performance of Greedy for Maximum Coverage. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{balkanski_et_al:LIPIcs.ICALP.2026.20,
author = {Balkanski, Eric and Chatzitheodorou, Jason and Sentenac, Flore},
title = {{On the Average-Case Performance of Greedy for Maximum Coverage}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {20:1--20:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.20},
URN = {urn:nbn:de:0030-drops-264099},
doi = {10.4230/LIPIcs.ICALP.2026.20},
annote = {Keywords: Maximum Coverage, Greedy Algorithm, Average-Case Analysis, Differential Equation Method, Random Graphs}
}
Document
Track A: Algorithms, Complexity and Games
Geometric Optimization Parameterized by Piercing Complexity
Abstract
Packing and Covering problems with geometric regions in the plane have been extensively studied and several notions of "complexity" of the regions involved have been developed and exploited to obtain good approximation algorithms. Examples of such complexity measures are VC-dimension, union complexity, shallow-cell complexity, fatness, etc. While these restrictions lead to constant-factor approximation algorithms in many cases, they typically do not lead to PTASs. In fact, several geometric Set Cover and Discrete Independent Set variants remain APX-hard even when these parameters are small, as demonstrated in earlier work by Chan and Grant (Exact algorithms and APX-hardness results for geometric packing and covering problems. Comput. Geom., 2014), and by Har-Peled and Quanrud (Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs. SIAM J. Comput., 2017). A key feature of these hardness constructions is that many pairs of regions in the input pierce one another.
Motivated by this observation, we initiate a systematic study of geometric families parameterized by their piercing complexity. A connected region A is said to pierce a connected region B if B ⧵ A has more than one connected component; we consider instances in which every region is pierced by at most a constant number of others. This framework smoothly interpolates between the classical non-piercing case-where local-search PTASs are known due to Raman and Ray (Constructing Planar Support for Non-Piercing Regions, Discret. Comput. Geom., 2020), and the fully general case, where APX-hardness persists.
Our main contribution is to show that bounded-piercing families admit efficient approximation schemes for fundamental geometric optimization problems. For regions in the plane with a constant piercing bound, we obtain PTASs for the (unweighted) Discrete Independent Set and Set Cover problems, and constant-factor approximation algorithms for their weighted variants. These results strictly generalize the known PTASs for non-piercing families and yield improved guarantees for several long-standing special cases, including Independent Set and Set Cover with axis-parallel rectangles under bounded piercing.
Overall, our work identifies piercing complexity as a robust and expressive topological parameter-distinct from geometric notions such as density or fatness-and demonstrates that bounding this parameter yields a broad family of geometric instances for which PTASs become achievable.
Cite as
Aritra Banik, Rajiv Raman, and Saurabh Ray. Geometric Optimization Parameterized by Piercing Complexity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{banik_et_al:LIPIcs.ICALP.2026.21,
author = {Banik, Aritra and Raman, Rajiv and Ray, Saurabh},
title = {{Geometric Optimization Parameterized by Piercing Complexity}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {21:1--21:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.21},
URN = {urn:nbn:de:0030-drops-264102},
doi = {10.4230/LIPIcs.ICALP.2026.21},
annote = {Keywords: Geometric set cover, geometric discrete independent set, approximation algorithms, PTAS, parameterized complexity, piercing complexity, non-piercing regions, independent set, axis-parallel rectangles}
}
Document
Track A: Algorithms, Complexity and Games
Expander Decomposition with Almost Optimal Overhead
Abstract
We present the first polynomial-time algorithm for computing a near-optimal flow-expander decomposition. Given a graph G and a parameter ϕ, our algorithm removes at most a ϕlog^{1+o(1)}n fraction of edges so that every remaining connected component is a ϕ-flow-expander (a stronger guarantee than being a ϕ-cut-expander). This achieves overhead log^{1+o(1)}n, nearly matching the Ω(log n) graph-theoretic lower bound that already holds for cut-expander decompositions, up to a log^{o(1)}n factor. Prior polynomial-time algorithms required removing O(ϕlog^{1.5}n) and O(ϕlog²n) fractions of edges to guarantee ϕ-cut-expander and ϕ-flow-expander components, respectively.
Cite as
Nikhil Bansal, Arun Jambulapati, and Thatchaphol Saranurak. Expander Decomposition with Almost Optimal Overhead. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bansal_et_al:LIPIcs.ICALP.2026.22,
author = {Bansal, Nikhil and Jambulapati, Arun and Saranurak, Thatchaphol},
title = {{Expander Decomposition with Almost Optimal Overhead}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {22:1--22:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.22},
URN = {urn:nbn:de:0030-drops-264113},
doi = {10.4230/LIPIcs.ICALP.2026.22},
annote = {Keywords: Graph algorithms, expander decomposition, flow expansion, sparse cuts}
}
Document
Track A: Algorithms, Complexity and Games
Solving Random Planted CSPs Below the n^{k/2} Threshold
Abstract
We present a family of algorithms to solve random planted instances of any k-ary Boolean constraint satisfaction problem (CSP). A randomly planted instance of a Boolean CSP is generated by (1) choosing an arbitrary planted assignment x^*, and then (2) sampling constraints from a particular "planting distribution" designed so that x^* will satisfy every constraint. Given an n variable instance of a k-ary Boolean CSP with m constraints, our algorithm runs in time n^O(𝓁) for a choice of a parameter 𝓁, and succeeds in outputting a satisfying assignment if m ⩾ O(n)⋅(n/𝓁)^{k/2 - 1} log n. This generalizes the poly(n)-time algorithm of [Vitaly Feldman et al., 2015], the case of 𝓁 = O(1), to larger runtimes, and matches the constraint number vs. runtime trade-off established for refuting random CSPs by [Prasad Raghavendra et al., 2017].
Our algorithm is conceptually different from the recent algorithm of [Venkatesan Guruswami et al., 2023], which gave a poly(n)-time algorithm to solve semirandom CSPs with m ⩾ Õ(n^{k/2}) constraints by exploiting conditions that allow a basic SDP to recover the planted assignment x^* exactly. Instead, we forego certificates of uniqueness and recover x^* in two steps: we first use a degree-O(𝓁) Sum-of-Squares SDP to find some x̂ that is o(1)-close to x^*, and then we use a second rounding procedure to recover x^* from x̂.
Cite as
Arpon Basu, Jun-Ting Hsieh, Andrew D. Lin, and Peter Manohar. Solving Random Planted CSPs Below the n^{k/2} Threshold. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{basu_et_al:LIPIcs.ICALP.2026.23,
author = {Basu, Arpon and Hsieh, Jun-Ting and Lin, Andrew D. and Manohar, Peter},
title = {{Solving Random Planted CSPs Below the n^\{k/2\} Threshold}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {23:1--23:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.23},
URN = {urn:nbn:de:0030-drops-264127},
doi = {10.4230/LIPIcs.ICALP.2026.23},
annote = {Keywords: Random CSPs, Sparse Learning Parity with Noise}
}
Document
Track A: Algorithms, Complexity and Games
Compressing Suffix Trees by Path Decompositions
Abstract
The suffix tree is arguably the most fundamental data structure on strings: introduced by Weiner (SWAT 1973) and McCreight (JACM 1976), it allows solving a myriad of computational problems on strings in linear time. Motivated by its large space usage, subsequent research focused first on reducing its size by a constant factor via Suffix Arrays, and later on reaching space proportional to the size of the compressed string. Modern compressed indexes, such as the r-index (Gagie et al., JACM 2020), fit in space proportional to r, the number of runs in the Burrows-Wheeler transform (a strong and universal repetitiveness measure). These advances, however, came with a price: while modern compressed indexes boast optimal bounds in the RAM model, they are often orders of magnitude slower than uncompressed counterparts in practice due to catastrophic cache locality. This reality gap highlights that Big-O complexity in the RAM model has become a misleading predictor of real-world performance, leaving a critical question unanswered: can we design compressed indexes that are efficient in the I/O model of computation?
We answer this in the affirmative by introducing a new Suffix Array sampling technique based on particular path decompositions of the suffix tree. We prove that sorting the suffix tree leaves by specific priority functions induces a decomposition where the number of distinct paths (each corresponding to a string suffix) is bounded by r. This allows us to solve indexed pattern matching efficiently in the I/O model using a Suffix Array sample of size at most r, strictly improving upon the (tight) 2r bound of Suffixient Arrays, another recent compressed Suffix Array sampling technique.
Experiments confirm that this theoretical I/O efficiency translates to practice in pangenomic applications: our index locates pattern occurrences using less space and orders of magnitude less time than the r-index when performing pattern matching on repetitive DNA collections. Beyond this, our contributions are twofold: (i) unlike Suffixient Arrays, our technique supports most standard suffix tree operations in O(r) space on top of the text while matching the I/O complexity of uncompressed suffix trees; and (ii) we establish a general framework where any valid path decomposition induces a Suffix Array sampling whose size is a new strong repetitiveness measure; we provide a universal mechanism for locating all pattern occurrences for each such path decomposition.
Cite as
Ruben Becker, Davide Cenzato, Travis Gagie, Ragnar Groot Koerkamp, Sung-Hwan Kim, Giovanni Manzini, and Nicola Prezza. Compressing Suffix Trees by Path Decompositions. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 24:1-24:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{becker_et_al:LIPIcs.ICALP.2026.24,
author = {Becker, Ruben and Cenzato, Davide and Gagie, Travis and Groot Koerkamp, Ragnar and Kim, Sung-Hwan and Manzini, Giovanni and Prezza, Nicola},
title = {{Compressing Suffix Trees by Path Decompositions}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {24:1--24:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.24},
URN = {urn:nbn:de:0030-drops-264139},
doi = {10.4230/LIPIcs.ICALP.2026.24},
annote = {Keywords: Text indexing, suffix tree, I/O-efficient, Compressed Data Structures}
}
Document
Track A: Algorithms, Complexity and Games
Permutation Patterns in Streams
Abstract
Permutation patterns and pattern avoidance are central, well-studied concepts in combinatorics and computer science. Given two permutations τ and π, the pattern matching problem (PPM) asks whether τ contains π. This problem arises in various contexts in computer science and statistics and has been studied extensively in exact-, parameterized-, approximate-, property-testing- and other formulations.
In this paper, we study pattern matching in a streaming setting, when the input τ is revealed sequentially, one element at a time. There is extensive work on the space complexity of various statistics in streams of integers. The novelty of our setting is that the input stream is a permutation, which allows inferring some information about future inputs. Our algorithms crucially take advantage of this fact, while existing lower bound techniques become difficult to apply.
We show that the complexity of the problem changes dramatically depending on the pattern π. The space requirement is:
- Θ(klog{n}) for the monotone patterns π = 12…k, or π = k…21,
- 𝒪(√{nlog{n}}) for π ∈ {312,132},
- 𝒪(√n log n) for π ∈ {231,213},
- Θ̃_π(n) for all other π. If τ is an arbitrary sequence of integers (not necessary a permutation), we show that the complexity is Θ̃_π(n) in all except the first (monotone) cases.
Cite as
Benjamin Aram Berendsohn. Permutation Patterns in Streams. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{berendsohn:LIPIcs.ICALP.2026.25,
author = {Berendsohn, Benjamin Aram},
title = {{Permutation Patterns in Streams}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {25:1--25:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.25},
URN = {urn:nbn:de:0030-drops-264144},
doi = {10.4230/LIPIcs.ICALP.2026.25},
annote = {Keywords: permutations, pattern matching, streaming}
}
Document
Track A: Algorithms, Complexity and Games
Fast Decremental Tree Sums in Forests
Abstract
We study two fundamental decremental dynamic graph problems. In both problems, we need to maintain a vertex-weighted forest of size n under edge deletions, weight updates, and a certain information-retrieval query. Both problems can be solved in 𝒪(log n) time per update/query using standard dynamic forest data structures like top trees - even if additionally edge insertions are allowed. We investigate whether the deletion-only problem can be solved faster.
First, we consider tree-sum queries, where we ask for the sum of vertex weights in one of the connected components (i.e., trees) in the forest. We give a data structure with 𝒪(n) preprocessing time and 𝒪(log^* n) time per operation, based on a micro-macro tree decomposition (Alstrup et al., 1997). If the forest is unweighted (i.e., all weights are 1 and cannot be changed), then the operation time can be improved to 𝒪(1).
Additionally, we give an asymptotically universally optimal algorithm. More specifically, our algorithm works in the group model, and processes m operations on an initial forest F in running time 𝒪(OPT(F, m)). Here OPT(F, m) is the number of weight additions and subtractions that a best possible algorithm performs to handle a worst-case instance for a fixed initial forest F and a fixed number m of operations. We achieve this with a combination of the aforementioned decomposition technique, precomputation of optimal data structures for very small instances, and some insights into the behavior of OPT. Note that even the worst-case complexity of this algorithm remains unknown to us.
Second, we consider subtree-sum queries. Here, the forest is rooted, and a query subtree-sum(v) returns the sum of weights in the subtree rooted at v. An easy reduction from the well-known prefix sum problem shows that the general, weighted version of the problem requires Θ(n log n) time for n operations. Interestingly, we prove that the Ω(n log n) complexity lower bound still holds even if weight updates are disallowed. On the other hand, we show that the unweighted version can be solved with 𝒪((log n)/(log log n)) time per operation, and this is tight.
Cite as
Benjamin Aram Berendsohn and Marek Sokołowski. Fast Decremental Tree Sums in Forests. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 26:1-26:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{berendsohn_et_al:LIPIcs.ICALP.2026.26,
author = {Berendsohn, Benjamin Aram and Soko{\l}owski, Marek},
title = {{Fast Decremental Tree Sums in Forests}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {26:1--26:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.26},
URN = {urn:nbn:de:0030-drops-264151},
doi = {10.4230/LIPIcs.ICALP.2026.26},
annote = {Keywords: dynamic graphs, connectivity, group model, universal optimality}
}
Document
Track A: Algorithms, Complexity and Games
Plane Strong Connectivity Augmentation
Abstract
We investigate the problem of strong connectivity augmentation within plane oriented graphs. We show that deciding whether a plane oriented graph D can be augmented with (any number of) arcs X such that D+X is strongly connected, but still plane and oriented, is NP-hard. The hardness also holds for the planar variant. This question becomes trivial within plane (or planar) digraphs, like most connectivity augmentation problems without a budget constraint. The budgeted variant, Plane Strong Connectivity Augmentation (PSCA) considers a plane oriented graph D along with some integer k, and asks for an X of size at most k ensuring that D+X is strongly connected, while remaining plane and oriented. Our main result is a fixed-parameter tractable algorithm for PSCA, running in time 2^O(k) n² log n. The cornerstone of our procedure is a structural result showing that, for any fixed k, each face admits a bounded number of partial solutions "dominating" all others. Then, our algorithm for PSCA combines face-wise branching with a randomized reduction to the polynomial Minimum Dijoin problem, yielding a Monte-Carlo FPT algorithm, which we derandomize. To the best of our knowledge, this is the first FPT algorithm for a (hard) connectivity augmentation problem constrained by planarity.
Cite as
Stéphane Bessy, Daniel Gonçalves, Amadeus Reinald, and Dimitrios M. Thilikos. Plane Strong Connectivity Augmentation. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bessy_et_al:LIPIcs.ICALP.2026.27,
author = {Bessy, St\'{e}phane and Gon\c{c}alves, Daniel and Reinald, Amadeus and Thilikos, Dimitrios M.},
title = {{Plane Strong Connectivity Augmentation}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {27:1--27:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.27},
URN = {urn:nbn:de:0030-drops-264163},
doi = {10.4230/LIPIcs.ICALP.2026.27},
annote = {Keywords: Connectivity augmentation, Directed graphs, Parameterized complexity}
}
Document
Track A: Algorithms, Complexity and Games
The Stochastic Block Model Has the Overlap Graph Property for Modularity
Abstract
The overlap gap property (OGP) is a statement about the geometry of near-optimal solutions. Exhibiting OGP implies failure of a class of local algorithms; and has been observed to coincide with conjectured algorithmic limits in problems with statistical computational gap.
We consider the Stochastic Block Model (SBM), where the graph has a planted partition with k equal-size blocks which form the "communities", and where, for parameters p > q, vertices within the same community connect with probability p, while vertices in different communities connect with probability q, independently across pairs of vertices. Modularity-based clustering algorithms have become ubiquitous in applications. This article studies theoretical limits of local algorithms based on the modularity score on the SBM.
We establish that modularity exhibits OGP on the SBM. This rules out a class of local algorithms based on modularity for recovery in the SBM, and shows slow mixing time for a related Markov Chain. Theoretically this is one of the few instances where OGP has been established for a "planted" model, as most such analyses to date consider the "null" model.
As part of our analysis, we extend a result by Bickel and Chen 2009, who established that with high probability, the modularity optimal partition of SBM is o(n) local moves away from the planted partition, where n is the graph size. We show that, with high probability, any partition with modularity score sufficiently near the optimal value is close to the planted partition.
Cite as
Shankar Bhamidi, David Gamarnik, Remco van der Hofstad, Nelly Litvak, Paweł Prałat, Fiona Skerman, and Yasmin Tousinejad. The Stochastic Block Model Has the Overlap Graph Property for Modularity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhamidi_et_al:LIPIcs.ICALP.2026.28,
author = {Bhamidi, Shankar and Gamarnik, David and van der Hofstad, Remco and Litvak, Nelly and Pra{\l}at, Pawe{\l} and Skerman, Fiona and Tousinejad, Yasmin},
title = {{The Stochastic Block Model Has the Overlap Graph Property for Modularity}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {28:1--28:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.28},
URN = {urn:nbn:de:0030-drops-264177},
doi = {10.4230/LIPIcs.ICALP.2026.28},
annote = {Keywords: community detection, average-case complexity, overlap gap property, modularity, Louvain, stochastic block model}
}
Document
Track A: Algorithms, Complexity and Games
A 4.509-Approximation Algorithm for Generalized Min Sum Set Cover
Abstract
We study the generalized min-sum set cover (GMSSC) problem, where given a collection of hyperedges E with arbitrary covering requirements {k_e ∈ ℤ^+ : e ∈ E}, the objective is to find an ordering of the vertices that minimizes the total cover time of the hyperedges. A hyperedge e is considered covered at the first time when k_e of its vertices appear in the ordering.
We present a 4.509-approximation algorithm for GMSSC, improving upon the previous best-known guarantee of 4.642 [Nikhil Bansal et al., 2021]. Our approach retains the general LP-based framework of Bansal, Batra, Farhadi, and Tetali [Nikhil Bansal et al., 2021] but provides an improved analysis that narrows the gap toward the lower bound of 4-approximation assuming P≠NP. Our analysis takes advantage of the constraints of the linear program in a nontrivial way, along with new lower-tail bounds for the sums of independent Bernoulli random variables, which could be of independent interest.
Cite as
Amey Bhangale and Yezhou Zhang. A 4.509-Approximation Algorithm for Generalized Min Sum Set Cover. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhangale_et_al:LIPIcs.ICALP.2026.29,
author = {Bhangale, Amey and Zhang, Yezhou},
title = {{A 4.509-Approximation Algorithm for Generalized Min Sum Set Cover}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {29:1--29:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.29},
URN = {urn:nbn:de:0030-drops-264185},
doi = {10.4230/LIPIcs.ICALP.2026.29},
annote = {Keywords: Generalized Min Sum Set Cover, Approximation Algorithm, Min latency set cover, Linear programming, Knapsack cover inequalities}
}
Document
Track A: Algorithms, Complexity and Games
Optimal Inapproximability of Generalized Linear Equations over a Finite Group
Abstract
Constraint satisfaction problems (CSPs) consist of a set of variables taking values from some finite domain and a set of local constraints on these variables. The objective is to find an assignment to the variables that maximizes the fraction of satisfied constraints.
In this work, we study the CSP where the constraints are generalized linear equations over a finite group G. More specifically, for a given S ⊆ G, the constraints in this CSP are of the form addition of the values to the variables (similarly, product for non-abelian groups) belongs to the set S. We give an approximation algorithm for this problem on satisfiable instances and show that it is optimal for certain S assuming 𝐏≠ NP.
This natural predicate is one of the very few known predicates that are approximation resistant on almost satisfiable instances, assuming 𝐏≠ NP, but admits a non-trivial approximation algorithm on satisfiable instances.
Cite as
Amey Bhangale and Yezhou Zhang. Optimal Inapproximability of Generalized Linear Equations over a Finite Group. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 30:1-30:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhangale_et_al:LIPIcs.ICALP.2026.30,
author = {Bhangale, Amey and Zhang, Yezhou},
title = {{Optimal Inapproximability of Generalized Linear Equations over a Finite Group}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {30:1--30:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.30},
URN = {urn:nbn:de:0030-drops-264193},
doi = {10.4230/LIPIcs.ICALP.2026.30},
annote = {Keywords: Constraint satisfaction problems, inapproximability, approximation algorithms, non-abelian groups, Fourier analysis}
}
Document
Track A: Algorithms, Complexity and Games
An Algorithmic Proof of Kruskal’s Tensor Decomposition Theorem
Abstract
A famous theorem of Kruskal gives the simplest and arguably most fundamental criterion under which a tensor is guaranteed a unique minimum-rank decomposition. Kruskal’s condition requires that the sum of the Kruskal ranks {k_i}_{i=1}^m of the components satisfies ∑_{i∈[m]} k_i ≥ 2r + m - 1, where r denotes the rank and m the order of the tensor. However, Kruskal’s original proof and subsequent simplifications/generalizations have remained non-constructive. With the sole exception of the case (k₁ = r, k₂ = r, k₃ = 2), attributed to Jennrich - no algorithm has been established for decomposing tensors under the Kruskal condition without additional assumptions. In fact, whether there exists an efficient algorithm for decomposing a tensor under the Kruskal condition was explicitly posed as an open problem in the work of Bhaskara et al. (COLT 2014). Even slight variations of the Jennrich special case, such as the (r, r-1, 3) case, have remained algorithmically open; specifically, no sub-exponential time bound was known.
In this work, we make progress on this problem by giving an elementary, constructive proof of Kruskal’s Theorem for general m-way tensors. Concretely, we give a randomized algorithm that decomposes any tensor satisfying the Kruskal condition by utilizing random projections to map the problem into a geometry of intersecting hyperplanes via a MinRank instance. Specifically for 3-way tensors satisfying k₁+k₂+k₃ = 2r+2, the algorithm achieves a runtime of n^O(k) where k = min(k₁,k₂,k₃). Thus, we extend smoothly beyond the Jennrich special case, achieving polynomial-time complexity for any family of tensors that satisfies the Kruskal condition, provided the least Kruskal rank is bounded.
Cite as
Vishwas Bhargava, Leonard J. Schulman, and Shiri Sivan. An Algorithmic Proof of Kruskal’s Tensor Decomposition Theorem. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 31:1-31:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhargava_et_al:LIPIcs.ICALP.2026.31,
author = {Bhargava, Vishwas and Schulman, Leonard J. and Sivan, Shiri},
title = {{An Algorithmic Proof of Kruskal’s Tensor Decomposition Theorem}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {31:1--31:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.31},
URN = {urn:nbn:de:0030-drops-264206},
doi = {10.4230/LIPIcs.ICALP.2026.31},
annote = {Keywords: Tensor decomposition, Kruskal’s theorem, tensor rank, MinRank, algebraic algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Constant Rate Isometric Embeddings of Hamming Metric into Edit Metric
Abstract
A function φ: {0,1}^n → {0,1}^N is called an isometric embedding of the n-dimensional Hamming metric space to the N-dimensional edit metric space if, for all x, y ∈ {0,1}ⁿ, the Hamming distance between x and y is equal to the edit distance between φ(x) and φ(y). The rate of such an embedding is defined as the ratio n/N.
It is well known in the literature how to construct isometric embeddings with a rate of Ω(1/log n). However, achieving even near-isometric embeddings with a positive constant rate has remained elusive until now.
In this paper, we present an isometric embedding with a rate of 1/8 by discovering connections to synchronization strings, which were studied in the context of insertion-deletion codes (Haeupler-Shahrasbi [JACM'21]). At a technical level, we introduce a framework for obtaining high-rate isometric embeddings using a novel object called a misaligner. We speculate that, with sufficient computational resources, our framework could potentially yield isometric embeddings with a rate of 1/5.
As an immediate consequence of our constant rate isometric embedding, we improve known conditional lower bounds for the closest pair problem and the discrete 1-center problem in the edit metric and NP-hardness of approximation results for clustering problems and the Steiner tree problem in the edit metric, but now with optimal dependency on the dimension. Furthermore, we obtain optimal lower bounds for the gap edit distance problem in the two-player randomized communication complexity model.
We complement our results by showing that no isometric embedding φ:{0,1}^n → {0,1}^N can have rate greater than 15/32 for all positive integers n. En route to proving this upper bound, we uncover fundamental structural properties necessary for every Hamming-to-edit isometric embedding. We also prove similar upper and lower bounds for embeddings over larger alphabets.
Finally, we consider embeddings φ:Σ_in^n → Σ_out^N between different input and output alphabets, where the rate is given by (n log|Σ_in|)/(Nlog|Σ_out|). In this setting, we show that the rate can be made arbitrarily close to 1.
Cite as
Sudatta Bhattacharya, Sanjana Dey, Elazar Goldenberg, Mursalin Habib, Bernhard Haeupler, Karthik C. S., and Michal Koucký. Constant Rate Isometric Embeddings of Hamming Metric into Edit Metric. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhattacharya_et_al:LIPIcs.ICALP.2026.32,
author = {Bhattacharya, Sudatta and Dey, Sanjana and Goldenberg, Elazar and Habib, Mursalin and Haeupler, Bernhard and Karthik C. S. and Kouck\'{y}, Michal},
title = {{Constant Rate Isometric Embeddings of Hamming Metric into Edit Metric}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {32:1--32:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.32},
URN = {urn:nbn:de:0030-drops-264215},
doi = {10.4230/LIPIcs.ICALP.2026.32},
annote = {Keywords: Edit distance, Hamming distance, metric embeddings, synchronization strings, fine-grained complexity}
}
Document
Track A: Algorithms, Complexity and Games
Visibility Queries in Simple Polygons
Abstract
Given a simple polygon P with n vertices, we consider the problem of constructing a data structure for visibility queries: for any query point q ∈ P, compute the visibility polygon of q in P. To obtain O(log n + k) query time, where k is the size of the visibility polygon of q, the previous best result requires O(n³) space. In this paper, we propose a new data structure that uses O(n^{2+ε}) space, for any ε > 0, while achieving the same query time. If only O(n²) space is available, the best known result provides O(log² n + k) query time. We improve this to O(log n log log n + k) time. When restricted to o(n²) space, the only previously known approach, aside from the O(n)-time algorithm that computes the visibility polygon without preprocessing, is an O(n)-space data structure that supports O(k log n)-time queries. We construct a data structure using O(n log n) space that answers visibility queries in O(n^{1/2+ε} + k) time. In addition, for the special case in which q lies on the boundary of P, we build a data structure of O(n log n) space supporting O(log² n + k) query time; alternatively, we achieve O(log n + k) query time using O(n^{1+ε}) space. To achieve our results, we propose a new method for decomposing simple polygons, which may be of independent interest.
Cite as
Sujoy Bhore, Chih-Hung Liu, Anurag Murty Naredla, Yakov Nekrich, Eunjin Oh, André van Renssen, Frank Staals, Haitao Wang, and Jie Xue. Visibility Queries in Simple Polygons. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 33:1-33:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.ICALP.2026.33,
author = {Bhore, Sujoy and Liu, Chih-Hung and Naredla, Anurag Murty and Nekrich, Yakov and Oh, Eunjin and van Renssen, Andr\'{e} and Staals, Frank and Wang, Haitao and Xue, Jie},
title = {{Visibility Queries in Simple Polygons}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {33:1--33:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.33},
URN = {urn:nbn:de:0030-drops-264222},
doi = {10.4230/LIPIcs.ICALP.2026.33},
annote = {Keywords: simple polygons, visibility polygons, visibility queries, polygon decompositions}
}
Document
Track A: Algorithms, Complexity and Games
Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes
Abstract
We study two fundamental geometric problems on a dynamic set of n axis-parallel boxes in d-dimensional space. The maximum depth problem asks for the largest number of boxes that contain a common point, whereas Klee’s measure problem asks for the volume of the union of the boxes. We present fully dynamic exact data structures for both problems achieving Õ(n^{(d-1)/2}) amortized update time. This update time is optimal for an exact dynamic algorithm, up to logarithmic factors, assuming the Combinatorial k-Clique Hypothesis. Previously, matching bounds were established only for d = 1 [Imai and Asano, J. Algo.'83], and for d = 2 [Suri, Xue, Yang, and Zhu, SoCG'25].
Our approach integrates a classic grid-based partition framework with a novel charging analysis that controls the cost of structure-sensitive offline routines within each cell. This argument allows us to perform a global aggregation of the update time, by circumventing the worst-case costs associated with individual cell updates. We believe this technique may be of independent interest for other dynamic geometric problems.
Cite as
Sujoy Bhore, Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu. Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.ICALP.2026.34,
author = {Bhore, Sujoy and Suri, Subhash and Xue, Jie and Yang, Xiongxin and Zhu, Jiumu},
title = {{Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {34:1--34:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.34},
URN = {urn:nbn:de:0030-drops-264230},
doi = {10.4230/LIPIcs.ICALP.2026.34},
annote = {Keywords: dynamic algorithms, maximum depth}
}
Document
Track A: Algorithms, Complexity and Games
Simpler and Improved Replacement Path Coverings
Abstract
An important tool in the design of fault-tolerant graph data structures are (L,f)-replacement path coverings (RPCs). An RPC is a family 𝒢 of subgraphs of a given graph G such that, for every set F of at most f edges, there is a subfamily 𝒢_F ⊆ 𝒢 with the following properties.
1) No subgraph in 𝒢_F contains an edge of F.
2) For each pair of vertices s,t that have a shortest path in G-F with at most L edges, one such path also exists in some subgraph in 𝒢_F. The covering value of the RPC is the total number |𝒢| of subgraphs. The query time is the time needed to compute the subfamily 𝒢_F given the set F.
Weimann and Yuster [TALG'13] devised a randomized RPC with covering value Õ(fL^f) and query time Õ(f² L^f). This was derandomized by Karthik and Parter [TALG'24], who also reduced the query time to Õ(f² L). Their approach uses some heavy algebraic machinery involving error-correcting codes and an increased covering value of O((cfL log n)^{f+1}) for some constant c > 1. We instead devise a much simpler derandomization via conditional expectations that lowers the covering value back to Õ(fL^{f+o(1)}) and decreases the query time to Õ(f^{5/2} L^o(1)), assuming f = o(log L).
We also investigate the optimal covering value of any (L,f)-replacement path covering (deterministic or randomized) for different parameter ranges. We provide a new randomized construction as well as improving a known lower bound, also by Karthik and Parter. For example, for f = o(log L), we give an RPC with Õ((L/f)^f L^o(1)) subgraphs and show that this is tight up to the L^o(1) term.
Cite as
Davide Bilò, Shiri Chechik, Keerti Choudhary, Sarel Cohen, and Martin Schirneck. Simpler and Improved Replacement Path Coverings. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bilo_et_al:LIPIcs.ICALP.2026.35,
author = {Bil\`{o}, Davide and Chechik, Shiri and Choudhary, Keerti and Cohen, Sarel and Schirneck, Martin},
title = {{Simpler and Improved Replacement Path Coverings}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {35:1--35:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.35},
URN = {urn:nbn:de:0030-drops-264243},
doi = {10.4230/LIPIcs.ICALP.2026.35},
annote = {Keywords: derandomization, fault tolerance, replacement path coverings, sensitivity data structures}
}
Document
Track A: Algorithms, Complexity and Games
Kronecker Scaling of Tensors with Applications to Arithmetic Circuits and Algorithms
Abstract
We show that sufficiently low tensor rank for the balanced tripartitioning tensor P_d(x,y,z) = ∑_{A,B,C ∈ binom([3d],d):A∪ B∪ C = [3d]} x_A y_B z_C for a large enough constant d implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser’s formula.
Under the same low-rank assumption, we obtain exponential-time improvements over the state of the art for a wide variety of related counting and decision problems.
Our main methodological contribution is that the tensors P_n have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of P_d for constant d. We prove this with a new technique relying on Steinitz’s lemma, which we hence call Steinitz balancing.
As a consequence of our methods, we show that the mentioned low-rank assumption (and hence the improved algorithms) is implied by Strassen’s asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress.
Cite as
Andreas Björklund, Petteri Kaski, Tomohiro Koana, and Jesper Nederlof. Kronecker Scaling of Tensors with Applications to Arithmetic Circuits and Algorithms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 36:1-36:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bjorklund_et_al:LIPIcs.ICALP.2026.36,
author = {Bj\"{o}rklund, Andreas and Kaski, Petteri and Koana, Tomohiro and Nederlof, Jesper},
title = {{Kronecker Scaling of Tensors with Applications to Arithmetic Circuits and Algorithms}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {36:1--36:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.36},
URN = {urn:nbn:de:0030-drops-264258},
doi = {10.4230/LIPIcs.ICALP.2026.36},
annote = {Keywords: tensor rank, Kronecker powers, arithmetic circuits, permanent, parameterized algorithms}
}
Document
Track A: Algorithms, Complexity and Games
The Expiration Streaming Model: Diameter, k-Center, Counting, Sampling, and Friends
Abstract
An important thread in the study of data-stream algorithms focuses on settings where stream items are active only for a limited time. We introduce a new expiration model, where each item arrives with its own arbitrary expiration time. The special case where items expire in the order that they arrive, which we call consistent expirations, contains the classical sliding-window model of Datar, Gionis, Indyk, and Motwani [SICOMP 2002] and its timestamp-based variant of Braverman and Ostrovsky [FOCS 2007].
Our first set of results explores the expiration streaming model and presents algorithms for several fundamental problems, including approximate counting, uniform sampling, and weighted sampling by efficiently tracking active items without explicitly storing them all. Naturally, these algorithms have many immediate applications, e.g., to range counting.
Our second and main set of results for the expiration model designs algorithms for the diameter and k-center problems, where items are points in a metric space. Our results significantly extend those known for the special case of sliding-window streams by Cohen-Addad, Schwiegelshohn, and Sohler [ICALP 2016], and obtain a strictly better approximation factor for the diameter in the important special case of high-dimensional Euclidean metrics. We develop new decomposition and coordination techniques along with a geometric dominance framework to filter out redundant points based on both temporal and spatial proximity.
Cite as
Lotte Blank, Sergio Cabello, Mohammad Taghi Hajiaghayi, Robert Krauthgamer, Sepideh Mahabadi, André Nusser, Jeff M. Phillips, and Jonas Sauer. The Expiration Streaming Model: Diameter, k-Center, Counting, Sampling, and Friends. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 37:1-37:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blank_et_al:LIPIcs.ICALP.2026.37,
author = {Blank, Lotte and Cabello, Sergio and Hajiaghayi, Mohammad Taghi and Krauthgamer, Robert and Mahabadi, Sepideh and Nusser, Andr\'{e} and Phillips, Jeff M. and Sauer, Jonas},
title = {{The Expiration Streaming Model: Diameter, k-Center, Counting, Sampling, and Friends}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {37:1--37:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.37},
URN = {urn:nbn:de:0030-drops-264269},
doi = {10.4230/LIPIcs.ICALP.2026.37},
annote = {Keywords: clustering, diameter, streaming, sliding window, sampling}
}
Document
Track A: Algorithms, Complexity and Games
Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees
Abstract
We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph G and a constraint function D, we ask for a (minimum-cost) spanning tree T such that for each vertex v, T achieves a degree specified by D(v). Specifically, we consider three kinds of constraint functions ordered by their generality - D may either assign to each vertex a list of admissible degrees, an upper bound on the degree, or a specific degree. Using a combination of novel techniques and state-of-the-art machinery, we obtain an almost-complete overview of the fine-grained complexity of these problems taking into account the most classical structural graph parameters of the input graph G. In particular, we present SETH-tight upper and lower bounds for these problems when parameterized by pathwidth and cutwidth, an ETH-tight algorithm parameterized by clique-width, and a nearly SETH-tight algorithm parameterized by treewidth.
In order to obtain our upper bound for clique-width, we develop a novel technique of double representation through "requirement shifting". Using this technique, we also obtain an ETH-tight single-exponential XP algorithm for the Exact Leaf Spanning Tree problem parameterized by clique-width, which settles the final remaining open case for clique-width from the classical Cut and Count of Cygan et al. [FOCS 2011, TALG 2022]. This shows the versatility of our technique and its potential applicability to other problems as well. Additionally, in order to establish our lower and upper bounds we introduce a number of tools which may be of independent interest, including lazy coloring and "asymptotic" SETH-based reductions for structural parameters.
Cite as
Narek Bojikian, Alexander Firbas, Robert Ganian, Hung P. Hoang, and Krisztina Szilágyi. Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bojikian_et_al:LIPIcs.ICALP.2026.38,
author = {Bojikian, Narek and Firbas, Alexander and Ganian, Robert and Hoang, Hung P. and Szil\'{a}gyi, Krisztina},
title = {{Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {38:1--38:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.38},
URN = {urn:nbn:de:0030-drops-264272},
doi = {10.4230/LIPIcs.ICALP.2026.38},
annote = {Keywords: Parameterized complexity, Structural parameters, Clique-width, fine-grained complexity, Spanning tree}
}
Document
Track A: Algorithms, Complexity and Games
Tight Bounds for Feedback Vertex Set Parameterized by Clique-Width
Abstract
We introduce a new form of acyclicity representation in labeled graphs, and present three applications thereof. Our main result is an algorithm that, given a graph G and a k-clique expression of G, in time 𝒪(6^kn^c) counts modulo 2 the number of feedback vertex sets of G of each size. We achieve this through dynamic programming on the clique expression with an involved subroutine for merging partial solutions at union nodes in the expression. In the usual way this results in a one-sided error Monte-Carlo algorithm for solving the decision problem in the same time. We complement these by a matching lower bound under the Strong Exponential-Time Hypothesis (SETH). This closes an open question that appeared multiple times in the literature [ESA 23, ICALP 24, IPEC 25], and significantly improves the dependence on k compared to the 𝒪(15^k 2^((ω+1)k) k^c n) time algorithm due to Bergougnoux and Kanté [TCS 2019] at the cost of being randomized.
We also present an algorithm that, given a graph G and a tree decomposition of width k of G, in time 𝒪(3^kn^c) counts modulo 2 the number of feedback vertex sets of G of each size. This matches the known SETH-tight bound for the decision version, which was obtained using the celebrated cut-and-count technique [FOCS 11, TALG 22]. Unlike other applications of cut-and-count, which use the isolation lemma to reduce a decision problem to counting solutions modulo 2, this bound was obtained via counting other objects, leaving open the complexity of counting solutions modulo 2.
Finally, we present a one-sided error Monte-Carlo algorithm that, given a graph G and a k-clique expression of G, in time 𝒪(18^k n^c) decides the existence of a connected feedback vertex set of size t in G. We provide a matching lower bound under SETH.
Cite as
Narek Bojikian and Stefan Kratsch. Tight Bounds for Feedback Vertex Set Parameterized by Clique-Width. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 39:1-39:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bojikian_et_al:LIPIcs.ICALP.2026.39,
author = {Bojikian, Narek and Kratsch, Stefan},
title = {{Tight Bounds for Feedback Vertex Set Parameterized by Clique-Width}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {39:1--39:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.39},
URN = {urn:nbn:de:0030-drops-264280},
doi = {10.4230/LIPIcs.ICALP.2026.39},
annote = {Keywords: Feedback Vertex Set, Treewidth, Clique-width, SETH}
}
Document
Track A: Algorithms, Complexity and Games
Fast Shortest Path in Graphs with Sparse Signed Tree Models and Applications
Abstract
A signed tree model of a graph G is a compact binary structure consisting of a rooted binary tree whose leaves are bijectively mapped to the vertices of G, together with 2-colored edges xy, called transversal pairs, interpreted as bicliques or anti-bicliques whose sides are the leaves of the subtrees rooted at x and at y. We design an algorithm that, given such a representation of an unweighted n-vertex graph G with p transversal pairs, and given a source v ∈ V(G), computes a shortest-path tree rooted at v in G in time O(p log n). A wide variety of graph classes are such that for all n, their n-vertex graphs admit signed tree models with O(n) transversal pairs: for instance, those of bounded symmetric difference (hence, in particular, those of bounded flip-width, merge-width, twin-width, and degeneracy), more generally of bounded sd-degeneracy, as well as interval graphs.
As applications of our Single-Source Shortest Path algorithm and new techniques, we
- improve the runtime of the fixed-parameter algorithm for first-order model checking on graphs given with a witness of low merge-width from cubic [Dreier & Toruńczyk, STOC '25] to quadratic;
- give an O(n² log n)-time algorithm for All-Pairs Shortest Path on graphs given with a witness of low merge-width, generalizing a result known for twin-width [Twin-Width III, SICOMP '24];
- significantly extend and simplify an O(n² log n)-time algorithm for multiplying two n × n matrices A, B of bounded twin-width in [Twin-Width V, STACS '23]: now A solely has to be an adjacency matrix of a graph of bounded twin-width and B can be arbitrary;
- give an O(n² log² n)-time algorithm for All-Pairs Shortest Path on graphs of bounded twin-width, bypassing the need for contraction sequences in [Twin-Width III, SICOMP '24; Bannach et al. STACS '24];
- give an O(n^{7/3} log² n)-time algorithm for All-Pairs Shortest Path on graphs of symmetric difference O(n^{1/3}). The second and the last two items imply the same for Diameter, Radius, Eccentricity, Wiener Index, etc. The last three items do not assume any witness to be given as part of the input.
Cite as
Édouard Bonnet, Colin Geniet, Eun Jung Kim, and Sungmin Moon. Fast Shortest Path in Graphs with Sparse Signed Tree Models and Applications. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 40:1-40:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bonnet_et_al:LIPIcs.ICALP.2026.40,
author = {Bonnet, \'{E}douard and Geniet, Colin and Kim, Eun Jung and Moon, Sungmin},
title = {{Fast Shortest Path in Graphs with Sparse Signed Tree Models and Applications}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {40:1--40:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.40},
URN = {urn:nbn:de:0030-drops-264297},
doi = {10.4230/LIPIcs.ICALP.2026.40},
annote = {Keywords: Shortest path, tree model, twin-width, merge-width, symmetric difference}
}
Document
Track A: Algorithms, Complexity and Games
Connected Dominating Sets in Triangulations
Abstract
A dominating set of a graph G is connected if it induces a connected graph in G. For planar triangulations, it has been known since 1990 that every n-vertex triangulation admits a connected dominating set of size at most n/2 - 1, and no improvement to this bound was known for over three decades. We break this longstanding barrier by showing that every n-vertex triangulation has a connected dominating set of size at most 10n/21. Equivalently, every triangulation admits a spanning tree with at least 11n/21 leaves. Moreover, we present an algorithm that computes such a set in optimal linear time. Our result narrows the gap to the best known lower bound and has graph drawing applications, establishing a bound for one-bend free sets and improving the known bound for simultaneous planar embeddings.
Cite as
Prosenjit Bose, Vida Dujmović, Hussein Houdrouge, Pat Morin, and Saeed Odak. Connected Dominating Sets in Triangulations. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 41:1-41:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bose_et_al:LIPIcs.ICALP.2026.41,
author = {Bose, Prosenjit and Dujmovi\'{c}, Vida and Houdrouge, Hussein and Morin, Pat and Odak, Saeed},
title = {{Connected Dominating Sets in Triangulations}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {41:1--41:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.41},
URN = {urn:nbn:de:0030-drops-264300},
doi = {10.4230/LIPIcs.ICALP.2026.41},
annote = {Keywords: connected domination, triangulations, planar graphs, graph drawing, collinear sets}
}
Document
Track A: Algorithms, Complexity and Games
Classification of Local Optimization Problems in Directed Cycles
Abstract
We present a complete classification of the distributed computational complexity of local optimization problems in directed cycles for both the deterministic and the randomized LOCAL model. We show that for any local optimization problem Π (that can be of the form min-sum, max-sum, min-max, or max-min, for any local cost or utility function over some finite alphabet), and for any constant approximation ratio α, the task of finding an α-approximation of Π in directed cycles has one of the following complexities:
1) O(1) rounds in deterministic LOCAL, O(1) rounds in randomized LOCAL,
2) Θ(log^* n) rounds in deterministic LOCAL, O(1) rounds in randomized LOCAL,
3) Θ(log^* n) rounds in deterministic LOCAL, Θ(log^* n) rounds in randomized LOCAL,
4) Θ(n) rounds in deterministic LOCAL, Θ(n) rounds in randomized LOCAL. Moreover, for any given Π and α, we can determine the complexity class automatically, with an efficient (centralized, sequential) meta-algorithm, and we can also efficiently synthesize an asymptotically optimal distributed algorithm.
Before this work, similar results were only known for local search problems (e.g., locally checkable labeling problems). The family of local optimization problems is a strict generalization of local search problems, and it contains numerous commonly studied distributed tasks, such as the problems of finding approximations of the maximum independent set, minimum vertex cover, minimum dominating set, and minimum vertex coloring.
Cite as
Thomas Boudier, Fabian Kuhn, Augusto Modanese, Ronja Stimpert, and Jukka Suomela. Classification of Local Optimization Problems in Directed Cycles. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 42:1-42:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{boudier_et_al:LIPIcs.ICALP.2026.42,
author = {Boudier, Thomas and Kuhn, Fabian and Modanese, Augusto and Stimpert, Ronja and Suomela, Jukka},
title = {{Classification of Local Optimization Problems in Directed Cycles}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {42:1--42:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.42},
URN = {urn:nbn:de:0030-drops-264317},
doi = {10.4230/LIPIcs.ICALP.2026.42},
annote = {Keywords: LOCAL model, optimization, cycles, meta-algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Unique Decoding of Reed–Solomon and Related Codes for Semi-Adversarial Errors
Abstract
Motivated by recent developments in coding theory, particular in list-decoding, we introduce a new error model which we call semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols.
As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. For interleaved Reed-Solomon (IRS), folded Reed-Solomon (FRS) and univariate multiplicity codes, we design decoding algorithms running in near-linear time for most mixtures of random and adversarial errors. Our analysis matches the information-theoretic optimum for semi-adversarial errors.
Our algorithm for interleaved Reed-Solomon codes is an improved implementation of the decoding algorithm by Bleichenbacher-Kiayias-Yung (BKY) for fully random errors. We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors. Inspired by the BKY algorithm, we use novel interpolations to extend our approach to the settings of folded Reed-Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. Our new decoders for FRS and multiplicity codes replace the sophisticated root-finding step in traditional algorithms, such as the Guruswami-Wang algorithm, with a straightforward polynomial long division. Analysis of these algorithms requires more robust monomial-tracking arguments than IRS codes.
Cite as
Joshua Brakensiek, Yeyuan Chen, Manik Dhar, and Zihan Zhang. Unique Decoding of Reed–Solomon and Related Codes for Semi-Adversarial Errors. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brakensiek_et_al:LIPIcs.ICALP.2026.43,
author = {Brakensiek, Joshua and Chen, Yeyuan and Dhar, Manik and Zhang, Zihan},
title = {{Unique Decoding of Reed–Solomon and Related Codes for Semi-Adversarial Errors}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {43:1--43:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.43},
URN = {urn:nbn:de:0030-drops-264326},
doi = {10.4230/LIPIcs.ICALP.2026.43},
annote = {Keywords: coding theory, interleaved codes, Reed-Solomon codes, semi-random models, unique decoding}
}
Document
Track A: Algorithms, Complexity and Games
Multiplicative Error Set System Sparsification: A Simpler Proof via Chain Length Contraction
Abstract
The chain length of a set family 𝒮 ⊆ 2^[m] is the largest ascending sequence of sets in containment order in the union-closure of S. In this work, we provide a significantly simpler and more optimal characterization of the sparsifiability of set systems in terms of their chain length, improving on the work of Brakensiek and Guruswami [STOC 2025]. Our proof relies on a generalization of Karger’s [SODA 1993] famous contraction algorithm and its recent linear algebraic extensions [Khanna-Putterman-Sudan SODA 2024], and our resulting bounds show that, just as VC dimension characterizes the additive sparsifiability of a set system, chain length governs the multiplicative sparsifiability. As a corollary, we obtain improved bounds for weighted CSP sparsification.
Cite as
Joshua Brakensiek, Venkatesan Guruswami, and Aaron Putterman. Multiplicative Error Set System Sparsification: A Simpler Proof via Chain Length Contraction. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 44:1-44:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brakensiek_et_al:LIPIcs.ICALP.2026.44,
author = {Brakensiek, Joshua and Guruswami, Venkatesan and Putterman, Aaron},
title = {{Multiplicative Error Set System Sparsification: A Simpler Proof via Chain Length Contraction}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {44:1--44:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.44},
URN = {urn:nbn:de:0030-drops-264331},
doi = {10.4230/LIPIcs.ICALP.2026.44},
annote = {Keywords: constraint satisfaction problem, chain length, sparsification, VC dimension}
}
Document
Track A: Algorithms, Complexity and Games
Dynamic Rank, Basis, and Matching
Abstract
We study dynamic algorithms for maintaining fundamental algebraic properties of matrices, specifically, rank, basis, and full-rank submatrices, with applications to maximum matching on dynamic graphs. Prior dynamic algorithms for rank achieve subquadratic update times but scale with the matrix dimension n, and could not always maintain the corresponding objects such as a basis or maximum full-rank submatrix.
We present the first dynamic rank algorithms whose update time scales with the matrix rank r, achieving Õ(r^1.405) time per entry-update and Õ(r^1.528 + z) per column-update, where z is the number of changed entries. This extends to Õ(|M|^1.405) edge-update time to maintain the size |M| of a maximum matching. We also give dynamic algorithms for maintaining a column-basis subject to column-updates and a maximum full-rank submatrix subject to entry-updates.
Cite as
Jan van den Brand, Vishal Kumar, and Daniel J. Zhang. Dynamic Rank, Basis, and Matching. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 45:1-45:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vandenbrand_et_al:LIPIcs.ICALP.2026.45,
author = {van den Brand, Jan and Kumar, Vishal and Zhang, Daniel J.},
title = {{Dynamic Rank, Basis, and Matching}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {45:1--45:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.45},
URN = {urn:nbn:de:0030-drops-264342},
doi = {10.4230/LIPIcs.ICALP.2026.45},
annote = {Keywords: Dynamic Graph Algorithm, Dynamic Algebraic Algorithm, Dynamic Matrix Inverse, Rank, Matching}
}
Document
Track A: Algorithms, Complexity and Games
Computing Flows in Subquadratic Space
Abstract
Space complexity is a critical factor in various computational models, including streaming, parallel/distributed computing, and communication complexity. We study the space complexity of the minimum-cost flow problem, a generalization of the st-max flow problem, focusing on computing flows in subquadratic space. In the general case with arbitrary capacities, minimum cost and st-maximum flows can use up to Ω(n²) edges, so computing the flow on each edge (rather than just the size/cost) seems impossible in subquadratic space. Indeed, there are lower bounds proving quadratic space is needed to store the flow on every edge, which has been used to prove lower bounds on streaming algorithms. However, we show that these lower bounds can be circumvented, opening up improvements for streaming and communication complexity.
For a directed graph with integer capacities and costs bounded by W, we provide a Õ(n^1.5 log (W/ε))-space Õ(√n log(W/ε))-pass streaming algorithm, which during the last pass returns the flow on each edge up to an additive error of ε. Crucially, the algorithm does not return the flow at the end of the last pass but returns the flow on an edge, as the edge is read in the stream. This allows us to circumvent existing Ω(n²) space lower bounds. In the 2-party communication model, our algorithm implies Õ(n^1.5 log² W) bits of communication.
Cite as
Jan van den Brand, Zhao Song, and Albert Weng. Computing Flows in Subquadratic Space. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 46:1-46:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vandenbrand_et_al:LIPIcs.ICALP.2026.46,
author = {van den Brand, Jan and Song, Zhao and Weng, Albert},
title = {{Computing Flows in Subquadratic Space}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {46:1--46:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.46},
URN = {urn:nbn:de:0030-drops-264355},
doi = {10.4230/LIPIcs.ICALP.2026.46},
annote = {Keywords: Combinatorial Optimization, Continuous Optimization, Graph Algorithms, Streaming and Sketching}
}
Document
Track A: Algorithms, Complexity and Games
A Constant-Factor Approximation for Continuous Dynamic Time Warping in 2D
Abstract
Continuous Dynamic Time Warping (CDTW) is a robust similarity measure for polygonal curves that has recently found a variety of applications. Despite its practical use, not much is known about the algorithmic complexity of computing it in 2D, especially when one requires either an exact solution or strong approximation guarantees. We fill this gap by introducing a 5-approximation algorithm with running time O(n⁵) under the 1-norm.
This is the first constant-factor approximation for 2D CDTW with polynomial running time. We extend our algorithm to all polygonal norms on ℝ², which we subsequently use in order to achieve a (5+ε)-approximation with time complexity O(n⁵/ε^{1/2}) for CDTW in 2D under any fixed norm. The latter result in particular includes the usual Euclidean 2-norm.
Cite as
Kevin Buchin, Maike Buchin, Jan Erik Swiadek, and Sampson Wong. A Constant-Factor Approximation for Continuous Dynamic Time Warping in 2D. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 47:1-47:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{buchin_et_al:LIPIcs.ICALP.2026.47,
author = {Buchin, Kevin and Buchin, Maike and Swiadek, Jan Erik and Wong, Sampson},
title = {{A Constant-Factor Approximation for Continuous Dynamic Time Warping in 2D}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {47:1--47:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.47},
URN = {urn:nbn:de:0030-drops-264365},
doi = {10.4230/LIPIcs.ICALP.2026.47},
annote = {Keywords: Continuous Dynamic Time Warping, Curve Similarity, Geometric Approximation Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Static to Dynamic Correlation Clustering
Abstract
Correlation clustering is a well-studied problem, first proposed by Bansal, Blum, and Chawla [Mach. Learn. '04]. The input is an unweighted, undirected graph. The problem is to cluster the vertices so as to minimize the number of edges between vertices in different clusters and missing edges between vertices inside the same cluster. This problem has a wide application in data mining and machine learning. We introduce a general framework that transforms existing static correlation clustering algorithms into fully-dynamic ones that work against an adaptive adversary.
We show how to apply our framework to known efficient correlation clustering algorithms, starting from the classic 3-approximate Pivot algorithm from Ailon, Charikar and Newman [JACM'08]. Applied to the most recent sublinear 1.485-approximation algorithm from Cao, Cohen-Addad, Lee, Li, Lolck, Newman, Thorup, Vogl, Yan and Zhang [STOC'25] , we get an 1.485-approximation fully-dynamic algorithm that works with worst-case constant update time. The original static algorithm gets its approximation factor with constant probability, and we get the same against an adaptive adversary in the sense that for any given update step, not known to our algorithm, our solution is an 1.485-approximation with constant probability when we reach this update.
Most of previous dynamic algorithms, including the celebrated result from Behnezhad, Charikar, Ma and Tan [FOCS'19], had approximation factors around 3 in expectation, and they could only handle an oblivious adversary. A recent algorithm by Braverman, Dharangutte, Pai, Shah, and Wang [AISTATS'25] handles an adaptive adversary, but it has a large unspecified constant approximation ratio. This contrasts with our general transformation, which works with all the best approximation factors known for the static case.
Cite as
Nairen Cao, Vincent Cohen-Addad, Euiwoong Lee, Shi Li, David Rasmussen Lolck, Alantha Newman, Mikkel Thorup, Lukas Vogl, Shuyi Yan, and Hanwen Zhang. Static to Dynamic Correlation Clustering. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 48:1-48:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cao_et_al:LIPIcs.ICALP.2026.48,
author = {Cao, Nairen and Cohen-Addad, Vincent and Lee, Euiwoong and Li, Shi and Lolck, David Rasmussen and Newman, Alantha and Thorup, Mikkel and Vogl, Lukas and Yan, Shuyi and Zhang, Hanwen},
title = {{Static to Dynamic Correlation Clustering}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {48:1--48:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.48},
URN = {urn:nbn:de:0030-drops-264378},
doi = {10.4230/LIPIcs.ICALP.2026.48},
annote = {Keywords: Dynamic Algorithms, Correlation Clustering, Approximation Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
A Scalable and Unified Framework to Weighted Rank Aggregation
Abstract
The rank aggregation problem, seeks to combine multiple rank orderings of the same set of candidates into a single consensus ordering. Such problems arise in diverse domains, including web search, employment, college admissions, and voting. In this work we focus on the 1-median objective: given a set of m rankings over [n], the goal is to compute a ranking that minimizes the sum of its distances to all input rankings.
We study rank aggregation under several classical distance metrics: Ulam distance, Spearman’s footrule, Hamming distance, and Kendall-tau, as well as their weighted variants. Our contributions begin with a novel unified framework that identifies a key structural property: it suffices to focus on a small subset of rankings (of size three or five), where the corresponding local one-median provides a good approximation to the global median. This principle extends across these distance measures, yielding a general algorithmic framework for weighted rank aggregation.
Building on this, we present a new approximation algorithm for rank aggregation under the Ulam distance that scales in the Massively Parallel Computation (MPC) model. Our algorithm computes a (2-α)-approximation, for a constant α > 0, to the 1-median in a constant number of rounds, using local memory sublinear in n (the size of a ranking) and total memory near linear in n.
We further design new MPC approximation algorithms for Spearman’s footrule and for the element-weighted variants of Hamming and Kendall-tau distances. For each metric, we obtain a (2-ζ)-approximation, for a constant ζ > 0 (which may differ across metrics), to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory linear or near-linear in n.
Moreover, for the Ulam distance, where computing the 1-median is NP-hard [Fischer et al., ESA, 2025], we simplify and strengthen the analysis of Chakraborty et al. [ITCS 2023], obtaining an improved 1.968-approximation that further extends to the weighted setting.
Cite as
Amir Carmel, Debarati Das, and Tien-Long Nguyen. A Scalable and Unified Framework to Weighted Rank Aggregation. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 49:1-49:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{carmel_et_al:LIPIcs.ICALP.2026.49,
author = {Carmel, Amir and Das, Debarati and Nguyen, Tien-Long},
title = {{A Scalable and Unified Framework to Weighted Rank Aggregation}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {49:1--49:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.49},
URN = {urn:nbn:de:0030-drops-264385},
doi = {10.4230/LIPIcs.ICALP.2026.49},
annote = {Keywords: Rank aggregation, 1-median, Ulam distance, Spearman’s footrule, Kendall-tau, Hamming distance, weighted metrics, Massively Parallel Computation, Gromov product}
}
Document
Track A: Algorithms, Complexity and Games
Touring a Sequence of Orthogonal Polygons
Abstract
We study the problem of computing a shortest tour that visits a sequence of k polygons P₁,…,P_k with a total number of n vertices. A tour is an oriented curve such that there exist points p_i ∈ P_i for all i where p_i appears not after p_{i+1}. In a seminal paper, Dror, Efrat, Lubiw and Mitchell (STOC 2003) considered the problem under L₂ distance, and gave Õ(nk) and Õ(nk²) algorithms for disjoint and intersecting convex polygons, respectively. In this paper, we consider the orthogonal setting (with orthogonal polygons and Manhattan distance) and obtain the following results:
- a truly subquadratic Õ(n^{2-1/48}) algorithm when consecutive polygons in the sequence are disjoint;
- an Õ(n) algorithm for ortho-convex polygons when consecutive polygons are disjoint;
- an O(n) algorithm for axis-aligned rectangles;
- Õ(n²) and Õ(n^{1.5}k²) algorithms without restrictions. Our algorithms build on a wide range of techniques, including additively weighted Voronoi diagrams, rectangle decompositions, persistent data structures, and dynamic distance oracles for weighted planar graphs.
Cite as
Katrin Casel, Sándor Kisfaludi-Bak, Linda Kleist, Jeroen S.K. Lamme, Eunjin Oh, and Yanheng Wang. Touring a Sequence of Orthogonal Polygons. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 50:1-50:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{casel_et_al:LIPIcs.ICALP.2026.50,
author = {Casel, Katrin and Kisfaludi-Bak, S\'{a}ndor and Kleist, Linda and Lamme, Jeroen S.K. and Oh, Eunjin and Wang, Yanheng},
title = {{Touring a Sequence of Orthogonal Polygons}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {50:1--50:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.50},
URN = {urn:nbn:de:0030-drops-264391},
doi = {10.4230/LIPIcs.ICALP.2026.50},
annote = {Keywords: shortest path, subquadratic time, dynamic planar distance oracle}
}
Document
Track A: Algorithms, Complexity and Games
Well-Quasi-Ordering Eulerian Digraphs: Bounded Carving Width
Abstract
We prove that every class of Eulerian directed graphs of bounded carving width (equivalently, of bounded degree and treewidth) is well-quasi-ordered by strong immersion. In fact, we prove a stronger result, namely that every class of Eulerian directed graphs of bounded carving width, where every vertex is additionally labelled from a well-quasi-order, fixes a linear order on its incident edges, and may impose further restrictions on how the immersion is allowed to route paths through it, is well-quasi-ordered by an adequate notion of strong immersion. To this extent, we develop a framework seemingly suited to prove well-quasi-ordering for classes of Eulerian directed graphs by (strong) immersion and present a first meta theorem in that direction. We complement our results by observing that the class of Eulerian directed graphs of unbounded degree is not well-quasi-ordered by strong immersion, even if we assume the treewidth of the class to be at most two. We conclude with a dichotomy result, proving for a very restricted class of Eulerian directed graphs of unbounded degree that it is not well-quasi-ordered by strong immersion, but it is well-quasi-ordered by weak immersion.
Cite as
Dario Cavallaro, Ken-ichi Kawarabayashi, and Stephan Kreutzer. Well-Quasi-Ordering Eulerian Digraphs: Bounded Carving Width. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 51:1-51:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cavallaro_et_al:LIPIcs.ICALP.2026.51,
author = {Cavallaro, Dario and Kawarabayashi, Ken-ichi and Kreutzer, Stephan},
title = {{Well-Quasi-Ordering Eulerian Digraphs: Bounded Carving Width}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {51:1--51:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.51},
URN = {urn:nbn:de:0030-drops-264404},
doi = {10.4230/LIPIcs.ICALP.2026.51},
annote = {Keywords: algorithmic graph theory, structural graph theory, digraphs, immersions, well-quasi ordering}
}
Document
Track A: Algorithms, Complexity and Games
Witness-Sensitive Detection of Induced Diamonds
Abstract
We provide a fast witness-sensitive algorithm for detecting an induced diamond (a K₄ minus an edge) in an n-vertex graph containing t induced diamonds. Our algorithm runs in time Õ(min(n^2.425/t^0.25 + n², n^ω)) with high probability, improving upon the prior state of the art (witness-oblivious) algorithm that runs in time O(n^ω log n) [Vassilevska Williams, Wang, Williams, Yu, SODA 2014] whenever t ≥ n^{(3-ω)/3}, where ω < 2.372 is the matrix multiplication exponent.
Our key insight is that the size of a clique containing one of the triangles of an induced diamond plays a crucial role in detecting such a diamond. We say that a diamond is r-heavy if this size is at least r, and we provide a fast detection algorithm for r-heavy diamonds in Õ(r⋅(n/r)^ω + (n/r)³+ nr) time. When there are no r-heavy diamonds, we provide a different fast detection algorithm in Õ(MM(n,n,n√{r/t})) time, where MM(a,b,c) denotes the time to multiply an a × b matrix by a b × c matrix, which is conditionally optimal for r = Õ(1).
Our main technical contribution is in designing a refinement framework for sampling vectors, which allows sampling vertices for detecting diamonds in a manner that is adaptive to the structure of graphs with no r-heavy diamonds. We establish that our technique is of a wide applicability, by showing how it also allows for faster witness-sensitive algorithms for 4-SUM and for a special case of 4-cycles.
Cite as
Keren Censor-Hillel, Tomer Even, Virginia Vassilevska Williams, and Nathan Wallheimer. Witness-Sensitive Detection of Induced Diamonds. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 52:1-52:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{censorhillel_et_al:LIPIcs.ICALP.2026.52,
author = {Censor-Hillel, Keren and Even, Tomer and Vassilevska Williams, Virginia and Wallheimer, Nathan},
title = {{Witness-Sensitive Detection of Induced Diamonds}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {52:1--52:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.52},
URN = {urn:nbn:de:0030-drops-264419},
doi = {10.4230/LIPIcs.ICALP.2026.52},
annote = {Keywords: Induced diamond detection, Witness-sensitive algorithms, Matrix multiplication, Subgraph detection, Fine-grained complexity}
}
Document
Track A: Algorithms, Complexity and Games
Hardness and Approximation for Coloring Digraphs
Abstract
The dichromatic number χ(D) of a digraph is the minimum number k such that V(D) can be partitioned into k subsets, each inducing an acyclic digraph. The acyclic number α(D) is the cardinality of a largest induced acyclic subdigraph of D.
We study these problems from an approximation point of view. We begin with establishing that even when restricted to tournaments, approximating χ and α remain as challenging as their undirected counterparts on general graphs. Specifically, we establish that for every ε > 0, it is hard to approximate both α and χ up to a factor of n^{1-ε} even when restricted to tournaments.
We next consider approximate coloring of digraphs in special cases. We begin with establishing that we can color 𝓁-dicolorable digraphs using at most 𝓁 ⋅ n^{1-1/(𝓁)} colors in time O(n^{2𝓁}); in particular, we can color 2-dicolorable digraphs with 2√n colors in polynomial time. We then focus on bounding the dichromatic number of dense digraphs as a function of the independence number α of the underlying graph. We consider two special cases in this regard: digraphs with χ(D) ≤ 2 and digraphs that do not contain any directed triangle. For these cases, we present algorithms which generalize and improve existing tools and results.
Cite as
Parinya Chalermsook, Harmender Gahlawat, Felix Klingelhoefer, Alantha Newman, and Chaoliang Tang. Hardness and Approximation for Coloring Digraphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chalermsook_et_al:LIPIcs.ICALP.2026.53,
author = {Chalermsook, Parinya and Gahlawat, Harmender and Klingelhoefer, Felix and Newman, Alantha and Tang, Chaoliang},
title = {{Hardness and Approximation for Coloring Digraphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {53:1--53:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.53},
URN = {urn:nbn:de:0030-drops-264421},
doi = {10.4230/LIPIcs.ICALP.2026.53},
annote = {Keywords: Graph Algorithms, Hardness of Approximation, Polynomial Time Approximation Algorithms, Structural Graph Theory}
}
Document
Track A: Algorithms, Complexity and Games
Charting the Landscape of Diameter Computation on Geometric Intersection Graphs in the Plane
Abstract
Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in 2D, the problem is solvable in truly subquadratic time [Chan et al., 2025], while for other objects, including unit segments and equilateral triangles in 2D or unit balls and axis-parallel unit cubes in 3D, there is no truly subquadratic time algorithm under the Orthogonal Vector (OV) hypothesis [Bringmann et al., 2022].
We undertake a comprehensive study of computing the diameter of geometric intersection graphs for various types of objects. We discover many new irregularities, showing that the landscape is extremely nuanced: the source of hardness is a combination of the object type, the true diameter value, and how the objects intersect with each other. Our highlighted results for the 2D case include:
1) The diameter of non-degenerate, axis-aligned line segments can be computed in truly subquadratic time. Previous hardness result [Bringmann et al., 2022] for line segments applies only to degenerate instances. On the other hand, for the degenerate case, we show that a truly subquadratic time algorithm exists when the true diameter is constant.
2) An almost-linear-time algorithm for unit-square graphs of constant diameter. Previous algorithms [Duraj et al., 2024; Chan et al., 2025] rely on succinct representation assuming bounded VC-dimension; for such a strategy Ω(n^{7/4}) time is an inherent barrier.
3) An Õ(n^{4/3})-time algorithm to decide if the diameter of a unit-disk graph is at most 2. This improves upon the recent algorithm with running time Õ(n^{2-1/9}) [Chan et al., 2025].
4) Deciding if the diameter of intersection graphs of fat triangles or line segments is at most 2 is truly subquadratic-hard under fine-grained complexity assumptions. Previous lower bounds [Bringmann et al., 2022] only hold when deciding if diameter is at most 3. Our findings are presented in a pair of papers. This paper focuses solely on the 2D case, while the companion paper is devoted to higher-dimensional cases.
Cite as
Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, Hung Le, and Da Wei Zheng. Charting the Landscape of Diameter Computation on Geometric Intersection Graphs in the Plane. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chan_et_al:LIPIcs.ICALP.2026.54,
author = {Chan, Timothy M. and Chang, Hsien-Chih and Gao, Jie and Kisfaludi-Bak, S\'{a}ndor and Le, Hung and Zheng, Da Wei},
title = {{Charting the Landscape of Diameter Computation on Geometric Intersection Graphs in the Plane}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {54:1--54:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.54},
URN = {urn:nbn:de:0030-drops-264432},
doi = {10.4230/LIPIcs.ICALP.2026.54},
annote = {Keywords: String graphs, Fine-grained complexity}
}
Document
Track A: Algorithms, Complexity and Games
Suffix Random Access via Function Inversion: A Key for Asymmetric Streaming String Algorithms
Abstract
Many string processing problems can be phrased in the streaming setting, where the input arrives symbol by symbol and we have sublinear working space. The area of streaming algorithms for string processing has flourished since the seminal work of Porat and Porat [FOCS 2009].
Unfortunately, problems with efficient solutions in the classical setting often do not admit efficient solutions in the streaming setting. As a bridge between these two settings, Saks and Seshadhri [SODA 2013] introduced the asymmetric streaming model (see also [Andoni, Krauthgamer, and Onak; FOCS 2010]). Here, one is given read-only access to a (typically short) reference string R of length m, while a (typically long) text T arrives as a stream.
We provide a generic technique to reduce fundamental string problems in the asymmetric streaming model to the online read-only model, lifting several existing algorithms and generally improving upon the state of the art. Most notably, we obtain asymmetric streaming algorithms for exact and approximate pattern matching (under both the Hamming and edit distances), and for relative Lempel-Ziv compression, a popular scheme for measuring and exploiting redundancy in repetitive text collections.
At the heart of our approach lies a novel tool that facilitates efficient computation in the asymmetric streaming model: the suffix random access data structure. In its simplest variant, it maintains constant-time random access to the longest suffix of (the seen prefix of) T that occurs in R. Let τ be a parameter that denotes the size of the data structure. A straightforward approach maintains the data structure in {O}(m/τ) time per arriving symbol of T.
We drastically improve this tradeoff and reveal fundamental barriers via a bidirectional reduction between suffix random access and function inversion, a central problem in cryptography:
- By leveraging Fiat and Naor’s function inversion data structure [SIAM J. Comput. 2000], we achieve Õ(1+m³/τ⁶) update time. In particular, for τ = √m, we obtain Õ(1) update time, improving over the Ω(√m) bound of the straightforward solution.
- We establish an unconditional Ω̃(m/τ³) lower bound on the update time. Additionally, we show that achieving update time o(m³/τ⁷) would imply a breakthrough in function inversion.
On the way to our upper bound, we propose a variant of the string synchronizing sets ([Kempa and Kociumaka; STOC 2019]) with a local sparsity condition that, as we show, admits an efficient streaming construction algorithm. We believe that our framework and techniques will find broad applications in the development of small-space string algorithms.
Cite as
Panagiotis Charalampopoulos, Taha El Ghazi, Jonas Ellert, Paweł Gawrychowski, and Tatiana Starikovskaya. Suffix Random Access via Function Inversion: A Key for Asymmetric Streaming String Algorithms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 55:1-55:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{charalampopoulos_et_al:LIPIcs.ICALP.2026.55,
author = {Charalampopoulos, Panagiotis and El Ghazi, Taha and Ellert, Jonas and Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana},
title = {{Suffix Random Access via Function Inversion: A Key for Asymmetric Streaming String Algorithms}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {55:1--55:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.55},
URN = {urn:nbn:de:0030-drops-264440},
doi = {10.4230/LIPIcs.ICALP.2026.55},
annote = {Keywords: streaming algorithms, function inversion, string algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Faster Deterministic Streaming Vertex Coloring
Abstract
Graph coloring is a fundamental problem in computer science. In the semi-streaming model, an input graph G on n vertices and maximum degree Δ is presented as a stream of edges, and the goal is to compute a vertex coloring using a small number of colors while storing only Õ(n) bits of memory.
Recent work has revealed an exponential separation between randomized and deterministic approaches in this setting: while randomized algorithms can achieve a (Δ+1)-coloring in a single pass [Assadi, Chen, and Khanna, 2019], any single-pass deterministic algorithm requires exp(Δ^Ω(1)) colors [Assadi, Chen, and Sun, 2022]. Consequently, deterministic algorithms that use few colors must necessarily make multiple passes over the stream. Prior to this work, the best known deterministic trade-offs were: an O(Δ²)-coloring in 2 passes, an O(Δ)-coloring in O(log Δ) passes [Assadi, Chen, and Sun, 2022], and a (Δ+1)-coloring in O(log Δ ⋅ log log Δ) passes [Assadi, Chakrabarti, Ghosh, and Stoeckl, 2023]. It remained open whether better trade-offs - particularly with sub-logarithmic pass complexity and linear-in-Δ palette size - were achievable.
In this paper, we present a new deterministic semi-streaming algorithm that computes an O(Δ)-coloring in O(√{log Δ}) passes. This is the first deterministic streaming algorithm to achieve a coloring with palette size linear-in-Δ using sublogarithmic-in-Δ passes.
Cite as
Shiri Chechik, Hongyi Chen, and Tianyi Zhang. Faster Deterministic Streaming Vertex Coloring. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 56:1-56:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chechik_et_al:LIPIcs.ICALP.2026.56,
author = {Chechik, Shiri and Chen, Hongyi and Zhang, Tianyi},
title = {{Faster Deterministic Streaming Vertex Coloring}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {56:1--56:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.56},
URN = {urn:nbn:de:0030-drops-264453},
doi = {10.4230/LIPIcs.ICALP.2026.56},
annote = {Keywords: vertex coloring, streaming}
}
Document
Track A: Algorithms, Complexity and Games
Tight Regret Bounds for Fixed-Price Bilateral Trade
Abstract
We examine fixed-price mechanisms in bilateral trade through the lens of regret minimization. Our main results are twofold. (i) For independent values, a near-optimal Θ̃(T^{2/3}) tight bound for Global Budget Balance fixed-price mechanisms with two-bit/one-bit feedback. (ii) For correlated/adversarial values, a near-optimal Ω(T^{3/4}) lower bound for Global Budget Balance fixed-price mechanisms with two-bit/one-bit feedback, which improves the best known Ω(T^{5/7}) lower bound obtained in the work [Martino Bernasconi et al., 2024] and, up to polylogarithmic factors, matches the 𝒪̃(T^{3/4}) upper bound obtained in the same work. Our work in combination with the previous works [Nicolò Cesa{-}Bianchi et al., 2024; Nicolò Cesa-Bianchi et al., 2024; Yossi Azar et al., 2024; Martino Bernasconi et al., 2024] (essentially) gives a thorough understanding of regret minimization for fixed-price bilateral trade.
En route, we have developed two technical ingredients that might be of independent interest: (i) A novel algorithmic paradigm, called fractal elimination, to address one-bit feedback and independent values. (ii) A new lower-bound construction with novel proof techniques, to address the Global Budget Balance constraint and correlated values.
Cite as
Houshuang Chen, Yaonan Jin, Pinyan Lu, and Chihao Zhang. Tight Regret Bounds for Fixed-Price Bilateral Trade. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 57:1-57:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.ICALP.2026.57,
author = {Chen, Houshuang and Jin, Yaonan and Lu, Pinyan and Zhang, Chihao},
title = {{Tight Regret Bounds for Fixed-Price Bilateral Trade}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {57:1--57:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.57},
URN = {urn:nbn:de:0030-drops-264462},
doi = {10.4230/LIPIcs.ICALP.2026.57},
annote = {Keywords: bilateral trade, online learning, regret minimization, budget balance}
}
Document
Track A: Algorithms, Complexity and Games
Quantum Multi-Level Estimation of Functionals of Discrete Distributions
Abstract
We propose a quantum multi-level estimation framework for a functional ∑_{i=1}^n f(p_i) of a discrete distribution (p_i)_{i=1}^n. We partition the values p_i into logarithmically many intervals whose length decays exponentially. For each interval, we perform non-destructive singular value discrimination to isolate the relevant p_i, enabling adaptive estimation of the partial sum over this interval. Unlike previous variable-time approaches, our method avoids high control overhead and requires only constant extra ancilla qubits. As an application, we present efficient quantum estimators for the q-Tsallis entropy of discrete distributions. Specifically,
- For q > 1, we obtain a near-optimal quantum algorithm with query complexity Θ̃(1/ε^{max{1/(2(q-1)), 1}}), improving the prior best O(1/ε^{1+1/(q-1)}) due to Liu and Wang (SODA 2025; IEEE Trans. Inf. Theory 2026).
- For 0 < q < 1, we obtain a quantum algorithm with query complexity Õ(n^{1/q-1/2}/ε^{1/q}), exhibiting a quantum speedup over the near-optimal classical estimators due to Jiao, Venkat, Han, and Weissman (IEEE Trans. Inf. Theory 2017). Our results achieve, to our knowledge, the first near-optimal quantum estimators for parameterized q-entropy for non-integer q.
Cite as
Kean Chen, Minbo Gao, Tongyang Li, Qisheng Wang, and Xinzhao Wang. Quantum Multi-Level Estimation of Functionals of Discrete Distributions. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.ICALP.2026.58,
author = {Chen, Kean and Gao, Minbo and Li, Tongyang and Wang, Qisheng and Wang, Xinzhao},
title = {{Quantum Multi-Level Estimation of Functionals of Discrete Distributions}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {58:1--58:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.58},
URN = {urn:nbn:de:0030-drops-264473},
doi = {10.4230/LIPIcs.ICALP.2026.58},
annote = {Keywords: Quantum algorithms, functional estimation, entropy estimation, query complexity, Tsallis entropy}
}
Document
Track A: Algorithms, Complexity and Games
Strict Hierarchy for Quantum Channel Certification to Unitary
Abstract
We consider the problem of quantum channel certification to unitary, where one is given access to an unknown d-dimensional channel ℰ, and wants to test whether ℰ is equal to a target unitary channel or is ε-far from it in the diamond norm. We present optimal quantum algorithms for this problem, settling the query complexities in three access models with increasing power. Specifically, we show that:
1) Θ(d/ε²) queries suffice for incoherent access model, matching the lower bound due to Fawzi, Flammarion, Garivier, and Oufkir (COLT 2023).
2) Θ(d/ε) queries suffice for coherent access model, matching the lower bound due to Regev and Schiff (ICALP 2008).
3) Θ(√d/ε) queries suffice for source-code access model, matching the lower bound due to Jeon and Oh (npj Quantum Inf. 2026).
This demonstrates a strict hierarchy of complexities for quantum channel certification to unitary across various access models.
Cite as
Kean Chen, Qisheng Wang, and Zhicheng Zhang. Strict Hierarchy for Quantum Channel Certification to Unitary. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 59:1-59:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.ICALP.2026.59,
author = {Chen, Kean and Wang, Qisheng and Zhang, Zhicheng},
title = {{Strict Hierarchy for Quantum Channel Certification to Unitary}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {59:1--59:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.59},
URN = {urn:nbn:de:0030-drops-264480},
doi = {10.4230/LIPIcs.ICALP.2026.59},
annote = {Keywords: Quantum algorithms, quantum channels, quantum certification, query complexity, entanglement fidelity}
}
Document
Track A: Algorithms, Complexity and Games
Near-Tight Approximation Algorithms for Bottleneck Multiple Knapsack Problems
Abstract
In the bottleneck multiple knapsack problem, we are given a set of items and a set of knapsacks, where each item has a profit and a weight, and each knapsack has a capacity. Our goal is to assign items to knapsacks so as to maximize the minimum profit received by any knapsack subject to the capacity constraint. When all knapsacks have identical capacity, we give a (2/3 - ε)-approximation algorithm for any constant ε > 0. This result almost matches the (2/3 + ε) inapproximability bound for the bottleneck multiple subset sum problem (Caprara et al., 2000). When the knapsacks can have arbitrary capacities, we propose a (1/2 - ε)-approximation algorithm for any constant ε > 0. We also prove a hardness bound of (1/2 + ε) for any constant ε > 0.
Cite as
Lin Chen, Tingwei Hu, Yuchen Mao, Yong Chen, Lili Mei, An Zhang, Guangting Chen, and Guochuan Zhang. Near-Tight Approximation Algorithms for Bottleneck Multiple Knapsack Problems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 60:1-60:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.ICALP.2026.60,
author = {Chen, Lin and Hu, Tingwei and Mao, Yuchen and Chen, Yong and Mei, Lili and Zhang, An and Chen, Guangting and Zhang, Guochuan},
title = {{Near-Tight Approximation Algorithms for Bottleneck Multiple Knapsack Problems}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {60:1--60:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.60},
URN = {urn:nbn:de:0030-drops-264495},
doi = {10.4230/LIPIcs.ICALP.2026.60},
annote = {Keywords: Bottleneck multiple knapsack, approximation algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Sublinear-Query Relative-Error Testing of Halfspaces
Abstract
The relative-error property testing model was introduced in [Chen et al., 2024] to facilitate the study of property testing for "sparse" Boolean-valued functions, i.e. ones for which only a small fraction of all input assignments satisfy the function. In this framework, the distance from the unknown target function f that is being tested to a function g is defined as Vol(f△g)/Vol(f), where the numerator is the fraction of inputs on which f and g disagree and the denominator is the fraction of inputs that satisfy f.
Recent work [Chen et al., 2026] has shown that over the Boolean domain {0,1}ⁿ, any relative-error testing algorithm for the fundamental class of {halfspaces} (i.e. linear threshold functions) must make Ω(log n) oracle calls. In this paper we complement the [Chen et al., 2026] lower bound by showing that halfspaces can be relative-error tested over ℝⁿ under the standard N(0,I_n) Gaussian distribution using a sublinear number of oracle calls - in particular, substantially fewer than would be required for learning. Our results use a wide range of tools including Hermite analysis, Gaussian isoperimetric inequalities, and geometric results on noise sensitivity and surface area.
Cite as
Xi Chen, Anindya De, Yizhi Huang, Shivam Nadimpalli, Rocco A. Servedio, and Tianqi Yang. Sublinear-Query Relative-Error Testing of Halfspaces. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 61:1-61:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.ICALP.2026.61,
author = {Chen, Xi and De, Anindya and Huang, Yizhi and Nadimpalli, Shivam and Servedio, Rocco A. and Yang, Tianqi},
title = {{Sublinear-Query Relative-Error Testing of Halfspaces}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {61:1--61:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.61},
URN = {urn:nbn:de:0030-drops-264508},
doi = {10.4230/LIPIcs.ICALP.2026.61},
annote = {Keywords: Property testing, relative-error testing, halfspaces, Gaussian space}
}
Document
Track A: Algorithms, Complexity and Games
Relative-Error Unateness Testing
Abstract
The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [X. Chen et al., 2025; Chen et al., 2025; Chen et al., 2025]. In this paper we consider the problem of relative-error testing an unknown and arbitrary f: {0,1}ⁿ → {0,1} for the property of being a unate function, i.e. a function that is either monotone non-increasing or monotone non-decreasing in each of the n input variables.
Our first result is a one-sided non-adaptive algorithm for this problem that makes Õ(log(N)/ε) samples and queries, where N = |f^{-1}(1)| is the number of satisfying assignments of the function that is being tested and the value of N is given as an input parameter to the algorithm. Building on this algorithm, we next give a one-sided adaptive algorithm for this problem that does not need to be given the value of N and with high probability makes Õ(log(N)/ε) samples and queries.
We also give lower bounds for both adaptive and non-adaptive two-sided algorithms that are given the value of N up to a constant multiplicative factor. In the non-adaptive case, our lower bounds essentially match the complexity of the algorithm that we provide.
Cite as
Xi Chen, Diptaksho Palit, Kabir Peshawaria, William Pires, Rocco A. Servedio, and Yiding Zhang. Relative-Error Unateness Testing. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.ICALP.2026.62,
author = {Chen, Xi and Palit, Diptaksho and Peshawaria, Kabir and Pires, William and Servedio, Rocco A. and Zhang, Yiding},
title = {{Relative-Error Unateness Testing}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {62:1--62:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.62},
URN = {urn:nbn:de:0030-drops-264511},
doi = {10.4230/LIPIcs.ICALP.2026.62},
annote = {Keywords: Property Testing, Relative Error}
}
Document
Track A: Algorithms, Complexity and Games
Learning Multinomial Logits in O(n log n) Time
Abstract
A Multinomial Logit (MNL) model is composed of a finite universe of items [n] = {1,…,n}, each assigned a positive weight. A query specifies an admissible subset - called a slate - and the model chooses one item from that slate with probability proportional to its weight. This query model is also known as the Plackett-Luce model or conditional sampling oracle in the literature. Although MNLs have been studied extensively, a basic computational question remains open: given query access to slates, how efficiently can we learn weights so that, for every slate, the induced choice distribution is within total variation distance ε of the ground truth? This question is central to MNL learning and has direct implications for modern recommender system interfaces.
We provide two algorithms for this task, one with adaptive queries and one with non‑adaptive queries. Each algorithm outputs an MNL M̂ that induces, for each slate S, a distribution M̂_S on S that is within ε total variation distance of the true distribution. Our adaptive algorithm makes O(n/ε³ log n) queries, while our non-adaptive algorithm makes O(n²/ε³ log n log(n/ε)) queries. Both algorithms query only slates of size two and run in time proportional to their query complexity.
We complement these upper bounds with lower bounds of Ω(n/ε² log n) for adaptive queries and Ω(n²/ε² log n) for non‑adaptive queries, thus proving that our adaptive algorithm is optimal in its dependence on the support size n, while the non-adaptive one is tight within a log n factor.
Cite as
Flavio Chierichetti, Mirko Giacchini, Ravi Kumar, Silvio Lattanzi, Alessandro Panconesi, Erasmo Tani, and Andrew Tomkins. Learning Multinomial Logits in O(n log n) Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 63:1-63:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chierichetti_et_al:LIPIcs.ICALP.2026.63,
author = {Chierichetti, Flavio and Giacchini, Mirko and Kumar, Ravi and Lattanzi, Silvio and Panconesi, Alessandro and Tani, Erasmo and Tomkins, Andrew},
title = {{Learning Multinomial Logits in O(n log n) Time}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {63:1--63:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.63},
URN = {urn:nbn:de:0030-drops-264526},
doi = {10.4230/LIPIcs.ICALP.2026.63},
annote = {Keywords: Multinomial Logits, Conditional Samples, Discrete Choice Models, Recommender Systems}
}
Document
Track A: Algorithms, Complexity and Games
Odd-Cycle-Packing-Treewidth: On the Maximum Independent Set Problem in Odd-Minor-Free Graph Classes
Abstract
We introduce the tree-decomposition-based graph parameter Odd-Cycle-Packing-treewidth (OCP-tw) as a width parameter that asks to decompose a given graph into pieces of bounded odd cycle packing number. The parameter OCP-tw is monotone under the odd-minor-relation and we provide an analogue to the celebrated Grid Theorem of Robertson and Seymour for OCP-tw. That is, we identify two infinite families of grid-like graphs whose presence as odd-minors implies large OCP-tw and prove that their absence implies bounded OCP-tw. This structural result is constructive and implies a 2^poly(k) poly(n)-time parameterized poly(k)-approximation algorithm for OCP-tw.
Moreover, we show that the (weighted) Maximum Independent Set problem (MIS) can be solved in polynomial time on graphs of bounded OCP-tw. Finally, we lift the concept of OCP-tw to a parameter for matrices of integer programs. To this end, we show that our strategy can be applied to efficiently solve integer programs whose matrices have entries in {-1,0,1} and can be "tree-decomposed" into totally Δ-modular matrices with at most two non-zero entries per row.
Cite as
Mujin Choi, Maximilian Gorsky, Gunwoo Kim, Caleb McFarland, and Sebastian Wiederrecht. Odd-Cycle-Packing-Treewidth: On the Maximum Independent Set Problem in Odd-Minor-Free Graph Classes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{choi_et_al:LIPIcs.ICALP.2026.64,
author = {Choi, Mujin and Gorsky, Maximilian and Kim, Gunwoo and McFarland, Caleb and Wiederrecht, Sebastian},
title = {{Odd-Cycle-Packing-Treewidth: On the Maximum Independent Set Problem in Odd-Minor-Free Graph Classes}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {64:1--64:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.64},
URN = {urn:nbn:de:0030-drops-264533},
doi = {10.4230/LIPIcs.ICALP.2026.64},
annote = {Keywords: Odd-minor, treewidth, parameterized algorithm, graph minor, structural graph theory, Odd-Cycle-Packing-treewidth, Maximum Independent Set problem}
}
Document
Track A: Algorithms, Complexity and Games
Randomized k-Server in Polynomial Time
Abstract
We study the design of computationally efficient randomized algorithms for the k-server problem. Existing randomized algorithms with the best known competitive ratios are, on the one hand, inherently implicit and, on the other hand, employ a rounding scheme that maintains a distribution over exponentially many configurations. In this work, we introduce a derandomization framework that transforms any randomized k-server algorithm on a hierarchically separated tree into one that uses only O(log k) random bits for request sequences of arbitrary length - hence maintaining a distribution over only polynomially many server configurations. Leveraging this black-box derandomization, we obtain the first polynomial-time randomized k-server algorithm on arbitrary n-point metrics with a polylogarithmic competitive ratio. Our results also have implications for the advice complexity of the k-server problem.
Cite as
Christian Coester and Romain Cosson. Randomized k-Server in Polynomial Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{coester_et_al:LIPIcs.ICALP.2026.65,
author = {Coester, Christian and Cosson, Romain},
title = {{Randomized k-Server in Polynomial Time}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {65:1--65:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.65},
URN = {urn:nbn:de:0030-drops-264549},
doi = {10.4230/LIPIcs.ICALP.2026.65},
annote = {Keywords: k-server, online algorithms, computational complexity, randomized complexity, advice complexity}
}
Document
Track A: Algorithms, Complexity and Games
Online Monotone Metric Embeddings
Abstract
Metric embeddings into structured spaces, particularly hierarchically well-separated trees (HSTs), are a fundamental tool in the design of online algorithms. In the classical online embedding setting, points arrive sequentially and must be embedded irrevocably upon arrival, resulting in strong distortion lower bounds of Ω(min(n, log nlog Δ)), where n is the number of points and Δ their aspect ratio.
We propose a novel relaxation, online monotone metric embeddings, which allows distances between embedded points in the target space to decrease monotonically over time. Such relaxed embeddings remain compatible with many online algorithms. Moreover, this relaxation breaks existing lower bound barriers, enabling embeddings into HSTs with distortion O(log² n).
We also study a dynamic variant, where points may both arrive and depart, seeking distortion guarantees in terms of the maximum number l of simultaneously present points. For traditional embeddings, such bounds are impossible, and this limitation persists even for deterministic monotone embeddings. Surprisingly, probabilistic monotone embeddings allow for O(l log l) distortion, which is nearly optimal given an Ω(l) lower bound.
Cite as
Christian Coester and Yichen Huang. Online Monotone Metric Embeddings. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 66:1-66:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{coester_et_al:LIPIcs.ICALP.2026.66,
author = {Coester, Christian and Huang, Yichen},
title = {{Online Monotone Metric Embeddings}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {66:1--66:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.66},
URN = {urn:nbn:de:0030-drops-264550},
doi = {10.4230/LIPIcs.ICALP.2026.66},
annote = {Keywords: Online Algorithms, Metric Embeddings, k-Taxi}
}
Document
Track A: Algorithms, Complexity and Games
Chasing Small Sets Optimally Against Adaptive Adversaries
Abstract
We study deterministic online algorithms for the problem of chasing sets of cardinality at most k in a metric space, also known as metrical service systems and equivalent to width-k layered graph traversal. We resolve the 30-year-old gap of Ω(2^k)∩ O(k2^k) on the competitive ratio of this problem by giving an O(2^k)-competitive deterministic algorithm. This bound is optimal even among randomized algorithms against adaptive adversaries. We also (slightly) improve the deterministic lower bound to D_k, defined recursively by D₁ = 1 and D_{k+1} = 2D_k+√{8+8D_k}+3, which we conjecture to be exactly tight. For k = 3, we provide a matching upper bound of D₃. Our results imply slightly improved upper and lower bounds for distributed asynchronous collective tree exploration and for the k-taxi problem, respectively.
Our algorithm generalizes the classical doubling strategy, previously known to be optimal for k = 2. The previous best bound for general k was achieved by the generalized work function algorithm (WFA), and was known to be tight for WFA. Our improved bound therefore implies that WFA is sub-optimal for chasing small sets.
Cite as
Christian Coester and Alexa Tudose. Chasing Small Sets Optimally Against Adaptive Adversaries. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 67:1-67:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{coester_et_al:LIPIcs.ICALP.2026.67,
author = {Coester, Christian and Tudose, Alexa},
title = {{Chasing Small Sets Optimally Against Adaptive Adversaries}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {67:1--67:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.67},
URN = {urn:nbn:de:0030-drops-264568},
doi = {10.4230/LIPIcs.ICALP.2026.67},
annote = {Keywords: online algorithms, competitive analysis, chasing small sets, layered graph traversal, metrical service systems}
}
Document
Track A: Algorithms, Complexity and Games
Tracing AG Codes: Toward Meeting the Gilbert-Varshamov Bound
Abstract
One of the oldest problems in coding theory is to match the Gilbert-Varshamov bound with explicit binary codes. Over larger - yet still constant-sized - fields, algebraic-geometry codes are known to beat the GV bound. In this work, we leverage this phenomenon by taking traces of AG codes. Our hope is that the margin by which AG codes exceed the GV bound will withstand the parameter loss incurred by taking the trace from a constant field extension to the binary field. In contrast to concatenation, the usual alphabet-reduction method, our analysis of trace-of-AG (TAG) codes uses the AG codes’ algebraic structure throughout - including in the alphabet-reduction step.
Our main technical contribution is a Hasse-Weil–type theorem that is well-suited for the analysis of TAG codes. The classical theorem (and its Grothendieck trace-formula extension) are inadequate in this setting. Although we do not obtain improved constructions, we show that a constant-factor strengthening of our bound would suffice. We also analyze the limitations of TAG codes under our bound and prove that, in the high-distance regime, they are inferior to code concatenation. Our Hasse-Weil–type theorem holds in far greater generality than is needed for analyzing TAG codes. In particular, we derive new estimates for exponential sums.
Cite as
Gil Cohen, Dean Doron, Noam Goldgraber, and Tomer Manket. Tracing AG Codes: Toward Meeting the Gilbert-Varshamov Bound. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 68:1-68:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cohen_et_al:LIPIcs.ICALP.2026.68,
author = {Cohen, Gil and Doron, Dean and Goldgraber, Noam and Manket, Tomer},
title = {{Tracing AG Codes: Toward Meeting the Gilbert-Varshamov Bound}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {68:1--68:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.68},
URN = {urn:nbn:de:0030-drops-264579},
doi = {10.4230/LIPIcs.ICALP.2026.68},
annote = {Keywords: Coding Theory, Algebraic Geometry, Finite Fields, Exponential Sums}
}
Document
Track A: Algorithms, Complexity and Games
Online Matroid Embeddings
Abstract
We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. The existence of such embedding enables a reduction from the version of the matroid secretary problem where the matroid is unknown to the version where the matroid is known in advance. We establish the existence of such an embedding for binary matroids, and use it to relate variants of the binary matroid secretary problem to each other, showing that seemingly simpler problems are in fact equivalent to seemingly harder ones (up to constant-factors). Specifically, we show this to be the case for the version of the matroid secretary problem in which the binary matroid is not known in advance, and where it is known in advance. We also show that the version with known matroid structure is equivalent to the problem where weights are not fully adversarial but drawn from a known pairwise-independent distribution.
Cite as
Andrés Cristi, Paul Dütting, Robert Kleinberg, Renato Paes Leme, and Neel Patel. Online Matroid Embeddings. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 69:1-69:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cristi_et_al:LIPIcs.ICALP.2026.69,
author = {Cristi, Andr\'{e}s and D\"{u}tting, Paul and Kleinberg, Robert and Paes Leme, Renato and Patel, Neel},
title = {{Online Matroid Embeddings}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {69:1--69:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.69},
URN = {urn:nbn:de:0030-drops-264585},
doi = {10.4230/LIPIcs.ICALP.2026.69},
annote = {Keywords: Matroids, Secretary Problem, Online Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Incremental (k, z)-Clustering on Graphs
Abstract
Given a weighted undirected graph, a number of clusters k, and an exponent z, the goal in the (k, z)-clustering problem on graphs is to select k vertices as centers that minimize the sum of the distances raised to the power z of each vertex to its closest center. This problem includes the well-known k-median (z = 1) and k-means (z = 2) clustering problems. In the dynamic setting, the graph is subject to adversarial edge updates, and the goal is to maintain explicitly an exact (k, z)-clustering solution in the induced shortest-path metric.
Prior works by Bhattacharya, Costa, Garg, Lattanzi, and Parotsidis [FOCS 2024] and by Bhattacharya, Costa, and Farokhnejad [STOC 2025] consider the dynamic (k, z)-clustering problem for point sets in metric spaces. These algorithms support adversarial point insertions and deletions under a model with access to pairwise distances. This model differs significantly from the dynamic graph setting, where no oracle access is given to pairwise distances and a single edge update can affect many distances - making these approaches inefficient when applied to graphs. While efficient dynamic k-center approximation algorithms on graphs exist [Cruciani, Forster, Goranci, Nazari, and Skarlatos, SODA 2024], to the best of our knowledge, no prior work provides similar results for the dynamic (k,z)-clustering problem.
As the main result of this paper, we develop a randomized incremental (k, z)-clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of Õ(k m^{1+o(1)} + k^{1+1/(λ)} m), where λ ≥ 1 is an arbitrary fixed constant. Our incremental algorithm also achieves an amortized update time of Õ(k n^o(1) + k^{1+1/(λ)}) and consists of two stages. In the first stage, we maintain a constant-factor bicriteria approximate solution of size Õ(k) with a total update time of m^{1+o(1)} (independent of the parameter k) over all adversarial edge insertions. This first stage is an intricate adaptation of the bicriteria approximation algorithm by Mettu and Plaxton [Machine Learning 2004] to incremental graphs. One of our key technical results is that the radii in their algorithm can be assumed to be non-decreasing while the approximation ratio remains constant - a property that may be of independent interest.
In the second stage, we maintain a constant-factor approximate (k,z)-clustering solution on a dynamic weighted instance induced by the bicriteria approximate solution. For this subproblem, we employ a dynamic spanner algorithm together with a static (k,z)-clustering algorithm.
Cite as
Emilio Cruciani, Sebastian Forster, and Antonis Skarlatos. Incremental (k, z)-Clustering on Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 70:1-70:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cruciani_et_al:LIPIcs.ICALP.2026.70,
author = {Cruciani, Emilio and Forster, Sebastian and Skarlatos, Antonis},
title = {{Incremental (k, z)-Clustering on Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {70:1--70:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.70},
URN = {urn:nbn:de:0030-drops-264599},
doi = {10.4230/LIPIcs.ICALP.2026.70},
annote = {Keywords: (k, z)-clustering, k-median, k-means, dynamic graph algorithms}
}
Document
Track A: Algorithms, Complexity and Games
The Quantum Smooth Label Cover Problem Is Undecidable
Abstract
We show that the quantum smooth label cover problem is undecidable and RE-hard. This sharply contrasts the quantum unique label cover problem, which can be decided efficiently by a result of Kempe, Regev, and Toner (FOCS'08). On the other hand, our result aligns with the RE-hardness of the quantum label cover problem, which follows from the celebrated MIP^* = RE result of Ji, Natarajan, Vidick, Wright, and Yuen (ACM'21). Additionally, we show that the quantum oracularized smooth label cover problem is RE-hard. Our second result fits with the alternative quantum unique games conjecture recently proposed by Mousavi and Spirig (ITCS'25) on the RE-hardness of the quantum oracularized unique label cover problem. Our proof techniques include a quantum version of Feige’s reduction from 3SAT to 3SAT5 (STOC'96) for BCS-MIP^*-protocols, which may be of independent interest.
Cite as
Eric Culf, Kieran Mastel, Connor Paddock, and Taro Spirig. The Quantum Smooth Label Cover Problem Is Undecidable. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 71:1-71:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{culf_et_al:LIPIcs.ICALP.2026.71,
author = {Culf, Eric and Mastel, Kieran and Paddock, Connor and Spirig, Taro},
title = {{The Quantum Smooth Label Cover Problem Is Undecidable}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {71:1--71:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.71},
URN = {urn:nbn:de:0030-drops-264602},
doi = {10.4230/LIPIcs.ICALP.2026.71},
annote = {Keywords: Complexity Theory, Constraint Satisfaction Problems, Hardness of Approximation, Quantum Computing}
}
Document
Track A: Algorithms, Complexity and Games
On Tight FPT Time Approximation Algorithms for k-Clustering Problems
Abstract
Following recent advances in combining approximation algorithms with fixed-parameter tractability (FPT), we study FPT-time approximation algorithms for minimum-norm k-clustering problems, parameterized by the number k of open facilities.
For the capacitated setting, we give a tight (3+ε)-approximation for the general-norm capacitated k-clustering problem in FPT-time parameterized by k and ε. Prior to our work, such a result was only known for the capacitated k-median problem [Cohen-Addad and Li, 2019]. As a special case, our result yields an FPT-time 3-approximation for capacitated k-center. The problem has not been studied in the FPT-time setting, with the previous best known polynomial-time approximation ratio being 9 [An et al., 2015].
In the uncapacitated setting, we consider the top-cn norm k-clustering problem, where the goal of the problem is to minimize the top-cn norm of the connection distance vector. Our main result is a tight (1 + 2/(ec) + ε)-approximation algorithm for the problem with c ∈ (1/e, 1]. (For the case c ≤ 1/e, there is a simple tight (3+ε)-approximation.) Our framework can be easily extended to give a tight (3, 1 + 2/e + ε)-bi-criteria approximation for the (k-center, k-median) problem in FPT time, improving the previous best polynomial-time (4, 8) guarantee [Soroush Alamdari and David B. Shmoys, 2017].
All results are based on a unified framework: computing a (1+ε)-approximate solution using O((k log n)/ε) facilities S via LP rounding, sampling a few client representatives R based on the solution S, guessing a few pivots from S ∪ R and some radius information on the pivots, and solving the problem using the guesses. We believe this framework can lead to further results on k-clustering problems.
Cite as
Han Dai, Shi Li, and Sijin Peng. On Tight FPT Time Approximation Algorithms for k-Clustering Problems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 72:1-72:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dai_et_al:LIPIcs.ICALP.2026.72,
author = {Dai, Han and Li, Shi and Peng, Sijin},
title = {{On Tight FPT Time Approximation Algorithms for k-Clustering Problems}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {72:1--72:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.72},
URN = {urn:nbn:de:0030-drops-264613},
doi = {10.4230/LIPIcs.ICALP.2026.72},
annote = {Keywords: Approximation algorithms, Monotone symmetric norms, Clustering, Fixed parameter tractability}
}
Document
Track A: Algorithms, Complexity and Games
Online Correlation Clustering: Simultaneously Optimizing All 𝓁_p-Norms
Abstract
The 𝓁_p-norm objectives for correlation clustering present a fundamental trade-off between minimizing total disagreements (the 𝓁₁-norm) and ensuring fairness to individual nodes (the 𝓁_∞-norm). Surprisingly, in the offline setting it is possible to simultaneously approximate all 𝓁_p-norms with a single clustering. Can this powerful guarantee be achieved in an online setting? This paper provides the first affirmative answer. We present a single algorithm for the online-with-a-sample (AOS) model that, given a small constant fraction of the input as a sample, produces one clustering that is simultaneously O(log⁴ n)-competitive for all 𝓁_p-norms with high probability, O(log n)-competitive for the 𝓁_∞-norm with high probability, and O(1)-competitive for the 𝓁₁-norm in expectation. This work successfully translates the offline "all-norm" guarantee to the online world.
Our setting is motivated by a new hardness result that demonstrates a fundamental separation between these objectives in the standard random-order (RO) online model. Namely, while the 𝓁₁-norm is trivially O(1)-approximable in the RO model, we prove that any algorithm in the RO model for the fairness-promoting 𝓁_∞-norm must have a competitive ratio of at least Ω(n^{1/3}). This highlights the necessity of a different beyond-worst-case model. We complement our algorithm with lower bounds, showing our competitive ratios for the 𝓁₁- and 𝓁_∞- norms are nearly tight in the AOS model.
Cite as
Sami Davies, Benjamin Moseley, and Heather Newman. Online Correlation Clustering: Simultaneously Optimizing All 𝓁_p-Norms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 73:1-73:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{davies_et_al:LIPIcs.ICALP.2026.73,
author = {Davies, Sami and Moseley, Benjamin and Newman, Heather},
title = {{Online Correlation Clustering: Simultaneously Optimizing All 𝓁\underlinep-Norms}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {73:1--73:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.73},
URN = {urn:nbn:de:0030-drops-264620},
doi = {10.4230/LIPIcs.ICALP.2026.73},
annote = {Keywords: Online algorithms, correlation clustering, all-norms objective, beyond-worst-case analysis}
}
Document
Track A: Algorithms, Complexity and Games
Unsplittable Transshipments
Abstract
We introduce the Unsplittable Transshipment Problem in directed graphs with multiple sources and sinks. An unsplittable transshipment routes given supplies and demands using at most one path for each source-sink pair. Although they are a natural generalization of single source unsplittable flows, unsplittable transshipments raise interesting new challenges and require novel algorithmic techniques. As our main contribution, we give a nontrivial generalization of a seminal result of Dinitz, Garg, and Goemans (1999) by showing how to efficiently turn a given transshipment x into an unsplittable transshipment y with y_a < x_a+d_max for all arcs a, where d_max is the maximum demand (or supply) value. Further results include bounds on the number of rounds required to satisfy all demands, where each round consists of an unsplittable transshipment that routes a subset of the demands while respecting arc capacity constraints.
Cite as
Srinwanti Debgupta, Sarah Morell, and Martin Skutella. Unsplittable Transshipments. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 74:1-74:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{debgupta_et_al:LIPIcs.ICALP.2026.74,
author = {Debgupta, Srinwanti and Morell, Sarah and Skutella, Martin},
title = {{Unsplittable Transshipments}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {74:1--74:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.74},
URN = {urn:nbn:de:0030-drops-264634},
doi = {10.4230/LIPIcs.ICALP.2026.74},
annote = {Keywords: Network flow, unsplittable flow, flow augmentation}
}
Document
Track A: Algorithms, Complexity and Games
Coordinated Motion Planning Is FPT on Discretized Simple Polygons
Abstract
In the coordinated motion planning problem, we are given a graph together with the starting and destination vertices of k robots. At each time step, any subset of robots may move, each traversing an edge of the graph, provided that no two robots collide. The goal is to compute a schedule that routes all robots to their destinations while minimizing some objective function. In this paper, we focus on the well-studied objective of minimizing the total travel length of all robots. This problem is known to be NP-hard, and it has been shown to be fixed-parameter tractable (FPT), when parameterized by the number k of robots, on full grids (SoCG 2023) and on bounded-treewidth graphs (ICALP 2024).
We present a fixed-parameter algorithm for coordinated motion planning, parameterized by the number k of robots, on graphs arising from discretizations of simple polygons. Such graphs are of particular interest in real-world applications, where planar motion is often constrained to discretized representations of polygonal environments. Moreover, these graphs generalize rectangular grids; consequently, our result constitutes a significant step toward resolving the parameterized complexity of coordinated motion planning on subgrids and, ultimately, planar graphs - two prominent open problems in the field.
Cite as
Argyrios Deligkas, Eduard Eiben, Robert Ganian, and Iyad Kanj. Coordinated Motion Planning Is FPT on Discretized Simple Polygons. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 75:1-75:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{deligkas_et_al:LIPIcs.ICALP.2026.75,
author = {Deligkas, Argyrios and Eiben, Eduard and Ganian, Robert and Kanj, Iyad},
title = {{Coordinated Motion Planning Is FPT on Discretized Simple Polygons}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {75:1--75:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.75},
URN = {urn:nbn:de:0030-drops-264648},
doi = {10.4230/LIPIcs.ICALP.2026.75},
annote = {Keywords: coordinated motion planning, multi-agent path finding, parameterized complexity}
}
Document
Track A: Algorithms, Complexity and Games
Tight Bounds for Sampling q-Colorings via Coupling from the Past
Abstract
The Coupling from the Past (CFTP) paradigm is a canonical method for perfect sampling. For uniform sampling of proper q-colorings in graphs with maximum degree Δ, the bounding chains of [Huber, STOC '98] provide a systematic framework for efficiently implementing CFTP algorithms within the classical regime q ≥ (1+o(1))Δ². This was subsequently improved to q > 3Δ by [Bhandari and Chakraborty, STOC '20] and to q ≥ (8/3 + o(1))Δ by [Jain, Sah, and Sawhney, STOC '21].
In this work, we establish the asymptotically tight threshold for bounding-chain-based CFTP algorithms for graph colorings. We prove a lower bound showing that all such algorithms satisfying the standard contraction property require q ≥ 2.5Δ, and we present an efficient CFTP algorithm that achieves this asymptotically optimal threshold q ≥ (2.5 + o(1))Δ via an optimal design of bounding chains.
Cite as
Tianxing Ding, Hongyang Liu, Yitong Yin, and Can Zhou. Tight Bounds for Sampling q-Colorings via Coupling from the Past. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 76:1-76:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ding_et_al:LIPIcs.ICALP.2026.76,
author = {Ding, Tianxing and Liu, Hongyang and Yin, Yitong and Zhou, Can},
title = {{Tight Bounds for Sampling q-Colorings via Coupling from the Past}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {76:1--76:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.76},
URN = {urn:nbn:de:0030-drops-264657},
doi = {10.4230/LIPIcs.ICALP.2026.76},
annote = {Keywords: perfect sampling, coupling from the past, graph coloring, bounding chains, Markov chains}
}
Document
Track A: Algorithms, Complexity and Games
Learning-Augmented Online Algorithms for Nonclairvoyant Joint Replenishment Problem with Deadlines
Abstract
This paper considers using predictions in the context of the online Joint Replenishment Problem with Deadlines (JRP-D). Prior work includes asymptotically optimal competitive ratios of O(1) for the clairvoyant setting and O(√n) of the nonclairvoyant setting, where n is the number of items. The goal of this paper is to significantly reduce the competitive ratio for the nonclairvoyant case by leveraging predictions: when a request arrives, the true deadline of the request is not revealed, but the algorithm is given a predicted deadline.
The main result is an algorithm whose competitive ratio is O(min(η^{1/3} log^{2/3}(n), √η, √n)), where n is the number of item types and η ≤ n² quantifies how flawed the predictions are in terms of the number of "instantaneous item inversions." Thus, the algorithm is robust, i.e., it is never worse than the nonclairvoyant solution, and it is consistent, i.e., if the predictions exhibit no inversions, then the algorithm behaves similarly to the clairvoyant algorithm. Moreover, if the error is not too large, specifically η < o(n^{3/2}/log²(n)), then the algorithm obtains an asymptotically better competitive ratio than the nonclairvoyant algorithm. We also show that all deterministic algorithms falling in a certain reasonable class of algorithms have a competitive ratio of Ω(η^{1/3}), so this algorithm is nearly the best possible with respect to this error metric.
Cite as
Michael Dinitz, Jeremy Fineman, and Seeun William Umboh. Learning-Augmented Online Algorithms for Nonclairvoyant Joint Replenishment Problem with Deadlines. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 77:1-77:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dinitz_et_al:LIPIcs.ICALP.2026.77,
author = {Dinitz, Michael and Fineman, Jeremy and Umboh, Seeun William},
title = {{Learning-Augmented Online Algorithms for Nonclairvoyant Joint Replenishment Problem with Deadlines}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {77:1--77:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.77},
URN = {urn:nbn:de:0030-drops-264664},
doi = {10.4230/LIPIcs.ICALP.2026.77},
annote = {Keywords: Online algorithms, Joint Replenishment, Algorithms with Predictions}
}
Document
Track A: Algorithms, Complexity and Games
Improved Time-Space Tradeoffs for 3SUM-Indexing
Abstract
3SUM-Indexing is a preprocessing variant of the 3SUM problem that has recently received a lot of attention. The best known time-space tradeoff for the problem is T S³ = n⁶ (up to logarithmic factors), where n is the number of input integers, S is the length of the preprocessed data structure, and T is the running time of the query algorithm. This tradeoff was achieved in [Kopelowitz and Porat, 2019; Golovnev et al., 2020] using the Fiat-Naor generic algorithm for Function Inversion. Consequently, [Golovnev et al., 2020] asked whether this algorithm can be improved by leveraging the structure of 3SUM-Indexing.
In this paper, we exploit the structure of 3SUM-Indexing to give a time-space tradeoff of T S = n^{2.5}, which is better than the best known one in the range n^{3/2} ≪ S ≪ n^{7/4}. We further extend this improvement to the kSUM-Indexing problem - a generalization of 3SUM-Indexing - and to the related kXOR-Indexing problem, where addition is replaced with XOR. Additionally, we improve the best known time-space tradeoffs for the Jumbled Indexing problem, which is a well-known data structure problem related to 3SUM-Indexing.
Our improvement comes from an alternative way to apply the Fiat-Naor algorithm to 3SUM-Indexing. Specifically, we exploit the structure of the function to be inverted by decomposing it into "sub-functions" with certain properties. This allows us to apply an improvement to the Fiat-Naor algorithm (which is not directly applicable to 3SUM-Indexing), obtained in [Golovnev et al., 2023] in a much larger range of parameters. We believe that our techniques may be useful in additional application-dependent optimizations of the Fiat-Naor algorithm.
Cite as
Itai Dinur and Alexander Golovnev. Improved Time-Space Tradeoffs for 3SUM-Indexing. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 78:1-78:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dinur_et_al:LIPIcs.ICALP.2026.78,
author = {Dinur, Itai and Golovnev, Alexander},
title = {{Improved Time-Space Tradeoffs for 3SUM-Indexing}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {78:1--78:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.78},
URN = {urn:nbn:de:0030-drops-264674},
doi = {10.4230/LIPIcs.ICALP.2026.78},
annote = {Keywords: Data Structures, 3SUM, Function Inversion}
}
Document
Track A: Algorithms, Complexity and Games
On Randomness Complexity of 1-Private Protocols
Abstract
In the field of information-theoretic cryptography, randomness complexity is a key metric for protocols for private computation, that is, the number of random bits needed to realize the protocol. Although some general bounds are known, even for the relatively simple example of 1-private computation of n-party AND, the exact complexity is unknown.
We study two settings. First, we consider the model of Goyal, Ishai, and Song (Crypto '22) where helper parties without any inputs are allowed to assist in the computation. In this setting, we show that two random bits always suffice to compute an arbitrary Boolean circuit C 1-privately: a single designated inputless helper flips the two bits and privately distributes the derived one-time bits to the other helper parties and the input parties as they are needed. We give an explicit construction using seven helper parties per AND gate and three helper parties per XOR gate (plus the single global randomness dealer). Moreover, two random bits are necessary already for the AND functionality (by a reduction to the standard no-helper model together with the lower bound of Kushilevitz, Ostrovsky, Prouff, Rosén, Thillard and Vergnaud (TCC '19), and therefore the worst-case helper-party randomness complexity is exactly 2 bits.
Second, in the setting without helper parties, we improve the upper bound from Couteau and Rosén (Asiacrypt '22) on the (asymptotic) randomness complexity of n-party AND from 6 to 5 bits. That is, we give a 1-private protocol for computing the AND of n parties' inputs requiring 5 bits of randomness, for all n ≥ 6. Our construction, like that of Couteau and Rosén, uses a single party to flip the 5 bits and distribute the required derived values during the execution. Our approach to both problems is built around a more systematic exploration of techniques for recycling randomness across sub-computations.
As part of resolving the second problem, we isolate an exact local-independence combinatorial object called a Sliding-Window Independence Generator, or a SWIG. A (k,m)-SWIG is a linear generator from a k-bit seed to m ≥ k output bits, where every cyclic length-k sliding window chosen from m output bits is perfectly uniform. We give an explicit (k,m)-SWIG for every k ≥ 1 and every m ≥ k and use a (5,n-1)-SWIG in our no-helper AND protocol.
Cite as
Samuel Dittmer and Rafail Ostrovsky. On Randomness Complexity of 1-Private Protocols. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 79:1-79:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dittmer_et_al:LIPIcs.ICALP.2026.79,
author = {Dittmer, Samuel and Ostrovsky, Rafail},
title = {{On Randomness Complexity of 1-Private Protocols}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {79:1--79:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.79},
URN = {urn:nbn:de:0030-drops-264680},
doi = {10.4230/LIPIcs.ICALP.2026.79},
annote = {Keywords: limited independence, bounded independence, k-wise independence, H-wise independent sample spaces, small sample spaces, small probability spaces, local independence, locally independent sample spaces, sliding-window independence, cyclic-window independence, sliding-window independence generators, cyclic-consecutive test arrays, covering arrays, pseudorandom generators, derandomization, randomness complexity, 1-private protocols, secure multiparty computation, n-party AND, helper parties}
}
Document
Track A: Algorithms, Complexity and Games
Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics
Abstract
In the metric k-median problem we are given a finite metric space (X∪ Y, 𝐝) and the objective is to compute a set of k centers C ⊆ Y that minimizes ∑_{p ∈ X} min_{c ∈ C} 𝐝(p,c). In general metric spaces, the best polynomial time algorithm, which is due to Cohen-Addad, Grandoni, Lee, Schwiegelshohn, and Svensson [Vincent Cohen-Addad et al., 2025], computes a (2+ε)-approximation for arbitrary constant ε > 0. However, if the metric space has bounded doubling dimension, a near linear time (1+ε)-approximation algorithm is known due to the work of Cohen-Addad, Feldmann, and Saulpic [Vincent Cohen{-}Addad et al., 2021].
In this paper, we show that the (1+ε)-approximation algorithm can be generalized to the case when either X or Y has bounded doubling dimension (but the other set not). The case when X has bounded doubling dimension is motivated by the assumption that even though X is part of a high-dimensional space, it may be that it is close to a low-dimensional structure. The case when Y has bounded doubling dimension is perhaps more natural. It is motivated by specific clustering problems where the centers are low-dimensional. Specifically, our work in this setting implies the first near linear time approximation algorithm for the (k,𝓁)-median problem under discrete Fréchet distance when 𝓁 is constant. The latter problem is a version of the k-median problem under Fréchet distance when the input consists of time series of z reals and where the centers are time series of 𝓁 reals [Anne Driemel et al., 2016]. Previously, for this problem no (1+ε)-approximation algorithm with running time polynomial in k was known. We also introduce a novel complexity reduction for time series of real values that leads to a similar result for the case of discrete Fréchet distance.
In order to solve the case when Y has a bounded doubling dimension, we introduce a form of dimension reduction that replaces points from X by sets of points in Y. To solve the case when X has a bounded doubling dimension, we generalize Talwar’s decomposition [Kunal Talwar, 2004] of doubling metrics to our setting. The running time of our algorithms is 2^{2^t} Õ(n+m) where t = O(ddim log ddim/ε) and where ddim is the doubling dimension of X (resp. Y). The results also extend to the metric (uncapacitated) facility location problem. We believe that our techniques are likely applicable to other problems.
Cite as
Anne Driemel, Jan Höckendorff, Ioannis Psarros, Christian Sohler, and Di Yue. Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{driemel_et_al:LIPIcs.ICALP.2026.80,
author = {Driemel, Anne and H\"{o}ckendorff, Jan and Psarros, Ioannis and Sohler, Christian and Yue, Di},
title = {{Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {80:1--80:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.80},
URN = {urn:nbn:de:0030-drops-264693},
doi = {10.4230/LIPIcs.ICALP.2026.80},
annote = {Keywords: Approximation Algorithms, Doubling Spaces, Facility Location, k-Median, Discrete Fr\'{e}chet Distance}
}
Document
Track A: Algorithms, Complexity and Games
A Faster Directed Single-Source Shortest Path Algorithm
Abstract
This paper presents a new deterministic algorithm for the single-source shortest paths (SSSP) problem on real non-negative edge-weighted directed graphs, with running time O(m√{log n} + √{mnlog nlog log n}), which is O(m√{log nlog log n}) for sparse graphs. This improves the recent breakthrough result of O(m log^{2/3} n) time for directed SSSP algorithm [Duan, Mao, Mao, Shu, Yin 2025].
Cite as
Ran Duan, Xiao Mao, Xinkai Shu, and Longhui Yin. A Faster Directed Single-Source Shortest Path Algorithm. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 81:1-81:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{duan_et_al:LIPIcs.ICALP.2026.81,
author = {Duan, Ran and Mao, Xiao and Shu, Xinkai and Yin, Longhui},
title = {{A Faster Directed Single-Source Shortest Path Algorithm}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {81:1--81:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.81},
URN = {urn:nbn:de:0030-drops-264706},
doi = {10.4230/LIPIcs.ICALP.2026.81},
annote = {Keywords: Shortest Paths, Graph Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
A Fine-Grained Dichotomy for the Center Problem on Gromov Hyperbolic Graphs
Abstract
A vertex in a graph is called central if it minimizes its maximum distance to the other vertices. The radius of a graph G is the largest distance between a central vertex and the other vertices, and it is denoted by rad(G). In the center problem, we are asked to find a central vertex. We study the fine-grained complexity of the center problem on graphs with small Gromov hyperbolicity. Roughly, the Gromov hyperbolicity of a graph represents how close, locally, it is to a tree, from a metric point of view. It has applications in the design of approximation algorithms. In particular, there is a linear-time algorithm that for every δ-hyperbolic graph G outputs some vertex at distance at most rad(G) + 5δ to the other vertices [Chepoi et al, SoCG'08]. However, a linear-time algorithm for computing a central vertex is known only for 0-hyperbolic graphs, whereas its existence was ruled out for 2-hyperbolic graphs under the Hitting Set Conjecture of [Abboud et al, SODA'16]. Our main contribution in the paper is a linear-time algorithm for computing a central vertex in the class of 1/2-hyperbolic graphs. Furthermore, we rule out the existence of such an algorithm for 1-hyperbolic graphs, under the Hitting Set Conjecture, thus completely settling all the cases left open.
Cite as
Guillaume Ducoffe. A Fine-Grained Dichotomy for the Center Problem on Gromov Hyperbolic Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 82:1-82:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ducoffe:LIPIcs.ICALP.2026.82,
author = {Ducoffe, Guillaume},
title = {{A Fine-Grained Dichotomy for the Center Problem on Gromov Hyperbolic Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {82:1--82:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.82},
URN = {urn:nbn:de:0030-drops-264715},
doi = {10.4230/LIPIcs.ICALP.2026.82},
annote = {Keywords: Center problem, Gromov hyperbolicity, Fine-grained complexity in P, Graph algorithms}
}
Document
Track A: Algorithms, Complexity and Games
On the Hardness of Recognizing Graphs of Small Mim-Width and Its Variants
Abstract
The mim-width of a graph is a powerful structural parameter that, when bounded by a constant, allows several hard problems to be polynomial-time solvable - with a recent meta-theorem encompassing a large class of problems [SODA2023]. Since its introduction, several variants such as sim-width and omim-width were developed, along with a linear version of these parameters. It was recently shown that mim-width and all these variants are all paraNP-hard, a consequence of the NP-hardness of distinguishing between graphs of linear mim-width at most 1211 and graphs of sim-width at least 1216 [ICALP2025]. The complexity of recognizing graphs of small width, particularly those close to 1, remained open, despite their especially attractive algorithmic applications.
In this work, we show that the width recognition problems remain NP-hard even on small widths. Specifically, after introducing the novel parameter Omim-width sandwiched between omim-width and mim-width, we show that: (1) deciding whether a graph has sim-width = 1, omim-width = 1, or Omim-width = 1 is NP-hard, and the same is true for their linear variants; (2) the problems of deciding whether mim-width ≤ 2 or linear mim-width ≤ 2 are both NP-hard. Interestingly, our reductions are relatively simple and are from the Unrooted Quartet Consistency problem, which is of great interest in computational biology but is not commonly used in the theory of algorithms.
Cite as
Max Dupré la Tour, Manuel Lafond, and Ndiamé Ndiaye. On the Hardness of Recognizing Graphs of Small Mim-Width and Its Variants. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 83:1-83:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{duprelatour_et_al:LIPIcs.ICALP.2026.83,
author = {Dupr\'{e} la Tour, Max and Lafond, Manuel and Ndiaye, Ndiam\'{e}},
title = {{On the Hardness of Recognizing Graphs of Small Mim-Width and Its Variants}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {83:1--83:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.83},
URN = {urn:nbn:de:0030-drops-264725},
doi = {10.4230/LIPIcs.ICALP.2026.83},
annote = {Keywords: Mim-width, NP-hardness, Computational Biology}
}
Document
Track A: Algorithms, Complexity and Games
Faster and Simpler Greedy Algorithm for k-Median and k-Means
Abstract
Clustering problems such as k-means and k-median are staples of unsupervised learning, and many algorithmic techniques have been developed to tackle their numerous aspects.
In this paper, we focus on the class of greedy approximation algorithm, that attracted less attention than local-search or primal-dual counterparts. In particular, we study the recursive greedy algorithm developed by Mettu and Plaxton [SIAM J. Comp 2003]. We provide a simplification of the algorithm, allowing for faster implementation: our algorithm matches the state-of-the-art running time for computing a constant-factor approximation in Euclidean space and graph metrics, and, in addition, is the first near-linear-time to compute a polylogarithmic approximation in Euclidean space.
Cite as
Max Dupré la Tour and David Saulpic. Faster and Simpler Greedy Algorithm for k-Median and k-Means. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 84:1-84:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{duprelatour_et_al:LIPIcs.ICALP.2026.84,
author = {Dupr\'{e} la Tour, Max and Saulpic, David},
title = {{Faster and Simpler Greedy Algorithm for k-Median and k-Means}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {84:1--84:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.84},
URN = {urn:nbn:de:0030-drops-264735},
doi = {10.4230/LIPIcs.ICALP.2026.84},
annote = {Keywords: Clustering, k-means, approximation algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Faster Algorithms for k-Orthogonal Vectors in Low Dimension
Abstract
In the Orthogonal Vectors problem (OV), we are given two families A, B of subsets of {1,…,d}, each of size n, and the task is to decide whether there exists a pair a ∈ A and b ∈ B such that a ∩ b = ∅. Straightforward algorithms for this problem run in 𝒪(n² ⋅ d) or 𝒪(2^d ⋅ n) time, and assuming SETH, there is no 2^o(d)⋅ n^{2-ε} time algorithm that solves this problem for any constant ε > 0.
Williams (FOCS 2024) presented a 𝒪̃(1.35^d ⋅ n)-time algorithm for the problem, based on the succinct equality-rank decomposition of the disjointness matrix. In this paper, we present a combinatorial algorithm that runs in randomized time 𝒪̃(1.25^d ⋅ n). This can be improved to 𝒪(1.16^d ⋅ n) using computer-aided evaluations.
We also consider a more general k-Orthogonal Vectors problem, where given k families A_1,…,A_k of subsets of {1,…,d}, each of size n, the task is to find elements a_i ∈ A_i for every i ∈ {1,…,k} such that a₁ ∩ a₂ ∩ … ∩ a_k = ∅. We show that for every fixed k ⩾ 2, there exists ε_k > 0 such that the k-OV problem can be solved in time 𝒪(2^{(1 - ε_k)⋅d} ⋅ n). We also show that, asymptotically, this is the best we can hope for: for any ε > 0 there exists a k ⩾ 2 such that 2^{(1 - ε)⋅ d} ⋅ n^𝒪(1) time algorithm for k-Orthogonal Vectors would contradict the Set Cover Conjecture.
Cite as
Anita Dürr, Evangelos Kipouridis, Michael Lampis, and Karol Węgrzycki. Faster Algorithms for k-Orthogonal Vectors in Low Dimension. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{durr_et_al:LIPIcs.ICALP.2026.85,
author = {D\"{u}rr, Anita and Kipouridis, Evangelos and Lampis, Michael and W\k{e}grzycki, Karol},
title = {{Faster Algorithms for k-Orthogonal Vectors in Low Dimension}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {85:1--85:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.85},
URN = {urn:nbn:de:0030-drops-264747},
doi = {10.4230/LIPIcs.ICALP.2026.85},
annote = {Keywords: Orthogonal Vectors, Fine-grained Complexity, Exact Algorithms, Set Cover}
}
Document
Track A: Algorithms, Complexity and Games
Random Access in Grammar-Compressed Strings: Optimal Trade-Offs in Almost All Parameter Regimes
Abstract
A Random Access query to a string T asks for the character T[i] at a given position i ∈ [0..|T|). This fundamental task admits a straightforward solution with constant-time queries and 𝒪(n log σ) bits of space when T ∈ [0..σ)ⁿ. While this is the best one can achieve in the worst case, much research has focused on the compressed setting: if T is compressible, one can hope for a much smaller data structure that still answers Random Access queries efficiently.
In this work, we investigate the grammar-compressed setting, where T is represented by a context-free grammar that produces only T. Our main result is a general trade-off that optimizes Random Access time as a function of the string length n, the grammar size (the total length of productions) g, the alphabet size σ, the data structure size M, and the word size w ≥ Ω(log n) of the word RAM model. For any data structure size M satisfying glog n < Mw < nlog σ, we show an 𝒪(M)-size data structure that answers Random Access queries in time 𝒪(log((n log σ)/(Mw)) / log(Mw/(g log n))) . We also prove a matching unconditional lower bound that holds for all parameter regimes except very small grammars (g ≤ w^{1+o(1)} log n) and relatively small data structures (Mw ≤ g log n ⋅ w^o(1)). The lower bound applies to word-RAM query time and, more strongly, to the worst-case cell-probe complexity of nondeterministic or bounded-error randomized query algorithms.
Previous work focused on optimizing the query time as a function of n only, achieving 𝒪(log n) time using 𝒪(g) space [Bille, Landau, Raman, Sadakane, Satti, Weimann; SIAM J. Comput. 2015] and 𝒪((log n)/(log log n)) time using 𝒪(g log^ε n) space for any constant ε > 0 [Belazzougui, Cording, Puglisi, Tabei; ESA 2015], [Ganardi, Jeż, Lohrey; J. ACM 2021]. Our result improves upon these bounds (strictly for g = n^{1-o(1)}) and generalizes them beyond M ≤ 𝒪(g poly log n), yielding a smooth interpolation with the uncompressed setting of Mw = nlogσ bits.
Thus far, the only tight lower bound [Verbin and Yu; CPM 2013] was Ω((log n)/(log log n)) for w = Θ(log n), n^Ω(1) ≤ g ≤ n^{1-Ω(1), and M = g⋅log^Θ(1) n. In contrast, our result yields a tight bound that accounts for all relevant parameters and is valid for almost all parameter regimes.
Our bounds remain valid for run-length grammars, where production sizes use run-length encoding. This lets us recover (and, for strings with small run-length grammars, improve) the trade-offs achieved by block trees, formulated in terms of the LZ77 size z [Belazzougui, Cáceres, Gagie, Gawrychowski, Kärkkäinen, Navarro, Ordóñez, Puglisi, Tabei; J. Comput. Syst. Sci. 2021] and substring complexity δ [Kociumaka, Navarro, Prezza; IEEE Trans. Inf. Theory 2023].
Our data structure admits an efficient deterministic construction algorithm. Beyond Random Access, its variants also support substring extraction (with optimal additive overhead 𝒪((m log σ)/w) for a length-m substring, provided that M ≥ g), as well as rank and select queries.
All our results rely on novel grammar transformations that generalize contracting grammars [Ganardi; ESA 2021] and achieve the optimal trade-off between grammar size and height while enforcing extra structure crucial for constant-time navigation in the parse tree.
Cite as
Anouk Duyster and Tomasz Kociumaka. Random Access in Grammar-Compressed Strings: Optimal Trade-Offs in Almost All Parameter Regimes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 86:1-86:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{duyster_et_al:LIPIcs.ICALP.2026.86,
author = {Duyster, Anouk and Kociumaka, Tomasz},
title = {{Random Access in Grammar-Compressed Strings: Optimal Trade-Offs in Almost All Parameter Regimes}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {86:1--86:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.86},
URN = {urn:nbn:de:0030-drops-264755},
doi = {10.4230/LIPIcs.ICALP.2026.86},
annote = {Keywords: grammar-based compression, straight-line programs, random access problem}
}
Document
Track A: Algorithms, Complexity and Games
White-Box Adversarial Streaming Lower Bounds Beyond Two-Party Communication
Abstract
Streaming algorithms in adversarial settings have attracted considerable attention recently. We show that, in the white-box adversarial streaming model [Miklós Ajtai et al., 2022], the fundamental problem of estimating the F_p moment to within any constant factor requires Ω(n) memory. In this model, the internal state of the (randomized) streaming algorithm is visible to an adversary, who can exploit this information when constructing subsequent stream updates. As a corollary, we also obtain a white-box lower bound for the well-studied problem of estimating the maximum matching size in graphs.
[Miklós Ajtai et al., 2022] proved that two-party white-box communication protocols can be derandomized. This allows them to prove deterministic communication lower bounds and automatically derive white-box (communication and streaming) lower bounds. However, such two-party lower bounds can only rule out approximation of the F_p moment within a specific constant factor. Ruling out approximation within any constant factor typically requires proving a lower bound for a multi-party communication problem.
We show that white-box communication protocols involving any number of parties can be derandomized, provided they compute a total function. However, this derandomization fails entirely when extended to partial functions and, consequently, to approximation problems. We are therefore compelled to prove our moment estimation lower bound for the white-box model directly. Our proof introduces a novel hybrid technique that, instead of taking hybrids over input distributions, constructs hybrids over white-box adversaries.
Cite as
Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang. White-Box Adversarial Streaming Lower Bounds Beyond Two-Party Communication. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 87:1-87:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{efremenko_et_al:LIPIcs.ICALP.2026.87,
author = {Efremenko, Klim and Kol, Gillat and Saxena, Raghuvansh R. and Zhang, Zhijun},
title = {{White-Box Adversarial Streaming Lower Bounds Beyond Two-Party Communication}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {87:1--87:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.87},
URN = {urn:nbn:de:0030-drops-264760},
doi = {10.4230/LIPIcs.ICALP.2026.87},
annote = {Keywords: White-box streaming, moment estimation, hybrid argument}
}
Document
Track A: Algorithms, Complexity and Games
Recursive Jump Operators and Optimal Proof Systems
Abstract
We study the relationship between the existence of optimal proof systems and recursive jump operators, two central open problems in proof complexity. For a set L, an optimal proof system is a strongest proof system in terms of proof length, whereas a recursive jump operator uniformly transforms any proof system for L into a stronger one with respect to proof length, thereby witnessing non-optimality. It is clear that the existence of a recursive jump operator for L rules out optimal proof systems for L. Khaniki (FOCS 2024) is interested in the converse of this implication and explicitly poses the following question, where TAUT denotes the set of propositional tautologies.
- Q: Does the non-existence of optimal proof systems for TAUT imply the existence of recursive jump operators for TAUT?
We generalize and address this question from both a relativized and an unrelativized perspective. We show that proving a positive answer for Q is provably hard by constructing the following oracle.
- O: The polynomial-time hierarchy is infinite, TAUT has no optimal proof systems, and TAUT has no recursive jump operators.
This shows that Khaniki’s question can not be answered in the positive by relativizable means, even under the standard complexity-theoretic assumption that the polynomial-time hierarchy is infinite.
In contrast, we obtain positive results when the question Q is posed for sets different from TAUT. We prove that the existence of recursive jump operators is upward closed under ≤_m^p-reducibility, a result that so far was only known for the non-existence of optimal proof systems. Furthermore, we show that the sets known to have no optimal proof systems by Messner (STACS 1999) in fact admit recursive jump operators. Thus, essentially all sets currently known to have no optimal proof systems have recursive jump operators.
Cite as
Fabian Egidy. Recursive Jump Operators and Optimal Proof Systems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 88:1-88:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{egidy:LIPIcs.ICALP.2026.88,
author = {Egidy, Fabian},
title = {{Recursive Jump Operators and Optimal Proof Systems}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {88:1--88:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.88},
URN = {urn:nbn:de:0030-drops-264770},
doi = {10.4230/LIPIcs.ICALP.2026.88},
annote = {Keywords: Relativization, Oracles, Proof Complexity, Optimal Proof Systems, Jump Operators}
}
Document
Track A: Algorithms, Complexity and Games
Submodular Maximization over a Matroid k-Intersection: Multiplicative Improvement over Greedy
Abstract
We study the problem of maximizing a non-negative monotone submodular objective f subject to the intersection of k arbitrary matroid constraints. The natural greedy algorithm guarantees (k+1)-approximation for this problem, and the state-of-the-art algorithm only improves this approximation ratio to k. We give a (2k ln 2)/(1 + ln 2) + O(√k) < 0.819 k + O(√k) approximation algorithm for this problem. Our result is the first multiplicative improvement over the approximation ratio of the greedy algorithm for general k. We further show that our algorithm can be used to obtain roughly the same approximation ratio also for the more general problem in which the objective is not guaranteed to be monotone (the sublinear term in the approximation ratio becomes O(k^{2/3}) rather than O(√k) in this case).
All of our results hold also when the k-matroid intersection constraint is replaced with a more general matroid k-parity constraint. Furthermore, unlike the case in many of the previous works, our algorithms run in time that is independent of k and polynomial in the size of the ground set. Our algorithms are based on a hybrid greedy local search approach recently introduced by Singer and Thiery [Neta Singer and Theophile Thiery, 2025] for the weighted matroid k-intersection problem, which is a special case of the problem we consider. Leveraging their approach in the submodular setting requires several non-trivial insights and algorithmic modifications since the marginals of a submodular function f, which correspond to the weights in the weighted case, are not independent of the algorithm’s internal randomness. In the special weighted case studied by [Neta Singer and Theophile Thiery, 2025], our algorithms reduce to a variant of the algorithm of [Neta Singer and Theophile Thiery, 2025] with an improved approximation ratio of (k + 1) ln 2 + O(ε) < 0.694k + 0.694 + O(ε), compared to an approximation ratio of (k+1)/(2ln 2) ≈ 0.722k + 0.722 guaranteed by Singer and Thiery [Neta Singer and Theophile Thiery, 2025].
Cite as
Moran Feldman and Justin Ward. Submodular Maximization over a Matroid k-Intersection: Multiplicative Improvement over Greedy. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 89:1-89:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{feldman_et_al:LIPIcs.ICALP.2026.89,
author = {Feldman, Moran and Ward, Justin},
title = {{Submodular Maximization over a Matroid k-Intersection: Multiplicative Improvement over Greedy}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {89:1--89:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.89},
URN = {urn:nbn:de:0030-drops-264785},
doi = {10.4230/LIPIcs.ICALP.2026.89},
annote = {Keywords: Submodular function, matroid k-parity, matroid intersection, local search, greedy}
}
Document
Track A: Algorithms, Complexity and Games
New Convex Programming Technique for Nash Social Welfare and Scheduling
Abstract
We propose a new convex programming relaxation for the weighted Nash social welfare (NSW) problem that achieves a matching (e^{1/e} ≈ 1.445)-approximation via the rounding algorithm of Feng and Li. Unlike the exponential-size configuration LP used in prior work, our formulation can be converted into a compact linear program of polynomial size, incurring only an additive loss of ln(1+ε) in the objective. This allows the program to be solved directly using standard LP solvers, without the ellipsoid method or dual separation oracles.
In the unweighted case, we show that our convex program is equivalent to the restricted-spending Fisher market convex program of Cole and Gkatzelis, yielding a constructive proof that its integrality gap is exactly e^{1/e}. With a minor modification, our analysis also gives a simple proof of the e^{1/e} EF1 gap for the identical agent setting. Finally, we show that our convex programming technique extends to two unrelated machine scheduling problems, recovering the best-known approximation ratios with simpler analyses.
Cite as
Yuda Feng, Weijiang Hu, and Shi Li. New Convex Programming Technique for Nash Social Welfare and Scheduling. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 90:1-90:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{feng_et_al:LIPIcs.ICALP.2026.90,
author = {Feng, Yuda and Hu, Weijiang and Li, Shi},
title = {{New Convex Programming Technique for Nash Social Welfare and Scheduling}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {90:1--90:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.90},
URN = {urn:nbn:de:0030-drops-264797},
doi = {10.4230/LIPIcs.ICALP.2026.90},
annote = {Keywords: Nash Social Welfare, Convex Programming, Approximation Algorithms, Scheduling}
}
Document
Track A: Algorithms, Complexity and Games
Faster Weak Expander Decompositions and Approximate Max Flow
Abstract
We give faster algorithms for weak expander decompositions and approximate max flow on undirected graphs. First, we show that it is possible to "warm start" the cut-matching game when computing weak expander decompositions, avoiding the cost of the recursion depth. Our algorithm is also flexible enough to support weaker flow subroutines than previous algorithms.
Our second contribution is to streamline the recent non-recursive approximate max flow algorithm of Li, Rao, and Wang (SODA, 2025) and adapt their framework to use our new weak expander decomposition primitive. Consequently, we give an approximate max flow algorithm within a few logarithmic factors of the limit of expander decomposition-based approaches.
Cite as
Henry Fleischmann, George Z. Li, and Jason Li. Faster Weak Expander Decompositions and Approximate Max Flow. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 91:1-91:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fleischmann_et_al:LIPIcs.ICALP.2026.91,
author = {Fleischmann, Henry and Li, George Z. and Li, Jason},
title = {{Faster Weak Expander Decompositions and Approximate Max Flow}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {91:1--91:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.91},
URN = {urn:nbn:de:0030-drops-264800},
doi = {10.4230/LIPIcs.ICALP.2026.91},
annote = {Keywords: max flow, expander decompositions, congestion approximators, cut-matching game}
}
Document
Track A: Algorithms, Complexity and Games
Beyond Brooks: (Δ-1)-Coloring in Semi-Streaming
Abstract
Reed [J. Comb. Theory B, 1999] showed that graphs of maximum degree Δ ⩾ 10^14 without Δ-cliques are (Δ-1)-colorable. We design a one-pass semi-streaming algorithm for computing such a coloring. Additionally, we prove that any one-pass (Δ-k)-coloring algorithm for 0 ⩽ k < (Δ+1)/2 requires Ω(n(k+1)) space.
Cite as
Maxime Flin and Magnús M. Halldórsson. Beyond Brooks: (Δ-1)-Coloring in Semi-Streaming. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 92:1-92:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{flin_et_al:LIPIcs.ICALP.2026.92,
author = {Flin, Maxime and Halld\'{o}rsson, Magn\'{u}s M.},
title = {{Beyond Brooks: (\Delta-1)-Coloring in Semi-Streaming}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {92:1--92:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.92},
URN = {urn:nbn:de:0030-drops-264817},
doi = {10.4230/LIPIcs.ICALP.2026.92},
annote = {Keywords: Graph coloring, streaming algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Proving Algebraic Independence in Zero-Knowledge
Abstract
A set of multivariate polynomials is algebraically independent if they exhibit no non-trivial algebraic relations, and this notion is fundamental in algebra. When these polynomials are given as algebraic circuits, deciding algebraic independence has several applications in algebraic complexity theory. Over fields of zero (or exponentially large) characteristic, this problem is known to have an efficient randomized algorithm. Over finite fields of small characteristic, a sequence of works has culminated in showing that algebraic independence admits Arthur-Merlin proofs, in particular giving the complexity bound of AM∩coAM ([Guo et al., 2019]).
We improve the complexity of deciding algebraic independence over finite fields by showing that it admits zero-knowledge proofs, in particular giving the upper bound of NISZK ⊆ AM∩coAM, the class of problems admitting non-interactive statistical zero-knowledge proofs. This is achieved by arguing that algebraically independent polynomials yield maps whose output distribution has high-entropy, while algebraically dependent polynomials yield maps with low-entropy. We can then reduce to the question of approximating entropy, which is a known NISZK-complete problem.
We also more generally show that transcendence degree, which quantifies the independence of a set of possibly dependent polynomials, can be computed in NISZK.
Cite as
Michael A. Forbes and Andrei Staicu. Proving Algebraic Independence in Zero-Knowledge. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 93:1-93:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{forbes_et_al:LIPIcs.ICALP.2026.93,
author = {Forbes, Michael A. and Staicu, Andrei},
title = {{Proving Algebraic Independence in Zero-Knowledge}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {93:1--93:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.93},
URN = {urn:nbn:de:0030-drops-264820},
doi = {10.4230/LIPIcs.ICALP.2026.93},
annote = {Keywords: Algebraic Dependence, Non-Interactive Statistical Zero-Knowledge}
}
Document
Track A: Algorithms, Complexity and Games
When Does Sparsity Help for k-Independent Set in Hypergraphs and Other Boolean CSPs?
Abstract
Consider the fundamental task of finding independent sets of (constant) size k in a given n-node hypergraph. How much is the time complexity affected by the sparsity of the input, i.e., the number of hyperedges m? Turán’s theorem implies that the problem is trivial if m = O(n^{2-ε}) for some ε > 0. Above that threshold (i.e., if m = Θ(n^γ) for some γ ≥ 2), we give a perhaps surprising algorithm with running time O(min{ n^({ω/3}k) + m^{k/3}, n^k}) (for k divisible by 3), which is essentially conditionally optimal for all γ ≥ 2, assuming the k-clique and 3-uniform hyperclique hypotheses (here, ω ≤ 2.372 denotes the matrix multiplication exponent). In fact, we obtain a more detailed time complexity that is sensitive to the arity distribution of the hyperedges.
To study such phenomena in more generality, we study the time complexity of finding solutions of (constant) size k in sparse instances of Boolean constraint satisfaction problems, where n and m denote the number of variables and constraints, respectively. Our results include, among others:
- an essentially full classification of the influence of sparsity for Boolean constraint families of binary arity. Of particular technical interest is a conditionally tight algorithm for the family consisting of the binary NAND and the binary Implication constraints, with a running time of Θ(m^{ω k/6 ± c}).
- the identification of a large class of constraint families ℱ that exhibits a sharp phase transition: there is a threshold γ_ℱ such that the problem is trivial for m = O(n^{γ_ℱ-ε}), but requires essentially brute-force running time Θ(n^{k±c}) for m = Ω(n^{γ_ℱ}), assuming the 3-uniform hyperclique hypothesis. In general, we observe a rich landscape of time complexities. Notably, in many cases the combination of constraints display higher time complexity than either constraint alone.
Cite as
Timo Fritsch, Marvin Künnemann, Mirza Redzic, and Julian Stieß. When Does Sparsity Help for k-Independent Set in Hypergraphs and Other Boolean CSPs?. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 94:1-94:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fritsch_et_al:LIPIcs.ICALP.2026.94,
author = {Fritsch, Timo and K\"{u}nnemann, Marvin and Redzic, Mirza and Stie{\ss}, Julian},
title = {{When Does Sparsity Help for k-Independent Set in Hypergraphs and Other Boolean CSPs?}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {94:1--94:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.94},
URN = {urn:nbn:de:0030-drops-264836},
doi = {10.4230/LIPIcs.ICALP.2026.94},
annote = {Keywords: Multivariate algorithmics, fine-grained complexity theory, classification theorems, algorithmic hypergraph theory}
}
Document
Track A: Algorithms, Complexity and Games
Computing the (k+2)-Edge-Connected Components in k-Edge-Connected Digraphs in Subquadratic Time
Abstract
Computing edge-connected components in directed and undirected graphs is a fundamental and well-studied problem in graph algorithms. In a very recent breakthrough, Korhonen [STOC 2025] showed that for any fixed k, the k-edge connected components of an undirected graph can be computed in linear time. In contrast, the directed case remains significantly more challenging: linear-time algorithms are only known for k ≤ 3, and for any fixed k > 3, the best known bound for sparse or moderately dense graphs is still the O(mn)-time algorithm of Nagamochi and Watanabe (1993).
In this paper, we break the O(mn) barrier for all k = o(n^{1/4}/√{log{n}}). We present a randomized algorithm that computes the (k+2)-edge-connected components of a k-edge-connected directed graph in O(k² m √n log n) time, for any k. This constitutes the first improvement over the classic Nagamochi-Watanabe bound for any constant k > 3. Our approach introduces new structural insights into directed edge-cuts and combines these with both new and existing techniques. A central contribution of our work is a substantial simplification and generalization of the framework introduced in [Loukas Georgiadis et al., 2023], which achieved an Õ(m√m) bound for computing the 3-edge-connected components of a digraph. In addition, we develop a variant of our algorithm that achieves the same O(m √n log n) running time for computing the 4-edge-connected components of a general directed graph.
Cite as
Loukas Georgiadis, Evangelos Kipouridis, Evangelos Kosinas, Charis Papadopoulos, and Nikos Parotsidis. Computing the (k+2)-Edge-Connected Components in k-Edge-Connected Digraphs in Subquadratic Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 95:1-95:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{georgiadis_et_al:LIPIcs.ICALP.2026.95,
author = {Georgiadis, Loukas and Kipouridis, Evangelos and Kosinas, Evangelos and Papadopoulos, Charis and Parotsidis, Nikos},
title = {{Computing the (k+2)-Edge-Connected Components in k-Edge-Connected Digraphs in Subquadratic Time}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {95:1--95:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.95},
URN = {urn:nbn:de:0030-drops-264846},
doi = {10.4230/LIPIcs.ICALP.2026.95},
annote = {Keywords: Graph connectivity, edge-connected components, directed edge-cuts}
}
Document
Track A: Algorithms, Complexity and Games
A 9/4-Approximation for Directed Feedback Vertex Sets in Quasi-Transitive Digraphs
Abstract
We provide the first non-trivial approximation algorithm for the fundamental directed feedback vertex set (DFVS) problem in the class of quasi-transitive digraphs. This class of digraphs encompasses both dense and sparse classes of digraphs, for which specialized DFVS algorithms were proposed in the literature, like tournaments or transitive orientations of bounded treewidth graphs.
Our approximation algorithm can handle both dense graphs, as well as sparse graphs, by a single approach, which is based on carefully analysing the solutions to a linear programming relaxation of DFVS. It also handles the node-weighted DFVS problem, for which it computes a 9/4-approximation in polynomial time.
Along the way, we improve and simplify the best-known deterministic polynomial-time approximation algorithms for DFVS in tournaments (Cai et al., SICOMP 2001; Mnich et al., ESA 2016).
Cite as
Ebrahim Ghorbani and Matthias Mnich. A 9/4-Approximation for Directed Feedback Vertex Sets in Quasi-Transitive Digraphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 96:1-96:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ghorbani_et_al:LIPIcs.ICALP.2026.96,
author = {Ghorbani, Ebrahim and Mnich, Matthias},
title = {{A 9/4-Approximation for Directed Feedback Vertex Sets in Quasi-Transitive Digraphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {96:1--96:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.96},
URN = {urn:nbn:de:0030-drops-264852},
doi = {10.4230/LIPIcs.ICALP.2026.96},
annote = {Keywords: directed feedback vertex set, tournaments, quasi-transitive digraphs}
}
Document
Track A: Algorithms, Complexity and Games
Quantum Algorithms on Edge Lists: Hiding, Shuffling, and Cycle Finding
Abstract
The edge list model is arguably the simplest input model for graphs, where the graph is specified by a list of its edges. In this model, we study the quantum query complexity of three variants of the triangle finding problem. The first asks whether there exists a triangle containing a target edge and raises general questions about the hiding of a problem’s input among irrelevant data. The second asks whether there exists a triangle containing a target vertex and raises general questions about the shuffling of a problem’s input. The third asks whether there exists a triangle; this problem bridges the 3-distinctness and 3-sum problems, which have been extensively studied by both cryptographers and complexity theorists. We provide tight or nearly tight results for these problems as well as some first answers to the general questions they raise.
Furthermore, given any graph with low maximum degree, such as a typical random sparse graph, we prove that the quantum query complexity of finding a length-k cycle in its length-m edge list is m^{3/4-1/(2^{k+2}-4) ± o(1)}, which matches the best-known upper bound for the quantum query complexity of k-distinctness on length-m inputs up to an m^o(1) factor. We prove the lower bound by developing new techniques within Zhandry’s recording query framework [Zhandry, 2019] as generalized by Hamoudi and Magniez [Hamoudi and Magniez, 2023]. These techniques extend the framework to treat any non-product distribution that results from conditioning a product distribution on the absence of rare events. We prove the upper bound by adapting Belovs’s learning graph algorithm for k-distinctness [Belovs, 2012]. Finally, assuming a plausible conjecture concerning only cycle finding, we show that the lower bound can be lifted to an essentially tight lower bound on the quantum query complexity of k-distinctness, which is a long-standing open question.
Cite as
Amin Shiraz Gilani, Daochen Wang, Pei Wu, and Xingyu Zhou. Quantum Algorithms on Edge Lists: Hiding, Shuffling, and Cycle Finding. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 97:1-97:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gilani_et_al:LIPIcs.ICALP.2026.97,
author = {Gilani, Amin Shiraz and Wang, Daochen and Wu, Pei and Zhou, Xingyu},
title = {{Quantum Algorithms on Edge Lists: Hiding, Shuffling, and Cycle Finding}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {97:1--97:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.97},
URN = {urn:nbn:de:0030-drops-264867},
doi = {10.4230/LIPIcs.ICALP.2026.97},
annote = {Keywords: Quantum query complexity, graph algorithms, edge list model}
}
Document
Track A: Algorithms, Complexity and Games
Optimal Sequential Flows
Abstract
We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximise the flow through a graph where edge capacities can be changed over time by choosing a sequence of capacity labelings from a given finite set. Our method is based on a novel factorization theorem for finite semigroups that, applied to a suitable flow semigroup, allows to derive small witnesses. This generalises to multiple in/output vertices, as well as regular constraints.
Cite as
Hugo Gimbert, Corto Mascle, and Patrick Totzke. Optimal Sequential Flows. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 98:1-98:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gimbert_et_al:LIPIcs.ICALP.2026.98,
author = {Gimbert, Hugo and Mascle, Corto and Totzke, Patrick},
title = {{Optimal Sequential Flows}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {98:1--98:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.98},
URN = {urn:nbn:de:0030-drops-264875},
doi = {10.4230/LIPIcs.ICALP.2026.98},
annote = {Keywords: Network Flow, Sequential Flow, Semigroup Factorization}
}
Document
Track A: Algorithms, Complexity and Games
Quickly Excluding an Annotated Planar Graph
Abstract
We provide proofs certifying that the structure theorem for vertex sets of bounded bidimensionality holds with polynomial bounds. The bidimensionality of vertex sets is a common generalisation of both treewidth and the face-cover-number of vertex sets in planar graphs. As such, it plays a crucial role in extensions of Courcelle’s Theorem to H-minor-free graphs. Recently, bidimensionality and similar parameters have emerged as key for extensions of known parameterized algorithms for problems defined on a terminal set R. A prominent example for such a problem is Steiner Tree, which admits efficient algorithms on planar graphs whenever R can be covered with few faces.
Key to the algorithmic applications of bidimensionality is a structure theorem that explains how a graph G can be decomposed into pieces where the behaviour of R is highly controlled. One may see this structure theorem as a rooted analogue of Robertson and Seymour’s celebrated Grid Theorem. Combining recent advances in obtaining polynomial bounds in the Graph Minors framework with new techniques for handling annotated vertex sets, we show that all parameters in the structure theorem above admit polynomial bounds. As an application, we also provide a sketch showing how our techniques imply polynomial bounds for the structure theorem for graphs excluding an apex minor.
Cite as
Maximilian Gorsky, Evangelos Protopapas, and Sebastian Wiederrecht. Quickly Excluding an Annotated Planar Graph. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 99:1-99:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gorsky_et_al:LIPIcs.ICALP.2026.99,
author = {Gorsky, Maximilian and Protopapas, Evangelos and Wiederrecht, Sebastian},
title = {{Quickly Excluding an Annotated Planar Graph}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {99:1--99:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.99},
URN = {urn:nbn:de:0030-drops-264880},
doi = {10.4230/LIPIcs.ICALP.2026.99},
annote = {Keywords: Structural Graph Theory, Graph Minors, Annotated Graphs, Rooted Minors, Colorful Minors, Bidimensionality}
}
Document
Track A: Algorithms, Complexity and Games
The Price of Homogeneity Is Polynomial
Abstract
We provide explicit and polynomial bounds for the Homogeneous Wall Lemma which occurred for the first time implicitly in the 13th entry of Robertson and Seymour’s Graph Minors Series [JCTB 1990] and has since become a cornerstone in the algorithmic theory of graph minors.
A wall where each brick is assigned a set of colours is said to be homogeneous if each brick is assigned the same set of colours. The Homogeneous Wall Lemma says that there exists a function h that, given non-negative integers q and k and an h(q,k)-wall W where each brick is assigned a, possibly empty, subset of {1,…,q} contains a k-wall W' as a subgraph such that, if one assigns to each brick B of W' the union of the sets assigned to the bricks of W in its interior, then W' is homogeneous. It is well-known that h(q,k) ∈ k^𝒪(q). The Homogeneous Wall Lemma plays a key role in most applications of the Irrelevant Vertex Technique where an exponential dependency of h on q usually causes non-uniform dependencies on meta-parameters at best and additional exponential blow-ups at worst. By proving that h(q,k) ∈ 𝒪(q⁴⋅ k⁶), we provide a positive answer to a problem raised by Sau, Stamoulis, and Thilikos [ICALP 2020].
Cite as
Maximilian Gorsky, Michał T. Seweryn, and Sebastian Wiederrecht. The Price of Homogeneity Is Polynomial. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 100:1-100:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gorsky_et_al:LIPIcs.ICALP.2026.100,
author = {Gorsky, Maximilian and Seweryn, Micha{\l} T. and Wiederrecht, Sebastian},
title = {{The Price of Homogeneity Is Polynomial}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {100:1--100:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.100},
URN = {urn:nbn:de:0030-drops-264891},
doi = {10.4230/LIPIcs.ICALP.2026.100},
annote = {Keywords: Graph Minors, Grid Graph, Wall Graph, Homogeneous Wall, Colored Graph, Annotated Graph, Structural Graph Theory, Irrelevant Vertex Technique}
}
Document
Track A: Algorithms, Complexity and Games
Mutable Batch Arguments and Applications
Abstract
We put forth a new concept of mutability for batch arguments (BARGs), called mutable batch arguments. Our goal is to re-envision how we view and use BARGs. Traditionally, a BARG proof π is an immutable encoding of k NP witness ω_1, …, ω_k. A mutable BARG system captures the notion of computations over BARGs, where each proof string π is treated as a mutable encoding of original witnesses. We also study strong privacy notions for mutable BARGs, with the goal of hiding all non-trivial information about witnesses from a mutated proof. Such mutable BARGs are a naturally good fit for many privacy sensitive applications. Our main contributions include introducing the concept of mutable BARGs, identifying non-trivial classes of feasible mutations, designing mutable BARGs with varying capabilities satisfying mutation privacy from standard cryptographic assumptions, and enabling new applications while improving state-of-the-art known for many signature systems.
Cite as
Rishab Goyal. Mutable Batch Arguments and Applications. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 101:1-101:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{goyal:LIPIcs.ICALP.2026.101,
author = {Goyal, Rishab},
title = {{Mutable Batch Arguments and Applications}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {101:1--101:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.101},
URN = {urn:nbn:de:0030-drops-264906},
doi = {10.4230/LIPIcs.ICALP.2026.101},
annote = {Keywords: BARGs, Mutable proofs}
}
Document
Track A: Algorithms, Complexity and Games
Tight Bounds for Low-Error Frequency Moment Estimation and the Power of Multiple Passes
Abstract
Estimating the second frequency moment F₂ of a data stream up to a (1 ± ε) factor is a central problem in the streaming literature. For errors ε > Ω(1/√n), the tight bound Θ(log(ε² n)/ε²) was recently established by Braverman and Zamir. In this work, we complete the picture by resolving the remaining regime of small error, ε < 1/√n, showing that the optimal space complexity is Θ(min(n, 1/ε²)⋅(1 + |log(ε² n)|)) bits for all ε ≥ 1/n², assuming a sufficiently large universe. This closes the gap between the best known Ω(n) lower bound and the straightforward O(n log n) upper bound in that range, and shows that essentially storing the entire stream is necessary for high-precision estimation.
To derive this bound, we fully characterize the two-party communication complexity of estimating the size of a set intersection up to an arbitrary additive error ε n. In particular, we prove a tight Ω(n log n) lower bound for one-way communication protocols when ε < n^{-1/2-Ω(1)}, in contrast to classical O(n)-bit protocols that use two-way communication. Motivated by this separation, we present a two-pass streaming algorithm that computes the exact histogram of a stream with high probability using only O(n log log n) bits of space, in contrast to the Θ(n log n) bits required in one pass even to approximate F₂ with small error. This yields the first asymptotic separation between one-pass and O(1)-passes space complexity for small frequency moment estimation.
Cite as
Naomi Green-Maimon and Or Zamir. Tight Bounds for Low-Error Frequency Moment Estimation and the Power of Multiple Passes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 102:1-102:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{greenmaimon_et_al:LIPIcs.ICALP.2026.102,
author = {Green-Maimon, Naomi and Zamir, Or},
title = {{Tight Bounds for Low-Error Frequency Moment Estimation and the Power of Multiple Passes}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {102:1--102:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.102},
URN = {urn:nbn:de:0030-drops-264911},
doi = {10.4230/LIPIcs.ICALP.2026.102},
annote = {Keywords: streaming algorithms, frequency moments, communication complexity, multipass streaming}
}
Document
Track A: Algorithms, Complexity and Games
On the Pure Quantum Polynomial Hierarchy and Quantified Hamiltonian Complexity
Abstract
We prove several new results concerning the pure quantum polynomial hierarchy pureQPH. First, we show that QMA(2) ⊆ pureQΣ_2, i.e., two unentangled existential provers can be simulated by competing existential and universal provers. We further prove that pureQΣ_2 ⊆ QΣ_3 ⊆ NEXP. Second, we give an error reduction result for pureQPH, and, as a consequence, prove that pureQPH = QPH. A key ingredient in this result is an improved dimension-independent disentangler. Finally, we initiate the study of quantified Hamiltonian complexity, the quantum analogue of quantified Boolean formulae. We prove that the quantified pure sparse Hamiltonian problem is pureQΣ_i-complete. By contrast, other natural variants (pure/local, mixed/local, and mixed/sparse) admit nontrivial containments but fail to be complete under known techniques. For example, we show that the ∃∀-mixed local Hamiltonian problem lies in NP^QMA ∩ coNP^QMA.
Cite as
Sabee Grewal and Dorian Rudolph. On the Pure Quantum Polynomial Hierarchy and Quantified Hamiltonian Complexity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 103:1-103:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{grewal_et_al:LIPIcs.ICALP.2026.103,
author = {Grewal, Sabee and Rudolph, Dorian},
title = {{On the Pure Quantum Polynomial Hierarchy and Quantified Hamiltonian Complexity}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {103:1--103:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.103},
URN = {urn:nbn:de:0030-drops-264922},
doi = {10.4230/LIPIcs.ICALP.2026.103},
annote = {Keywords: quantum complexity theory, quantum polynomial hierarchy, pure quantum polynomial hierarchy, QPH, QMA(2), quantum proof systems, interactive proofs, quantified Hamiltonian complexity, local Hamiltonian problem, sparse Hamiltonians, disentanglers}
}
Document
Track A: Algorithms, Complexity and Games
Algorithms for Finite Group Epimorphism Testing
Abstract
The Group Epimorphism Problem (GpEpi) asks, given two finite groups G₁ and G₂, whether there exists a surjective group homomorphism, or epimorphism, from G₁ to G₂. When the input groups are given by their multiplication (Cayley) tables, the problem admits a quasipolynomial-time algorithm in general, but little is known about its complexity for structured classes of finite groups. In this paper, we study the computational complexity of GpEpi for several well-studied classes of finite groups.
Our main results are polynomial-time epimorphism tests for several classes of groups for which polynomial-time isomorphism testing was previously known:
- Groups with Abelian normal Hall subgroups with cyclic complement
- Groups with (product of) elementary Abelian normal Hall subgroup with elementary Abelian complement.
- Groups with some constraints on their Abelian chief factors.
Cite as
Joshua A. Grochow, Pranjal Srivastava, and Dhara Thakkar. Algorithms for Finite Group Epimorphism Testing. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 104:1-104:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{grochow_et_al:LIPIcs.ICALP.2026.104,
author = {Grochow, Joshua A. and Srivastava, Pranjal and Thakkar, Dhara},
title = {{Algorithms for Finite Group Epimorphism Testing}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {104:1--104:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.104},
URN = {urn:nbn:de:0030-drops-264933},
doi = {10.4230/LIPIcs.ICALP.2026.104},
annote = {Keywords: Group epimorphism problem, group-theoretic algorithms, polynomial-time algorithms, normal Hall subgroups}
}
Document
Track A: Algorithms, Complexity and Games
Better Diameter Bounds for Efficient Shortcuts and a Structural Criterion for Constructiveness
Abstract
All parallel algorithms for directed reachability and shortest paths crucially rely on efficient shortcut constructions. These constructions find directed paths and shortcut them by adding edges, with the goal to reduce the diameter of the graph. A long sequence of works has studied (efficient) shortcut constructions as well as impossibility results on the best diameter and therefore the best parallelism that can be achieved via this approach.
This paper introduces a new conceptual tool for this line of research in the form of a simple and natural structural criterion: A shortcut H for a graph G is certified if for any shortcut edge (u, v) ∈ H, there exists a vertex w such that the edges (u, w) and (w, v) are also in G ∪ H.
We show that this criterion captures constructiveness in the following sense: A shortcut H can be constructed in t time by repeatedly spending 𝓁 time on shortcutting a path of length 𝓁, if and only if, there exists a certified shortcut H' ⊇ H of size Õ(t). Furthermore, all known shortcut constructions with efficient algorithms can be extended to produce certified shortcuts of size Õ(m). On the other hand, for shortcut constructions for which attempts to find efficient implementations have failed, we can show that this is impossible.
We also obtain stronger diameter lower bounds for certified shortcuts and hopsets. For example, no certified shortcut construction with almost-linear size can reduce a graph’s diameter below n^{1/4-o(1)}. This seems to be the best bound one can hope for with current techniques.
Cite as
Bernhard Haeupler, Antti Roeyskoe, and Zhijun Zhang. Better Diameter Bounds for Efficient Shortcuts and a Structural Criterion for Constructiveness. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 105:1-105:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{haeupler_et_al:LIPIcs.ICALP.2026.105,
author = {Haeupler, Bernhard and Roeyskoe, Antti and Zhang, Zhijun},
title = {{Better Diameter Bounds for Efficient Shortcuts and a Structural Criterion for Constructiveness}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {105:1--105:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.105},
URN = {urn:nbn:de:0030-drops-264942},
doi = {10.4230/LIPIcs.ICALP.2026.105},
annote = {Keywords: Certified shortcuts, directed reachability, parallel shortest paths}
}
Document
Track A: Algorithms, Complexity and Games
Spiky Rank and Its Applications to Rigidity and Circuits
Abstract
We introduce spiky rank, a new matrix parameter that enhances blocky rank by combining the combinatorial structure of the latter with linear-algebraic flexibility. A spiky matrix is block-structured with diagonal blocks that are arbitrary rank-one matrices, and the spiky rank of a matrix is the minimum number of such matrices required to express it as a sum. This measure extends blocky rank to real matrices and is more robust for problems with both combinatorial and algebraic character.
Our conceptual contribution is as follows: we propose spiky rank as a well-behaved candidate matrix complexity measure and demonstrate its potential through applications. We show that large spiky rank implies high matrix rigidity, and that spiky rank lower bounds yield lower bounds for depth-2 ReLU circuits, the basic building blocks of neural networks. On the technical side, we establish tight bounds for random matrices and develop a framework for explicit lower bounds, applying it to Hamming distance matrices and spectral expanders. Finally, we relate spiky rank to other matrix parameters, including blocky rank, sparsity, and the γ₂-norm.
Cite as
Lianna Hambardzumyan, Konstantin Myasnikov, Artur Riazanov, Morgan Shirley, and Adi Shraibman. Spiky Rank and Its Applications to Rigidity and Circuits. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 106:1-106:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hambardzumyan_et_al:LIPIcs.ICALP.2026.106,
author = {Hambardzumyan, Lianna and Myasnikov, Konstantin and Riazanov, Artur and Shirley, Morgan and Shraibman, Adi},
title = {{Spiky Rank and Its Applications to Rigidity and Circuits}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {106:1--106:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.106},
URN = {urn:nbn:de:0030-drops-264954},
doi = {10.4230/LIPIcs.ICALP.2026.106},
annote = {Keywords: blocky rank, matrix rigidity, ReLU circuits, spiky rank}
}
Document
Track A: Algorithms, Complexity and Games
The Dirichlet Mechanism for Rounding with Strong Negative Correlation, with Applications
Abstract
Many optimization and scheduling problems can be abstracted in terms of a bipartite "assignment graph" G = (L ∪ R, E), where the goal is to select exactly one edge for each right-node. For example, a right-node may correspond to a job, and a left-node to a possible machine assignment. A common strategy to solve such problems is to obtain a fractional relaxation x_e for each edge e, and then have each right-node independently select an edge with probability x_e. However, this may cause the left-nodes to become unevenly loaded, leading to suboptimal solutions for some problems.
To address this, a number of algorithms for dependent rounding with strong negative correlation have been developed, e.g. Bansal, Srinivasan & Svensson (2021), Im & Shadloo (2020), Im & Li (2023), Harris (2024), Naor, Srinivasan & Wajc (2025). We introduce a new method for this, which we call the Dirichlet mechanism. It is based on having each left-node draw Dirichlet random variables for its edges, and then having each right-node select an edge based on these values. This achieves quantitatively stronger negative correlation than previous algorithms, and is also simpler since it avoids the need for a tie-breaking mechanism.
We illustrate the mechanism with improved approximation ratios for two problems. For oblivious online dependent rounding, we achieve a 0.68-approximation which improves upon the previous 0.652-approximation of Naor, Srinivasan & Wajc (2025). For the problem of scheduling jobs on unrelated machines to minimize weighted completion time, we achieve a 1.387-approximation which improves upon the 1.398-approximation of Harris (2024). (A recent algorithm of Li (2025) based on iterated rounding also provides a 1.36-approximation if the weights of each job are independent of machine.)
Cite as
David G. Harris, George Z. Li, Nitya Raju, and Renata Valieva. The Dirichlet Mechanism for Rounding with Strong Negative Correlation, with Applications. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 107:1-107:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{harris_et_al:LIPIcs.ICALP.2026.107,
author = {Harris, David G. and Li, George Z. and Raju, Nitya and Valieva, Renata},
title = {{The Dirichlet Mechanism for Rounding with Strong Negative Correlation, with Applications}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {107:1--107:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.107},
URN = {urn:nbn:de:0030-drops-264963},
doi = {10.4230/LIPIcs.ICALP.2026.107},
annote = {Keywords: Dirichlet distribution, copula, weighted completion time, online rounding}
}
Document
Track A: Algorithms, Complexity and Games
On (In)approximability of MaxMin Independent Set Reconfiguration
Abstract
In the Independent Set Reconfiguration problem under the Token Addition/Removal rule, given a graph G and two independent sets I and J of G, we want to transform I into J by adding and removing vertices, such that all the sets throughout the process are independent sets. Its approximate version called MaxMin Independent Set Reconfiguration aims to maximise the minimum size of the independent sets in the process above. We study the (in)approximability of this problem for general graphs as well as restricted graph classes. Firstly, on general graphs, we obtain a polynomial-time (n / log n)-factor approximation algorithm, complementing the PSPACE-hardness of n^Ω(1)-factor approximation due to Hirahara and Ohsaka [STOC 2024, ICALP 2024] and the NP-hardness of n^{1-ε}-factor approximation due to Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [TCS 2011]. Secondly, we present a polynomial-time approximation algorithm for degenerate graphs as well as FPT-approximation schemes for bounded-treewidth graphs and H-minor-free graphs. Lastly, we extend the above inapproximability results to bounded-degree graphs, graphs of bandwidth n^{1/2+Θ(1)}, and bipartite graphs.
Cite as
Hung P. Hoang, Naoto Ohsaka, Rin Saito, and Yuma Tamura. On (In)approximability of MaxMin Independent Set Reconfiguration. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 108:1-108:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hoang_et_al:LIPIcs.ICALP.2026.108,
author = {Hoang, Hung P. and Ohsaka, Naoto and Saito, Rin and Tamura, Yuma},
title = {{On (In)approximability of MaxMin Independent Set Reconfiguration}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {108:1--108:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.108},
URN = {urn:nbn:de:0030-drops-264974},
doi = {10.4230/LIPIcs.ICALP.2026.108},
annote = {Keywords: Combinatorial reconfiguration, independent set, approximation algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Local Samplers for Product Distributions
Abstract
We obtain several results on sampling product distributions in a local and randomness-efficient fashion:
1) Let D = (D_1,D_2,…,D_n) be a product distribution where the D_i have constant support and have dyadic probability masses (i.e., of the form a/2^b where a,b are integers). Then D can be sampled in constant time in the bit-probe model (equivalently, in NC⁰) and randomness complexity (h(D)+ε)n, up to an exponentially small statistical error. The dyadic requirement is necessary.
2) Every p-biased distribution can be sampled in constant time in the cell-probe model with randomness complexity h(p)n + √n ⋅ polylog(n), up to a polynomially small statistical distance.
3) We determine the tradeoffs between locality and statistical distance for sampling the 1/4-biased distribution using non-trivial randomness complexity (e.g., 1.99n). For 2 bit probes, essentially no non-trivial approximation is possible; for 3 bit probes, we give a sampler with 1/poly(n) statistical distance and show that this is best possible; finally, 4 bit probes suffice for exponentially small distance.
Our constructions rely on pseudorandom distributions that are bounded uniform on average. These distributions are obtained using various tools from low-density parity-check codes, and recent results on succinct and retrieval data structures by Hu, Liang, Yu, Zhang, and Zhou (STOC 2025).
Cite as
Jordan Horacsek, Chin Ho Lee, Igor Shinkar, Emanuele Viola, and Renfei Zhou. Local Samplers for Product Distributions. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 109:1-109:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{horacsek_et_al:LIPIcs.ICALP.2026.109,
author = {Horacsek, Jordan and Lee, Chin Ho and Shinkar, Igor and Viola, Emanuele and Zhou, Renfei},
title = {{Local Samplers for Product Distributions}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {109:1--109:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.109},
URN = {urn:nbn:de:0030-drops-264981},
doi = {10.4230/LIPIcs.ICALP.2026.109},
annote = {Keywords: Sampling, Succinct data structures, Pseudorandomness}
}
Document
Track A: Algorithms, Complexity and Games
Equivalence Between Coding and Complexity Lower Bounds
Abstract
The classical coding theorem in Kolmogorov complexity [Levin, 1974] states that if a string x is sampled with probability ≥ δ by an algorithm with prefix-free domain, then 𝖪(x) ≤ log(1/δ) + O(1). Motivated by applications in algorithms, average-case complexity, learning, and cryptography, computationally efficient variants of this result have been established for several recently introduced probabilistic measures of time-bounded Kolmogorov complexity, including rKt [Zhenjian Lu and Igor C. Oliveira, 2021] and pK^t [Zhenjian Lu et al., 2022]. However, establishing a coding theorem for classical (non-probabilistic) notions of time-bounded Kolmogorov complexity, such as Kt complexity [Leonid A. Levin, 1984], remains a longstanding open problem despite its significance. In particular, the current status of coding results reveals a fundamental gap in our understanding of the role of randomness in data compression.
In this work, we make progress by establishing the first equivalence between coding for Kt complexity and complexity lower bounds. Specifically, we show that weak coding for polynomial-time samplable distributions with bounds of the form Kt(x) ≤ (1/δ ⋅ |x|)^ε for all ε > 0 holds if and only if EXP ≠ BPP. Building on this equivalence, we show that similar characterizations hold for non-deterministic and zero-error variants of Kt complexity, demonstrating that coding is equivalent to a corresponding complexity separation in each case. We complement these results by establishing additional equivalences involving the computational hardness of approximating time-bounded Kolmogorov complexity, along with an unconditional lower bound on the complexity of approximating zero-error time-bounded Kolmogorov complexity.
These results reveal novel connections between coding (the existence of succinct encodings), complexity separations (e.g., NEXP versus BPP), and meta-complexity (the complexity of deciding if a succinct encoding exists). In particular, our work provides a new perspective on frontier questions in complexity theory and explains why coding theorems exist for rKt and pK^t but remain unknown for other measures of time-bounded Kolmogorov complexity. Finally, our results determine the minimal hardness assumptions sufficient for coding in different settings.
Cite as
Jinqiao Hu, Zhenjian Lu, and Igor C. Oliveira. Equivalence Between Coding and Complexity Lower Bounds. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 110:1-110:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hu_et_al:LIPIcs.ICALP.2026.110,
author = {Hu, Jinqiao and Lu, Zhenjian and Oliveira, Igor C.},
title = {{Equivalence Between Coding and Complexity Lower Bounds}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {110:1--110:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.110},
URN = {urn:nbn:de:0030-drops-264991},
doi = {10.4230/LIPIcs.ICALP.2026.110},
annote = {Keywords: meta-complexity, lower bounds, Kolmogorov complexity}
}
Document
Track A: Algorithms, Complexity and Games
On the (Classical and Quantum) Fine-Grained Complexity of Approximate CVP and Max-Cut
Abstract
We show a linear-size reduction from gap Max-2-Lin(2) (a generalization of the approximate Maximum Cut, or gap Max-Cut, problem) to γ-CVP_p for γ = O(1) and finite p ≥ 1, as well as a no-go theorem against poly-sized non-adaptive quantum reductions from k-SAT to CVP₂. This implies three headline results:
(i) Faster algorithms for γ-CVP_p are also faster algorithms for Max-2-Lin(2) and Max-Cut. Depending on the approximation regime, even a 2^{0.78n}-time or 2^{0.3n}-time algorithm would improve upon state-of-the-art algorithms such as Williams' 2004 algorithm [TCS 2005] or Arora, Barak, and Steurer’s 2010 algorithm [JACM 2015]. This provides evidence that γ-CVP_p for γ = O(1) requires exponential time, improving upon the previous exponential lower-bound for γ-CVP₂ with γ < 3 by Bennett, Golovnev, and Stephens-Davidowitz [FOCS 2017].
(ii) A new almost 2^{(1/2 + ε/4ς + o(1)) n}-time classical algorithm and a new almost 2^{(1/3 + ε/6ς + o(1)) n}-time quantum algorithm for (1-ε, 1-ς)-gap Max-Cut. This algorithm is faster than the algorithm of Arora, Barak and Steurer [JACM 2015], as well as the algorithm of Williams [TCS 2005], and the algorithm of Manurangsi and Trevisan [APPROX 2018] when c₀ ε < ς < c₁ ε for constants c₀, c₁.
(iii) If the Quantum Strong Exponential Time Hypothesis (QSETH) can be used to show a 2^{δ n}-time lower-bound for Max-Cut, Max-2-Lin(2), or CVP₂ for any constant δ > 0, it must be via an adaptive quantum reduction unless NP ⊆ pr-QSZK. This illuminates some difficulties in characterizing the hardness of approximate constraint satisfaction problems and shows that the post-quantum security of lattice-based cryptography likely cannot be supported by QSETH. This result complements the no-go results of Aggarwal and Kumar [FOCS 2023], who showed that the classical security of lattice-based cryptography likely cannot be supported by the classical Strong Exponential Time Hypothesis (SETH).
Cite as
Jeremy Ahrens Huang, Young Kun Ko, and Chunhao Wang. On the (Classical and Quantum) Fine-Grained Complexity of Approximate CVP and Max-Cut. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 111:1-111:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{huang_et_al:LIPIcs.ICALP.2026.111,
author = {Huang, Jeremy Ahrens and Ko, Young Kun and Wang, Chunhao},
title = {{On the (Classical and Quantum) Fine-Grained Complexity of Approximate CVP and Max-Cut}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {111:1--111:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.111},
URN = {urn:nbn:de:0030-drops-265001},
doi = {10.4230/LIPIcs.ICALP.2026.111},
annote = {Keywords: fine-grained complexity, instance compression, quantum algorithms, approximation algorithms, CVP, Max-Cut, Min-UnCut, Max-2-Lin, approximation-preserving reductions}
}
Document
Track A: Algorithms, Complexity and Games
Towards Tight Robust Coresets for k-Medians Clustering
Abstract
This paper considers coresets for the robust k-medians problem with m outliers, and new constructions in various metric spaces are obtained. Specifically, for metric spaces with a bounded VC or doubling dimension d, the coreset size is O(m) + Õ(kdε^{-2}), which is optimal up to logarithmic factors. For Euclidean spaces, the coreset size is O(mε^{-1}) + Õ(min{k^{4/3}ε^{-2}, kε^{-3}}), improving upon a recent result by Jiang and Lou (ICALP 2025). These results also extend to robust (k,z)-clustering, yielding, for VC and doubling dimension, a coreset size of O(m) + Õ(kdε^{-2z}) with the optimal linear dependence on m. This extended result improves upon the earlier work of Huang et al. (SODA 2025). The techniques introduce novel dataset decompositions, enabling chaining arguments to be applied jointly across multiple components.
Cite as
Lingxiao Huang, Zhenyu Jiang, Yi Li, and Xuan Wu. Towards Tight Robust Coresets for k-Medians Clustering. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 112:1-112:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{huang_et_al:LIPIcs.ICALP.2026.112,
author = {Huang, Lingxiao and Jiang, Zhenyu and Li, Yi and Wu, Xuan},
title = {{Towards Tight Robust Coresets for k-Medians Clustering}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {112:1--112:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.112},
URN = {urn:nbn:de:0030-drops-265013},
doi = {10.4230/LIPIcs.ICALP.2026.112},
annote = {Keywords: robust coresets, clustering, k-medians}
}
Document
Track A: Algorithms, Complexity and Games
Incremental k-Lowest Planes and Planar k-Nearest Neighbor with Optimal Query Time
Abstract
In a set of planes in ℝ³, the k-lowest planes query asks for the k lowest planes pierced by a vertical line q. In this paper we describe a semi-dynamic insertion-only data structure that answers k-lowest planes queries in optimal O(log n+k) time. Our data structure uses O(n) space, where n is the number of stored planes, and supports insertions in O(log⁸ n) amortized time. This result provides the first query optimal data structures for several fundamental problems: - An insertion-only structure that answers 3D halfspace range reporting queries on a set of n points in O(log n+k) time, where k is the number of reported points. - An insertion-only structure that answers 3D vertical ray shooting queries on a set of n planes in O(log n+k) time, where k is the number of reported planes. - An insertion-only structure that answers planar k-nearest neighbor queries in O(log n + k) time for any prescribed k (specified at query time). - An insertion-only structure that answers planar circular range reporting queries in time O(log n + k), where k is the number of reported points. For all of the above problems the query bound O(log n + k) is optimal, even in the static scenario. All of the above structures use linear O(n) space and support insertions in O(log⁸ n) amortized time.
We also obtain a query optimal static structure for a 4D problem. That is, one can compute in near-linear time a data structure that answers weighted halfspace range reporting queries in O(log n + k) time, where k is the number of reported points. The structure uses near-linear space.
Cite as
John Iacono, Yakov Nekrich, and Martin P. Seybold. Incremental k-Lowest Planes and Planar k-Nearest Neighbor with Optimal Query Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 113:1-113:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{iacono_et_al:LIPIcs.ICALP.2026.113,
author = {Iacono, John and Nekrich, Yakov and Seybold, Martin P.},
title = {{Incremental k-Lowest Planes and Planar k-Nearest Neighbor with Optimal Query Time}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {113:1--113:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.113},
URN = {urn:nbn:de:0030-drops-265027},
doi = {10.4230/LIPIcs.ICALP.2026.113},
annote = {Keywords: Data Structures, Dynamic Data Structures, k Nearest-Neighbor Queries}
}
Document
Track A: Algorithms, Complexity and Games
Canonical Labelling of Random Regular Graphs
Abstract
We prove that whenever d = d(n) → ∞ and n-d → ∞ as n → ∞, then with high probability for any non-trivial initial colouring, the colour refinement algorithm distinguishes all vertices of the random regular graph 𝒢_{n,d}. This, in particular, implies that with high probability 𝒢_{n,d} admits a canonical labelling computable in time O(min{n^ω, nd²+ndlog n}), where ω < 2.372 is the matrix multiplication exponent.
Cite as
Mikhail Isaev, Tamás Makai, Brendan D. McKay, Paweł Prałat, Jane Tan, and Maksim Zhukovskii. Canonical Labelling of Random Regular Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 114:1-114:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{isaev_et_al:LIPIcs.ICALP.2026.114,
author = {Isaev, Mikhail and Makai, Tam\'{a}s and McKay, Brendan D. and Pra{\l}at, Pawe{\l} and Tan, Jane and Zhukovskii, Maksim},
title = {{Canonical Labelling of Random Regular Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {114:1--114:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.114},
URN = {urn:nbn:de:0030-drops-265039},
doi = {10.4230/LIPIcs.ICALP.2026.114},
annote = {Keywords: random graphs, regular graphs, colour refinement, canonical labelling, graph isomorphism}
}
Document
Track A: Algorithms, Complexity and Games
A Linear Bound for the Size of the Finite Terminal Assembly of a Directed Non-Cooperative Tile Assembly System
Abstract
Introduced in [Erik Winfree, 1998], the abstract tile assembly model (aTAM) is a model of DNA self-assembly. Most of the studies focus on cooperative aTAM where a form of synchronization between the tiles is possible. Simulating Turing machines is achievable in this context. Few results and constructions are known for the non-cooperative case (a variant of Wang tilings [Hao Wang, 1961] where assemblies do not need to cover the whole plane and some mismatches may occur). For example, assembly of a square of width n is done with 2n-1 tiles types whereas only Θ(log(n)/(log log(n))) are required for the cooperative case [Leonard M. Adleman et al., 2001].
Introduced by P.-É. Meunier in [Meunier, 2015], efficient paths are a non-trivial construction for non-cooperative aTAM designed with n different tile types and reaching a distance linearly greater than n. Improved in [Pierre-Étienne Meunier and Damien Regnault, 2019], efficient paths were shown to be able to reach a distance of nlog(n). Assembling them relies heavily on a form of "non-determinism". Indeed, the set of tiles may produce different finite terminal assemblies but they all contain the same efficient path. In this paper, we prove that this non-determinism is strictly necessary for assembling the efficient paths of [Pierre-Étienne Meunier and Damien Regnault, 2019]. More formally, we show that if the terminal assembly of a directed non-cooperative tile assembly system (a model where only one terminal assembly is produced) is finite then its width and length are linear in the number of tiles. This result also implies that the construction of a square of width n using 2n-1 tiles types is asymptotically optimal. Moreover, we hope that the techniques introduced here will lead to a better comprehension of the non-directed case.
Cite as
Sergiu Ivanov and Damien Regnault. A Linear Bound for the Size of the Finite Terminal Assembly of a Directed Non-Cooperative Tile Assembly System. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ivanov_et_al:LIPIcs.ICALP.2026.115,
author = {Ivanov, Sergiu and Regnault, Damien},
title = {{A Linear Bound for the Size of the Finite Terminal Assembly of a Directed Non-Cooperative Tile Assembly System}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {115:1--115:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.115},
URN = {urn:nbn:de:0030-drops-265044},
doi = {10.4230/LIPIcs.ICALP.2026.115},
annote = {Keywords: Models of computation, DNA self-assembly, aTAM, Complexity}
}
Document
Track A: Algorithms, Complexity and Games
A Tight Double-Exponential Lower Bound for High-Multiplicity Bin Packing
Abstract
Consider a high-multiplicity Bin Packing instance I with d distinct item types. In 2014, Goemans and Rothvoss gave an algorithm with runtime (|I|²)^O(d) for this problem [SODA'14], where |I| denotes the encoding length of the instance I. Although Jansen and Klein [SODA'17] later developed an algorithm that improves upon this runtime in a special case, it has remained a major open problem by Goemans and Rothvoss [J.ACM'20] whether the doubly exponential dependency on d is necessary.
We solve this open problem by showing that unless the Exponential Time Hypothesis (ETH) fails, there is no algorithm solving the high-multiplicity Bin Packing problem in time (|I|²)^o(d). To prove this, we introduce a novel reduction from 3-SAT. The core of our construction is efficiently encoding all information from a 3-SAT instance with n variables into an ILP with O(log n) variables and constraints.
This result confirms that the Goemans and Rothvoss algorithm is essentially best-possible for Bin Packing parameterized by the number d of item sizes in the context of XP time algorithms.
Cite as
Klaus Jansen, Felix Ohnesorge, and Lis Pirotton. A Tight Double-Exponential Lower Bound for High-Multiplicity Bin Packing. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 116:1-116:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jansen_et_al:LIPIcs.ICALP.2026.116,
author = {Jansen, Klaus and Ohnesorge, Felix and Pirotton, Lis},
title = {{A Tight Double-Exponential Lower Bound for High-Multiplicity Bin Packing}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {116:1--116:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.116},
URN = {urn:nbn:de:0030-drops-265051},
doi = {10.4230/LIPIcs.ICALP.2026.116},
annote = {Keywords: Bin Packing, Lower Bound, Computational Complexity, ETH}
}
Document
Track A: Algorithms, Complexity and Games
A Quantum Time-Space Tradeoff for Directed st-Connectivity
Abstract
Directed st-connectivity (DSTCON) is the problem of deciding if there exists a directed path between a pair of distinguished vertices s and t in an input directed graph. This problem appears in many algorithmic applications, and is also a fundamental problem in complexity theory, due to its NL-completeness. We show that for any S ≥ log²(n), there is a quantum algorithm for dstcon using space S and time T ≤ 2^{1/2 log(n) log(n/S) + o(log²(n))}, which is an (up to quadratic) improvement over the best classical algorithm for any S = o(√n). Of the S total space used by our algorithm, only O(log²(n)) is quantum space - the rest is classical. This effectively means that we can trade off classical space for quantum time.
Cite as
Stacey Jeffery and Galina Pass. A Quantum Time-Space Tradeoff for Directed st-Connectivity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 117:1-117:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jeffery_et_al:LIPIcs.ICALP.2026.117,
author = {Jeffery, Stacey and Pass, Galina},
title = {{A Quantum Time-Space Tradeoff for Directed st-Connectivity}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {117:1--117:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.117},
URN = {urn:nbn:de:0030-drops-265063},
doi = {10.4230/LIPIcs.ICALP.2026.117},
annote = {Keywords: Quantum algorithms, time-space tradeoffs, directed st-connectivity, switching networks, space-bounded computation}
}
Document
Track A: Algorithms, Complexity and Games
The Compressed Oracle Is A Worthy (Multiplicative) Adversary
Abstract
The compressed oracle technique, introduced in the context of quantum cryptanalysis, is the latest method for proving quantum query lower bounds, and has had an impressive number of applications since its introduction, due in part to the ease of importing classical lower bound intuition into the quantum setting via this method. Previously, the main quantum query lower bound methods were the polynomial method, the adversary method, and the multiplicative adversary method, and their relative powers were well understood. In this work, we situate the compressed oracle technique within this established landscape, by showing that it is a special case of the multiplicative adversary method. To accomplish this, we introduce a simplified restriction of the multiplicative adversary method, the MLADV method, that remains powerful enough to capture the polynomial
method and exhibit a strong direct product theorem, but is much simpler to reason about. We show that the compressed oracle technique is also captured by the MLADV method. This might make the MLADV method a promising direction in the current quest to extend the compressed oracle technique to non-product distributions.
Cite as
Stacey Jeffery and Sebastian Zur. The Compressed Oracle Is A Worthy (Multiplicative) Adversary. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 118:1-118:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jeffery_et_al:LIPIcs.ICALP.2026.118,
author = {Jeffery, Stacey and Zur, Sebastian},
title = {{The Compressed Oracle Is A Worthy (Multiplicative) Adversary}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {118:1--118:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.118},
URN = {urn:nbn:de:0030-drops-265077},
doi = {10.4230/LIPIcs.ICALP.2026.118},
annote = {Keywords: Quantum query complexity, compressed oracle, adversary method}
}
Document
Track A: Algorithms, Complexity and Games
Deterministic Monotone Min-Plus Product and Convolution
Abstract
The Monotone Min-Plus Product problem is a useful primitive that has seen many algorithmic applications over the past decade. It also generalizes various other structured Min-Plus products studied in the literature, such as Bounded Difference Min-Plus Product and Bounded Integer Min-Plus Product. In this problem, we are given two n× n integer matrices A and B, where each row of B is a monotone non-decreasing sequence of integers from {1,…,n}, and the goal is to compute their Min-Plus product, defined as the n× n matrix C with C_{i,j} = min_k {A_{i,k} + B_{k,j}}. The fastest known algorithm for this task [Chi, Duan, Xie, and Zhang, STOC'22] runs in n^{(ω+3)/2 + o(1)} = 𝒪(n^2.686) time, significantly improving over the brute-force cubic algorithm. However, its main disadvantage is that it requires randomization, which is then inherited by all downstream applications.
Our main result is a deterministic algorithm for Monotone Min-Plus product with the same running time n^{(ω+3)/2 + o(1)} = 𝒪(n^2.686) as its randomized counterpart, improving upon the previous deterministic bound 𝒪(n^{2.875}) [Gu, Polak, Vassilevska Williams, and Xu, ICALP'21]. Our derandomization also applies to previously studied extensions and variants (e.g., [Dürr, IPL'23]), including rectangular matrices, bounded range [n^μ], and column-monotone matrices. As an immediate consequence, we derandomize state-of-the-art algorithms for multiple problems, including Language Edit Distance, RNA Folding, Optimum Stack Generation, unweighted Tree Edit Distance, Batched Range Mode, and Approximate All-Pairs Shortest Paths.
Our techniques also yield a deterministic algorithm for the Monotone Min-Plus Convolution problem that runs in n^{1.5 + o(1)} time, nearly matching the best-known randomized time complexity 𝒪̃(n^1.5) [Chi, Duan, Xie, and Zhang, STOC'22]. This algorithm can be used to derandomize state-of-the-art algorithms for Jumbled Indexing for binary strings and several variants of Knapsack.
Cite as
Ce Jin, Jaewoo Park, Barna Saha, and Yinzhan Xu. Deterministic Monotone Min-Plus Product and Convolution. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 119:1-119:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jin_et_al:LIPIcs.ICALP.2026.119,
author = {Jin, Ce and Park, Jaewoo and Saha, Barna and Xu, Yinzhan},
title = {{Deterministic Monotone Min-Plus Product and Convolution}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {119:1--119:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.119},
URN = {urn:nbn:de:0030-drops-265085},
doi = {10.4230/LIPIcs.ICALP.2026.119},
annote = {Keywords: Min-plus product, min-plus convolution, monotone matrices, fine-grained complexity, deterministic algorithms}
}
Document
Track A: Algorithms, Complexity and Games
Going Beyond Twin-Width? CSPs with Unbounded Domain and Few Variables
Abstract
We study connections between parameterized complexity, universal algebra, and structural graph parameters. Our starting point is the constraint satisfaction problem over instances with few variables but unbounded domain size (udCSP). Surprisingly, many upper and lower bounds in parameterized complexity can be expressed as solving such udCSPs. Prominent examples include the FPT algorithms for Boolean MinCSP [Eun Jung Kim et al., 2025], Directed Multicut with three cut requests [Meike Hatzel et al., 2023], and the canonical W[1]-hardness construction Paired Min Cut [Dániel Marx and Igor Razgon, 2009]. We represent constraints over unbounded domains by a set of unary maps ℳ into a finite base language Γ, situating udCSP(Γ, ℳ) in the algebraic terra incognita between finite and infinite domains. We present a novel algebraic theory that explains the parameterized complexity of problems such as Paired Min Cut, 𝓁-Chain Sat, and Coupled Min Cut, and unifies disparate FPT algorithms through the lens of twin-width. In particular, we simplify key steps in existing algorithms, e.g., for Boolean MinCSP, via a clean reduction to udCSP. We specifically concentrate on udCSP(Γ,ℳ) restricted to monotone maps Mo, where we identify the crucial connector polymorphism: its presence implies FPT for binary relations (via dynamic programming based on twin-width), while its absence entails W[1]-hardness. Extending this to higher-arity relations is related to the notoriously difficult task of finding a generalisation of twin-width to non-binary structures. As a step in this direction, inspired by our algebraic framework, we introduce a new structural parameter, projected grid-rank, and show that it coincides with the connector property, and agrees with twin-width for binary structures. More strongly, we show that for structures of bounded arity and bounded projected grid-rank, all binary projections have bounded twin-width. This width measure may thus be of independent interest for any problem currently hinging on generalizations of twin-width.
Cite as
Peter Jonsson, Victor Lagerkvist, Jorke M. de Vlas, and Magnus Wahlström. Going Beyond Twin-Width? CSPs with Unbounded Domain and Few Variables. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 120:1-120:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jonsson_et_al:LIPIcs.ICALP.2026.120,
author = {Jonsson, Peter and Lagerkvist, Victor and de Vlas, Jorke M. and Wahlstr\"{o}m, Magnus},
title = {{Going Beyond Twin-Width? CSPs with Unbounded Domain and Few Variables}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {120:1--120:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.120},
URN = {urn:nbn:de:0030-drops-265092},
doi = {10.4230/LIPIcs.ICALP.2026.120},
annote = {Keywords: Constraint satisfaction problems, parameterized complexity, twin-width, universal algebra}
}
Document
Track A: Algorithms, Complexity and Games
Faster Algorithms for (2k-1)-Stretch Distance Oracles
Abstract
The seminal distance oracles of Thorup and Zwick [STOC 2001, JACM 2005] provide optimal stretch/space tradeoffs. However, their O(mn^{1/k}) construction time is not optimal, and they posed the question of whether a faster construction time is possible (especially for small k). In this paper, we present the first improvement upon their construction algorithm in graphs that are not super sparse, i.e., when m = Ω(n^{1+1/k+ε}), for any ε > 0. Moreover, our construction improves upon the O(n²)-time construction of Baswana and Kavitha [FOCS 2006, SICOMP 2010], for every k > 2. By achieving the first subquadratic construction for 2 < k < 6, we resolve the open problem posed by Wulff-Nilsen [SODA 2012] of whether such subquadratic-time constructions exist.
Wulff-Nilsen [SODA 2012] targeted nearly linear construction times and presented algorithms running in Õ(m + n^{1+f(k)}) time, which is near-linear whenever the graph density m exceeds the threshold n^{1+f(k)}. We obtain improved bounds on f(k) for all k > 3, and thus expand the regime of graph densities for which nearly linear construction times are achievable.
In addition, for unweighted graphs, we present several new algorithms for constructing (2k - 1,β)-oracles that improve upon the results of Baswana, Gaur, Sen, and Upadhyay [ICALP 2008].
Our results are achieved through the development of several new algorithmic tools, which may be of independent interest. One of our main technical contributions is a hierarchy of parameterized distance oracles, which plays a central role in our fast construction algorithms.
Cite as
Avi Kadria and Liam Roditty. Faster Algorithms for (2k-1)-Stretch Distance Oracles. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 121:1-121:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kadria_et_al:LIPIcs.ICALP.2026.121,
author = {Kadria, Avi and Roditty, Liam},
title = {{Faster Algorithms for (2k-1)-Stretch Distance Oracles}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {121:1--121:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.121},
URN = {urn:nbn:de:0030-drops-265105},
doi = {10.4230/LIPIcs.ICALP.2026.121},
annote = {Keywords: Fine-grained complexity, Graph algorithms, shortest cycle, girth approximations}
}
Document
Track A: Algorithms, Complexity and Games
Hardness, Tractability and Density Thresholds of Finite Pinwheel Scheduling Variants
Abstract
The k-Visits problem is a recently introduced finite version of Pinwheel Scheduling [Kanellopoulos et al., SODA 2026]. Given the deadlines of n tasks, the problem asks whether there exists a schedule of length kn executing each task exactly k times, with no deadline expiring between consecutive visits (executions) of each task. In this work we prove that 2-Visits is strongly NP-complete even when the maximum multiplicity of the input is equal to 2, settling an open question from [Kanellopoulos et al., 2026] and contrasting the tractability of 2-Visits for simple sets. On the other hand, we prove that 2-Visits is in RP when the number of distinct deadlines is constant, thus making progress on another open question regarding the parameterization of 2-Visits by the number of numbers. We then generalize all existing positive results for 2-Visits to a version of the problem where some tasks must be visited once and some other tasks twice, while providing evidence that some of these results are unlikely to transfer to 3-Visits. Lastly, we establish bounds for the density thresholds of k-Visits, analogous to the (5/6)-threshold of Pinwheel Scheduling [Kawamura, STOC 2024]; in particular, we show a √2-1/2≈ 0.9142 lower bound for the density threshold of 2-Visits and prove that the density threshold of k-Visits approaches 5/6≈ 0.8333 for k → ∞.
Cite as
Sotiris Kanellopoulos, Giorgos Mitropoulos, Christos Pergaminelis, and Thanos Tolias. Hardness, Tractability and Density Thresholds of Finite Pinwheel Scheduling Variants. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 122:1-122:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kanellopoulos_et_al:LIPIcs.ICALP.2026.122,
author = {Kanellopoulos, Sotiris and Mitropoulos, Giorgos and Pergaminelis, Christos and Tolias, Thanos},
title = {{Hardness, Tractability and Density Thresholds of Finite Pinwheel Scheduling Variants}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {122:1--122:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.122},
URN = {urn:nbn:de:0030-drops-265115},
doi = {10.4230/LIPIcs.ICALP.2026.122},
annote = {Keywords: Pinwheel Scheduling, Perpetual Scheduling, NP-Completeness, Parameterized Complexity}
}
Document
Track A: Algorithms, Complexity and Games
How Hard Is It to Verify a Classical Shadow?
Abstract
Classical shadows are succinct classical representations of quantum states which allow one to encode a set of properties P of a quantum state ρ, while only requiring measurements on logarithmically many copies of ρ in the size of P. In this work, we initiate the study of verification of classical shadows, denoted classical shadow validity (CSV), from the perspective of computational complexity, which asks: Given a classical shadow S, how hard is it to verify that S predicts the measurement statistics of a quantum state? We first show that even for the elegantly simple classical shadow protocol of [Huang, Kueng, Preskill, Nature Physics 2020] utilizing local Clifford measurements, CSV is QMA-complete. This hardness continues to hold for the high-dimensional extension of said protocol due to [Mao, Yi, and Zhu, PRL 2025]. In contrast, we show that for the HKP and MYZ protocols utilizing global Clifford measurements, CSV can be "dequantized" for low-Frobenius norm observables, i.e., solved in randomized poly-time with standard sampling assumptions. Finally, we show that CSV for exponentially many observables is complete for a quantum generalization of the second level of the polynomial hierarchy, yielding the first natural complete problem for such a class.
Cite as
Georgios Karaiskos, Dorian Rudolph, Johannes Jakob Meyer, Jens Eisert, and Sevag Gharibian. How Hard Is It to Verify a Classical Shadow?. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 123:1-123:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{karaiskos_et_al:LIPIcs.ICALP.2026.123,
author = {Karaiskos, Georgios and Rudolph, Dorian and Meyer, Johannes Jakob and Eisert, Jens and Gharibian, Sevag},
title = {{How Hard Is It to Verify a Classical Shadow?}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {123:1--123:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.123},
URN = {urn:nbn:de:0030-drops-265121},
doi = {10.4230/LIPIcs.ICALP.2026.123},
annote = {Keywords: classical shadows, quantum complexity theory, QMA, quantum polynomial hierarchy}
}
Document
Track A: Algorithms, Complexity and Games
An Õ(n^{3/7}) Round Parallel Algorithm for Matroid Bases
Abstract
We study the parallel (adaptive) complexity of the classic problem of finding a basis in an n-element matroid, given access via an independence oracle. In this model, the algorithm may submit polynomially many independence queries in each round, and the central question is: how many rounds are necessary and sufficient to find a basis?
Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988; hereafter KUW) initiated this study, showing that O(√n) adaptive rounds suffice for any matroid, and that Ω̃(n^{1/3}) rounds are necessary even for partition matroids. This left a substantial gap that persisted for nearly four decades, until Khanna, Putterman, and Song (FOCS 2025; hereafter KPS) achieved Õ(n^{7/15}) rounds, the first improvement since KUW.
In this work, we make another conceptual advance beyond KPS, giving a new algorithm that finds a matroid basis in Õ(n^{3/7}) rounds. We develop a structural and algorithmic framework that brings a new lens to the analysis of random circuits, moving from reasoning about individual elements to understanding how dependencies span multiple elements simultaneously. Specifically, our framework introduces three new ideas:
1) A new subset-based decomposition that provides precise guarantees on how random circuits intersect groups of elements, yet remains computable in few adaptive rounds.
2) A new method for identifying and removing redundant elements in bulk, based on short circuit witnesses that certify redundancy across large portions of the matroid.
3) An adaptive early-stopping strategy that uses the evolving structure of the matroid to decide when to contract or delete, preventing wasted rounds.
Each of these contributions, in isolation, already yields meaningful improvements over the round complexity achieved in KPS; their combination enables our main result of Õ(n^{3/7}) rounds.
As further consequences, incorporating our improved basis-finding algorithm into known reductions yields an Õ(n^{17/21})-round parallel algorithm for matroid intersection, as well as an Õ(n^{3/7})-round parallel algorithm for approximate monotone submodular maximization under a matroid constraint.
Cite as
Sanjeev Khanna, Aaron Putterman, and Junkai Song. An Õ(n^{3/7}) Round Parallel Algorithm for Matroid Bases. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 124:1-124:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{khanna_et_al:LIPIcs.ICALP.2026.124,
author = {Khanna, Sanjeev and Putterman, Aaron and Song, Junkai},
title = {{An Õ(n^\{3/7\}) Round Parallel Algorithm for Matroid Bases}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {124:1--124:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.124},
URN = {urn:nbn:de:0030-drops-265130},
doi = {10.4230/LIPIcs.ICALP.2026.124},
annote = {Keywords: parallel algorithms, matroids}
}
Document
Track A: Algorithms, Complexity and Games
Optimal Parallel Basis Finding in Graphic and Related Matroids
Abstract
We study the parallel complexity of finding a basis of a graphic matroid under independence-oracle access. Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988) initiated the study of this problem and established two algorithms for finding a spanning forest: one running in O(log m) rounds with m^{Θ(log m)} queries, and another, for any d ∈ ℤ^+, running in O(m^{2/d}) rounds with Θ(m^d) queries. A key open question they posed was whether one could simultaneously achieve polylogarithmic rounds and polynomially many queries.
We give a deterministic algorithm that uses O(log m) adaptive rounds and poly(m) non-adaptive queries per round to return a spanning forest on m edges, and complement this result with a matching Ω(log m) lower bound for any (even randomized) algorithm with poly(m) queries per round. Thus, the adaptive round complexity for graphic matroids is characterized exactly, settling this long-standing problem.
Beyond graphs, we show that our framework also yields an O(log m)-round, poly(m)-query algorithm for any binary matroid satisfying a smooth circuit counting property, implying, among others, an optimal O(log m)-round parallel algorithms for finding bases of cographic matroids. Finally, we conjecture a natural strengthening of known circuit-counting bounds for the much broader class of regular matroids and even an extension to so-called max-flow min-cut matroids; assuming it, our algorithm achieves the same O(log m) rounds and poly(m) queries for all such matroids - which includes graphic and cographic matroids as special cases.
Cite as
Sanjeev Khanna, Aaron Putterman, and Junkai Song. Optimal Parallel Basis Finding in Graphic and Related Matroids. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 125:1-125:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{khanna_et_al:LIPIcs.ICALP.2026.125,
author = {Khanna, Sanjeev and Putterman, Aaron and Song, Junkai},
title = {{Optimal Parallel Basis Finding in Graphic and Related Matroids}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {125:1--125:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.125},
URN = {urn:nbn:de:0030-drops-265143},
doi = {10.4230/LIPIcs.ICALP.2026.125},
annote = {Keywords: parallel algorithms, matroids}
}
Document
Track A: Algorithms, Complexity and Games
Preprocessed 3SUM for Unknown Universes with Subquadratic Space
Abstract
We consider the classic 3SUM problem: given sets of integers A, B, C, determine whether there is a tuple (a, b, c) ∈ A × B × C satisfying a + b = c. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets A, B of size n, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form A', B', C', where A' ⊆ A and B' ⊆ B. A series of results have achieved Õ(n²) preprocessing time, Õ(n²) space, and query time improving progressively from Õ(n^{1.9}) [Timothy M. Chan and Moshe Lewenstein, 2015] to Õ(n^{11/6}) [Timothy M. Chan et al., 2023] to Õ(n^{1.5}) [Kasliwal et al., 2025]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes?
We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving Õ(n^{1.5 + ε}) query time and Õ(n^{2 - 2ε/3}) space complexity. Furthermore, our algorithm has Õ(n²) preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least n^{2-o(1)} under the 3SUM Hypothesis.
Cite as
Yael Kirkpatrick, John Kuszmaul, Surya Mathialagan, and Virginia Vassilevska Williams. Preprocessed 3SUM for Unknown Universes with Subquadratic Space. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 126:1-126:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kirkpatrick_et_al:LIPIcs.ICALP.2026.126,
author = {Kirkpatrick, Yael and Kuszmaul, John and Mathialagan, Surya and Vassilevska Williams, Virginia},
title = {{Preprocessed 3SUM for Unknown Universes with Subquadratic Space}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {126:1--126:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.126},
URN = {urn:nbn:de:0030-drops-265158},
doi = {10.4230/LIPIcs.ICALP.2026.126},
annote = {Keywords: Graph Algorithms, Diameter, Distance Oracle, Approximation Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
New Diameter Approximations via Distance Oracle Techniques
Abstract
Computing the diameter of a graph is a problem of great interest both in general algorithms research and specifically within fine-grained complexity, where it is a cornerstone hard problem. As computing the exact diameter in m-edge graphs requires m^{2-o(1)} time under the Strong Exponential Time Hypothesis, much work has gone into approximating this parameter. Recent work has achieved a full conditional lower bound tradeoff curve for both directed and undirected graphs [Dalirrooyfard, Li and Vassilevska W., FOCS'21]. However, the best known upper bounds do not match the lower bounds. In particular, the best known approximation scheme for undirected graph diameter [Cairo-Grossi-Rizzi, SODA 2016] has not been improved. Moreover, this scheme is randomized and no similar deterministic scheme is known.
Another fundamental field of research in shortest paths computation is the construction of approximate distance oracles. Thorup and Zwick [JACM'05] provided the first such distance oracle with constant query time and (conditionally) optimal space, and in the years since many advances have led to a vast toolbox of techniques and data structures.
These two areas of research seem natural to combine since they both concern approximating shortest paths. However, the known diameter approximation algorithms only use a small subset of the techniques used in distance oracles research. In this work we show that in fact approximate diameter and distance oracles are intricately connected.
We first demonstrate a strong connection between the current best known diameter approximation scheme of Cairo, Grossi and Rizzi ("CGR") and the (2k-1)-approximate distance oracle of Thorup and Zwick. This allows us to derandomize the CGR algorithm and obtain the first deterministic diameter approximation tradeoff.
We further derandomize other central techniques in the field of distance oracles and use them to achieve new deterministic diameter approximation algorithms, including a simpler 3/2-approximation with no additive error and a new 5/3-approximation, the first new step in the diameter approximation tradeoff in almost a decade. Finally, we show how these new techniques can be used to derandomize many current best known results in various fields of shortest paths approximations.
Cite as
Yael Kirkpatrick, Liam Roditty, Richard Qi, and Virginia Vassilevska Williams. New Diameter Approximations via Distance Oracle Techniques. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 127:1-127:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kirkpatrick_et_al:LIPIcs.ICALP.2026.127,
author = {Kirkpatrick, Yael and Roditty, Liam and Qi, Richard and Vassilevska Williams, Virginia},
title = {{New Diameter Approximations via Distance Oracle Techniques}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {127:1--127:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.127},
URN = {urn:nbn:de:0030-drops-265169},
doi = {10.4230/LIPIcs.ICALP.2026.127},
annote = {Keywords: Graph Algorithms, Diameter, Distance Oracle, Approximation Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Online Preemptive Matching Revisited
Abstract
We study the online preemptive matching problem, in which the edges of a graph arrive sequentially and the algorithm must maintain a matching by accepting or rejecting arriving edges and possibly discarding previously accepted ones. We prove a new upper bound of 0.5661 on the competitive ratio achievable for the problem. This bound applies to arbitrary randomized algorithms, bipartite graphs and if we allow the algorithm to output a fractional solution.
Our result improves upon the strongest previously known upper bound of 2-√2 ≈ 0.585, due to Huang et al. [SODA'19]. Previous hardness constructions relied on edge sequences described by vertex arrivals where each arriving vertex reveals its edges to yet unvaried vertices. Under such sequences, Huang et al. showed that there exists a non-preemptive online algorithm with competitive ratio ∼0.567 (or 2-√2 for fractional solutions). Consequently, our hardness construction is the first result which shows hardness for instances where the optimal algorithm employs preemption.
Cite as
Peter Kiss and Mohammad Sharifi. Online Preemptive Matching Revisited. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 128:1-128:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kiss_et_al:LIPIcs.ICALP.2026.128,
author = {Kiss, Peter and Sharifi, Mohammad},
title = {{Online Preemptive Matching Revisited}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {128:1--128:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.128},
URN = {urn:nbn:de:0030-drops-265172},
doi = {10.4230/LIPIcs.ICALP.2026.128},
annote = {Keywords: Online Algorithms, Preemptive Algorithms, Approximate Maximum Matching}
}
Document
Track A: Algorithms, Complexity and Games
Thin Trees for near Minimum Cuts
Abstract
The strong thin tree conjecture states that every k-edge-connected graph G contains an O(1/k)-thin spanning tree, meaning a spanning tree which contains at most an O(1/k) fraction of the edges across each cut in G. This conjecture is still open despite significant effort; the best current result by Anari and Oveis Gharan shows the existence of an O(polylog log n/k)-thin tree.
In this work, we demonstrate that the conjecture is true if one only requires thinness for the set of η-near minimum cuts of the graph for η = 1/40, in other words, for the set of cuts with fewer than (1+1/40)k edges. Our approach constructs such a tree in polynomial time. To show this, we utilize the structure of near minimum cuts, and in particular the polygon representation of Benczúr and Goemans, to reduce to the previously solved problem of finding a spanning tree that is O(1/k)-thin for all sets in a laminar family.
Cite as
Nathan Klein, Neil Olver, and Zi Song Yeoh. Thin Trees for near Minimum Cuts. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 129:1-129:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{klein_et_al:LIPIcs.ICALP.2026.129,
author = {Klein, Nathan and Olver, Neil and Yeoh, Zi Song},
title = {{Thin Trees for near Minimum Cuts}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {129:1--129:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.129},
URN = {urn:nbn:de:0030-drops-265180},
doi = {10.4230/LIPIcs.ICALP.2026.129},
annote = {Keywords: Graph Theory, Thin Trees, Polygon Representation, Near Minimum Cuts}
}
Document
Track A: Algorithms, Complexity and Games
Lower Bounds on Pure Dynamic Programming for Connectivity Problems on Graphs of Bounded Path-Width
Abstract
We give unconditional parameterized complexity lower bounds on pure dynamic programming algorithms - as modeled by tropical circuits - for connectivity problems such as the Traveling Salesperson Problem. Our lower bounds are higher than the currently fastest algorithms that rely on algebra and give evidence that these algebraic aspects are unavoidable for competitive worst case running times.
Specifically, we study input graphs with a small width parameter such as treewidth and pathwidth and show that for any k there exists a graph G of pathwidth at most k and k^𝒪(1) vertices such that any tropical circuit calculating the optimal value of a Traveling Salesperson round tour uses at least 2^Ω(k log log k) gates. We establish this result by linking tropical circuit complexity to the nondeterministic communication complexity of specific compatibility matrices. These matrices encode whether two partial solutions combine into a full solution, and Raz and Spieker [Combinatorica 1995] previously proved a lower bound for this complexity measure.
Cite as
Kacper Kluk and Jesper Nederlof. Lower Bounds on Pure Dynamic Programming for Connectivity Problems on Graphs of Bounded Path-Width. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 130:1-130:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kluk_et_al:LIPIcs.ICALP.2026.130,
author = {Kluk, Kacper and Nederlof, Jesper},
title = {{Lower Bounds on Pure Dynamic Programming for Connectivity Problems on Graphs of Bounded Path-Width}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {130:1--130:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.130},
URN = {urn:nbn:de:0030-drops-265196},
doi = {10.4230/LIPIcs.ICALP.2026.130},
annote = {Keywords: Parameterized Complexity, treewidth, tropical circuits, connectivity problems}
}
Document
Track A: Algorithms, Complexity and Games
Persistence Meets Resistance: Doubling down on Hardness
Abstract
We present results on the approximate computation of stable invariants for filtrations of finite metric spaces in the context of persistent homology. We establish novel approximation algorithms in the setting of n-point metric spaces where the growth of the doubling dimension is in o(log n) and the diameter is bounded. In the 1-parameter case, by revisiting known techniques (greedy permutations) in a new way, we derive the first linear-time algorithms for the problem of computing additive ε-approximations of any stable barcode. By deriving bounds on the convergence rate and the approximation quality of uniform samples, we extend the approach to selected multiparameter filtrations. We show that for normalized measure bifiltrations, including the multicover and subdivision-Rips bifiltration, any stable invariant can be probabilistically approximated in time constant in n. The constants in the running times of our algorithms depend on the doubling dimension, the diameter and the success probability.
We further study the problem through the lens of fine-grained complexity and show that computing the rank of a matrix reduces to that of approximating the barcode of the Vietoris-Rips or Čech filtration. We present two variants of the reduction, one for sufficiently good additive approximations and the other for any constant factor multiplicative approximations.
Cite as
Benedikt Kolbe and Tim Mayr. Persistence Meets Resistance: Doubling down on Hardness. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 131:1-131:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kolbe_et_al:LIPIcs.ICALP.2026.131,
author = {Kolbe, Benedikt and Mayr, Tim},
title = {{Persistence Meets Resistance: Doubling down on Hardness}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {131:1--131:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.131},
URN = {urn:nbn:de:0030-drops-265209},
doi = {10.4230/LIPIcs.ICALP.2026.131},
annote = {Keywords: Persistent homology, approximations, lower bounds, matrix rank, doubling dimension, Vietoris-Rips complex, \v{C}ech complex, measure bifiltration, subdivision-Rips bifiltration, Rhomboid filtration}
}
Document
Track A: Algorithms, Complexity and Games
VP, VNP and Algebraic Branching Programs over Min-Plus Semirings
Abstract
Arithmetic circuit complexity studies the complexity of computing polynomials using only arithmetic operations such as addition, multiplication, subtraction, and division. Polynomials over rings of integers model counting problems. Similarly, polynomials over semirings such as tropical semirings model optimization problems. Circuits over semirings then model so called pure algorithms, algorithms that only use the operations in the semiring.
In this paper, we do a complexity-theoretic study of the power and limitations of circuits (which represent dynamic programs) over semirings:
- We define VNP over min-plus semirings, which can faithfully represent problems such as computing min-weight perfect matchings and min-weight Hamiltonian cycles where we have efficiently verifiable certificates. Unlike over rings, we complement the values in the certificate for free as complementation is impossible over min-plus semirings. We prove a dichotomy theorem that states that if we only complement logarithmically many values, this class is same as VP over min-plus semirings. If we complement super-logarithmically many values, then VNP ≠ VP.
- We consider constant-width ABPs (which are also called incremental dynamic programs that are restricted to use only a constant number of registers) and show that even simple problems like computing the min-weight 2-edge-matching is impossible with width 2 (or 2 registers). However, with width 3 (or 3 registers), such programs can compute everything. More generally, we show that constant-depth formulas are efficiently simulated by constant-width ABPs.
- We show that an exponential hypercube sum (min in the semiring) over even provably weak models such as width-2 ABPs and products of linear forms are the same as VNP.
Cite as
Balagopal Komarath, Harshil Mittal, and Jayalal Sarma. VP, VNP and Algebraic Branching Programs over Min-Plus Semirings. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 132:1-132:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{komarath_et_al:LIPIcs.ICALP.2026.132,
author = {Komarath, Balagopal and Mittal, Harshil and Sarma, Jayalal},
title = {{VP, VNP and Algebraic Branching Programs over Min-Plus Semirings}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {132:1--132:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.132},
URN = {urn:nbn:de:0030-drops-265217},
doi = {10.4230/LIPIcs.ICALP.2026.132},
annote = {Keywords: Min-plus semirings, Pure dynamic programming, Incremental dynamic programming, Registers, Algebraic branching programs, ABP width, Circuit depth, Algebraic formulas}
}
Document
Track A: Algorithms, Complexity and Games
Partially-Dynamic Maximum Flow in Dense Graphs
Abstract
We give the first algorithms that, with high probability, maintain (1-ε)-approximate s-t maximum flow in an n-vertex undirected, capacitated graph undergoing either only edge insertions or only edge deletions in total update time Õ_ε(n²). For dense graphs, this yields polylogarithmic amortized update time, which was previously only obtained for the special case of uncapacitated graphs undergoing edge insertions.
We develop the following two algorithms:
- For graphs undergoing deletions, we generalize the congestion-balancing framework from [Aaron Bernstein et al., 2020], which was developed for maximum matching. We then show that this framework can be simulated on cut sparsifiers, which yields significant speed-ups.
- For graphs undergoing insertions, we show that the sparsification techniques by Eppstein et al. [Eppstein et al., 1997] can be combined more directly with the techniques from Henzinger and Goranci [Goranci and Henzinger, 2023]. We thereby bypass the need to dynamize the more involved residual graph sparsification approach by Levin and Karger [Karger and Levine, 2015] suggested in [Goranci et al., 2025], and extend their result to capacitated graphs.
Cite as
Egor Kravchenko and Maximilian Probst Gutenberg. Partially-Dynamic Maximum Flow in Dense Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 133:1-133:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kravchenko_et_al:LIPIcs.ICALP.2026.133,
author = {Kravchenko, Egor and Probst Gutenberg, Maximilian},
title = {{Partially-Dynamic Maximum Flow in Dense Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {133:1--133:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.133},
URN = {urn:nbn:de:0030-drops-265226},
doi = {10.4230/LIPIcs.ICALP.2026.133},
annote = {Keywords: Maximum Flow, Dynamic Graph Algorithm, Data Structure}
}
Document
Track A: Algorithms, Complexity and Games
Sampling Colorings with Fixed Color Class Sizes
Abstract
In 1970, Hajnal and Szemerédi proved a conjecture of Erdős stating that any graph with maximum degree Δ admits an equitable (Δ+1)-coloring, that is, a coloring where color class sizes differ by at most 1. In 2007 Kierstead and Kostochka reproved their result and provided a polynomial-time algorithm which produces such a coloring. In this paper we study the problem of approximately sampling uniformly random equitable colorings. A series of works gives polynomial-time sampling algorithms for colorings without the color class constraint, the latest improvement being by Carlson and Vigoda for q ≥ 1.809 Δ. In this paper we give a polynomial-time sampling algorithm for equitable colorings when q > 2Δ. Moreover, our results extend to colorings with small deviations from equitable (and as a corollary, establishing their existence). The proof uses the framework of the geometry of polynomials for multivariate polynomials, and as a consequence establishes a multivariate local Central Limit Theorem for color class sizes of uniform random colorings.
Cite as
Aiya Kuchukova, Will Perkins, and Xavier Povill. Sampling Colorings with Fixed Color Class Sizes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 134:1-134:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kuchukova_et_al:LIPIcs.ICALP.2026.134,
author = {Kuchukova, Aiya and Perkins, Will and Povill, Xavier},
title = {{Sampling Colorings with Fixed Color Class Sizes}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {134:1--134:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.134},
URN = {urn:nbn:de:0030-drops-265231},
doi = {10.4230/LIPIcs.ICALP.2026.134},
annote = {Keywords: sampling, approximate counting, graph coloring, zero-freeness, Potts model, LCLT}
}
Document
Track A: Algorithms, Complexity and Games
On the Complexity of the Matching Problem of Regular Expressions with Backreferences
Abstract
Regular Expression Denial of Service (ReDoS) is a well-known type of algorithmic complexity attack, where an adversary supplies maliciously crafted strings to a regular expression matching engine, aiming to exhaust computational resources of systems. Even quadratic-time behavior in matching engines has been exploited in successful attacks, as exemplified by major outages at Stack Overflow (2016) and Cloudflare (2019). These incidents motivate a fundamental question: Is it possible to construct matching engines that run in linear or near-linear time in the length of the input string? For classical regular expressions (REGEX), Thompson’s construction yields a linear-time algorithm for fixed expressions. However, practical engines support powerful features such as backreferences, which allow capturing a substring and reusing it later. This feature strictly extends the expressive power of REGEX but unfortunately increases the risk of ReDoS attacks.
This paper investigates the fine-grained complexity of the string matching problem for regular expressions with backreferences (REWBs). Specifically, we consider r-use k-REWBs, i.e., REWBs with k variables such that, in any computation, the total number of backreference executions is at most r. On the hardness side, we show that the string matching problem for k-REWBs cannot be solved in O(n^{2k-ε}) time for any ε > 0 under the Strong Exponential Time Hypothesis (SETH), where n is the length of the input string. We also prove that this problem is W[2]-hard when parameterized by the length of the REWB expression, strengthening the previous W[1]-hardness result. Moreover, we prove that this problem for 2-use 2-REWBs cannot be solved in n^{1+o(1)} time unless the triangle detection problem can be solved in that time. On the algorithmic side, we present an O(n log² n)-time algorithm for 1-use REWBs. In particular, we focus on the ABCBD problem, which is the REWB matching problem for the form A(B)_xC∖xD where A, B, C, and D are fixed REGEXes. We also show that every 1-use REWB can be transformed into this canonical form. Our algorithm significantly improves upon the recent O(n²)-time algorithm for the ABCBD problem by Nogami and Terauchi (MFCS, 2025). Our algorithm is highly nontrivial and employs several techniques, including suffix trees, transition monoids of REGEXes, factorization forest data structures, and periodicity of strings.
Cite as
Soh Kumabe and Yuya Uezato. On the Complexity of the Matching Problem of Regular Expressions with Backreferences. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 135:1-135:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kumabe_et_al:LIPIcs.ICALP.2026.135,
author = {Kumabe, Soh and Uezato, Yuya},
title = {{On the Complexity of the Matching Problem of Regular Expressions with Backreferences}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {135:1--135:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.135},
URN = {urn:nbn:de:0030-drops-265241},
doi = {10.4230/LIPIcs.ICALP.2026.135},
annote = {Keywords: Pattern Matching, Regular Expression, Backreference, Fine-grained Complexity}
}
Document
Track A: Algorithms, Complexity and Games
Back in the Saddle: Toward Parallel Approximate Minimum-Cost Flow
Abstract
We present the first polylog-depth, nearly-linear-work parallel algorithm that achieves a (1+ε)-bicriteria approximation guarantee for undirected minimum-cost flow on expanders. Fix an undirected graph G = (V,E) with unit capacities, unit lengths, and conductance ϕ. For any feasible demand vector d and any ε ∈ (0,1) we compute, in Õ(|E|/(εϕ)) work and Õ(1/(εϕ)) depth, a flow f that routes d exactly while satisfying ‖f‖_∞ ≤ 1+ε and ‖f‖_1 ≤ (1+ε)min_{Bg = d, ‖g‖_∞ ≤ 1}}‖g‖_1.
This bicriteria guarantee simultaneously controls congestion and total cost, strengthens the previously studied notion of throughput error, and matches the best known ε-dependence for parallel maximum flow/transshipment on general graphs.
Our main contribution is a new saddle-point optimization method for mixed 𝓁_∞-𝓁_1 optimization. Concretely, we (i) formulate a two-term regression capturing minimum-cost flow as a saddle-point problem that couples 𝓁_∞ and 𝓁_1 terms, (ii) construct a small-magnitude area-convex regularizer tailored to the resulting primal–dual domain (building on Sherman’s area-convexity framework [Sherman, 2017]), and (iii) implement efficient δ-approximate maximization/minimization oracles so that Sherman’s extragradient iteration yields low iteration-count convergence.
Beyond the concrete expander result, our mixed 𝓁_∞-𝓁_1 optimization toolkit appears broadly applicable and suggests a promising route toward Õ(m/ε) work and Õ(1/ε) depth algorithms for approximate undirected minimum-cost flow on general graphs.
Cite as
Rasmus Kyng and Aurelio L. Sulser. Back in the Saddle: Toward Parallel Approximate Minimum-Cost Flow. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 136:1-136:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kyng_et_al:LIPIcs.ICALP.2026.136,
author = {Kyng, Rasmus and Sulser, Aurelio L.},
title = {{Back in the Saddle: Toward Parallel Approximate Minimum-Cost Flow}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {136:1--136:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.136},
URN = {urn:nbn:de:0030-drops-265255},
doi = {10.4230/LIPIcs.ICALP.2026.136},
annote = {Keywords: Approximate Min-Cost-Flow, Parallel, Area-Convexity, Expanders}
}
Document
Track A: Algorithms, Complexity and Games
Determining the Outerthickness of Graphs Is NP-Hard
Abstract
We give a short, self-contained, and easily verifiable proof that determining the outerthickness of a general graph is NP-hard. This resolves a long-standing open problem on the computational complexity of outerthickness.
Moreover, our hardness result applies to a more general covering problem P_{ℱ, k}, defined as follows. Let ℱ be a proper graph class. Let k ≥ 1 be an integer parameter. Given an undirected simple graph G = (V, E), the task is to cover the edge set E(G) by at most k subsets E₁,…,E_k such that each subgraph (V(G),E_i) for i ∈ [k] belongs to ℱ. Note that if ℱ is monotone (in particular, when ℱ is the class of all outerplanar graphs), any such cover can be converted into an edge partition by deleting overlaps; hence, in this case, covering and partitioning are equivalent.
Our result shows that for every proper graph class ℱ that satisfies all of the following conditions: (a) ℱ is closed under topological minors, (b) ℱ is closed under 1-sums, and (c) ℱ contains a cycle of length 3, the problem P_{ℱ, k} is NP-hard for every integer k ≥ 3. In particular:
- For ℱ equal to the class of all outerplanar graphs, our result settles the long-standing open problem on the complexity of determining outerthickness.
- For ℱ equal to the class of all planar graphs, our result complements Mansfield’s NP-hardness result (1983) for the thickness, which applies only to the case k = 2.
It is also worth noting that each of the three conditions above is necessary. If ℱ is the class of all eulerian graphs, then condition (a) fails. If ℱ is the class of all pseudoforests, then condition (b) fails. If ℱ is the class of all forests, then condition (c) fails. For each of these three classes ℱ, the problem P_{ℱ, k} is solvable in polynomial time for every integer k ≥ 3, showing that none of the three conditions can be dropped unless P = NP.
Cite as
Pin-Hsian Lee, Te-Cheng Liu, and Meng-Tsung Tsai. Determining the Outerthickness of Graphs Is NP-Hard. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 137:1-137:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lee_et_al:LIPIcs.ICALP.2026.137,
author = {Lee, Pin-Hsian and Liu, Te-Cheng and Tsai, Meng-Tsung},
title = {{Determining the Outerthickness of Graphs Is NP-Hard}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {137:1--137:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.137},
URN = {urn:nbn:de:0030-drops-265265},
doi = {10.4230/LIPIcs.ICALP.2026.137},
annote = {Keywords: outerthickness, outerplanar graphs, edge partition}
}
Document
Track A: Algorithms, Complexity and Games
Counting Perfect Matchings and Hamiltonian Cycles Faster
Abstract
We show that the hafnian of a symmetric 2n× 2n matrix of poly(n)-bit integers (which counts the number of perfect matchings of a 2n-vertex graph) and the number of Hamiltonian cycles of an n-vertex directed graph can be computed in time 2^{n-Ω(√n)}, improving and generalizing an earlier algorithm of Björklund, Kaski, and Williams (Algorithmica 2019) that runs in time 2^{n - Ω(√{n/log log n})}.
A key tool of our approach is the design of a data structure that supports fast evaluation of high-order derivatives of hafnian and Hamiltonian cycles, which integrates with the new approach on multivariate multipoint evaluation by Bhargava, Ghosh, Guo, Kumar, and Umans (FOCS 2022, JACM 2024).
Cite as
Baitian Li. Counting Perfect Matchings and Hamiltonian Cycles Faster. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 138:1-138:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{li:LIPIcs.ICALP.2026.138,
author = {Li, Baitian},
title = {{Counting Perfect Matchings and Hamiltonian Cycles Faster}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {138:1--138:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.138},
URN = {urn:nbn:de:0030-drops-265278},
doi = {10.4230/LIPIcs.ICALP.2026.138},
annote = {Keywords: permanent, hafnian, Hamiltonian cycle, Kakeya sets}
}
Document
Track A: Algorithms, Complexity and Games
Connectivity Oracle Under Vertex Failures by Shortcutting Unbreakable Decomposition
Abstract
We study connectivity oracle under vertex failures, one of the most fundamental graph data structures with many applications. We provide a new deterministic connectivity oracle that handles update in O(k⁶) time and answers query in O(k) time, while using 2^O(k²) n + O(k²n α_c(n)) space and k^O(k²) n + O(m + k³ n log² n + k⁶n log n) preprocessing time.
Although some previous works achieve k² ⋅ n^o(1) update time and O(k) query time [Long and Saranurak, 2022; Yaowei Long and Yunfan Wang, 2024], the update time is still n-dependent, while oracles that have n-independent update and query times [Michal Pilipczuk et al., 2022; Jan van den Brand and Thatchaphol Saranurak, 2019] cannot achieve optimal O(k) query time [Monika Henzinger et al., 2015] and often have Ω(n²) space and processing time. Our solution would be the first vertex-failure connectivity oracle that achieves O(k) query time with update time completely independent of n, while improving space usage and having competitive preprocessing time.
Cite as
Xizhe Li, Yaowei Long, David Pidugu, Thatchaphol Saranurak, and Benyu Wang. Connectivity Oracle Under Vertex Failures by Shortcutting Unbreakable Decomposition. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 139:1-139:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{li_et_al:LIPIcs.ICALP.2026.139,
author = {Li, Xizhe and Long, Yaowei and Pidugu, David and Saranurak, Thatchaphol and Wang, Benyu},
title = {{Connectivity Oracle Under Vertex Failures by Shortcutting Unbreakable Decomposition}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {139:1--139:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.139},
URN = {urn:nbn:de:0030-drops-265285},
doi = {10.4230/LIPIcs.ICALP.2026.139},
annote = {Keywords: Graphs, Fault tolerant, Oracles}
}
Document
Track A: Algorithms, Complexity and Games
Streaming Complexity Separations for Dense and Sparse Graphs
Abstract
We identify a sharp separation in the streaming space complexity of Maximum Cut when the algorithm must output an approximate cut (rather than only the approximate value). For dense graphs, we show that O(n/ε²) space is sufficient and that Ω(n) space is necessary. In contrast, for graphs with Θ(n/ε²) edges, the situation is markedly different: we show that the problem requires Ω(n log(ε² n)/ε²) space for any ε = ω(1/√n), which is tight for the full range of ε. We also give an Ω(n log n/ε²)-space lower bound against deterministic algorithms for outputting a (1-ε) approximation to the value of the maximum cut.
Using similar techniques we prove an analogous sharp separation in the streaming space complexity of Densest Subgraph and show that for every constant-arity CSP over a constant-size alphabet and the Similarity problem the space complexity in dense streams can be improved by shaving a logarithmic factor.
Cite as
Yang P. Liu, Hoai-An Nguyen, Noah G. Singer, and David P. Woodruff. Streaming Complexity Separations for Dense and Sparse Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 140:1-140:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{liu_et_al:LIPIcs.ICALP.2026.140,
author = {Liu, Yang P. and Nguyen, Hoai-An and Singer, Noah G. and Woodruff, David P.},
title = {{Streaming Complexity Separations for Dense and Sparse Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {140:1--140:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.140},
URN = {urn:nbn:de:0030-drops-265290},
doi = {10.4230/LIPIcs.ICALP.2026.140},
annote = {Keywords: streaming, maximum cut, complexity separations}
}
Document
Track A: Algorithms, Complexity and Games
Online Steiner Forest with Recourse
Abstract
In the online Steiner forest problem we are given a graph G, and a sequence of terminal pairs (u_i,v_i) which arrive in an online fashion. We are asked to maintain a low-cost subgraph in which each u_i is connected to v_i for all the pairs that have arrived so far. If we are not allowed to delete edges from our solution, then the best possible competitive ratio is Θ(log n). In this work, we initiate the study of low-recourse algorithms for online Steiner forest. We give an algorithm that maintains a constant-competitive solution and has an amortized recourse of O(log n), i.e., inserts and deletes O(log n) edges per demand on average.
Cite as
Yaowei Long, Sepideh Mahabadi, Sherry Sarkar, and Jakub Tarnawski. Online Steiner Forest with Recourse. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 141:1-141:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{long_et_al:LIPIcs.ICALP.2026.141,
author = {Long, Yaowei and Mahabadi, Sepideh and Sarkar, Sherry and Tarnawski, Jakub},
title = {{Online Steiner Forest with Recourse}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {141:1--141:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.141},
URN = {urn:nbn:de:0030-drops-265303},
doi = {10.4230/LIPIcs.ICALP.2026.141},
annote = {Keywords: Online algorithms with recourse, Steiner forest, Network design}
}
Document
Track A: Algorithms, Complexity and Games
Combinatorial Perpetual Scheduling: Existence and Computation of Low-Height Schedules
Abstract
This paper considers a framework for combinatorial variants of perpetual-scheduling problems. Given an independence system (E,ℐ), a schedule consists of an independent set I_t ∈ ℐ for every time step t ∈ ℕ, with the objective of fulfilling frequency requirements on the occurrence of elements in E. We focus specifically on combinatorial bamboo garden trimming, where elements accumulate height at growth rates g(e) for e ∈ E and are reset to zero when scheduled, with the goal of minimizing the maximum height attained by any element. We assume that g is normalized so that it is a convex combination of the incidence vectors of ℐ.
Using the integrality of the matroid-intersection polytope, we prove that, when (E,ℐ) is a matroid, it is possible to guarantee a maximum height of at most 2, which is optimal. We complement this existential result with efficient algorithms for specific matroid classes, achieving a maximum height of 2 for uniform and partition matroids, and 4 for graphic and laminar matroids. In contrast, we show that for general independence systems, the optimal guaranteed height is Θ(log |E|) and can be achieved by an efficient algorithm. For combinatorial pinwheel scheduling, where each element e ∈ E needs to occur in the schedule at least every a_e ∈ ℕ time steps, our results imply bounds on the density sufficient for schedulability.
Cite as
Mirabel Mendoza-Cadena, Arturo Merino, Mads Anker Nielsen, and Kevin Schewior. Combinatorial Perpetual Scheduling: Existence and Computation of Low-Height Schedules. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 142:1-142:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mendozacadena_et_al:LIPIcs.ICALP.2026.142,
author = {Mendoza-Cadena, Mirabel and Merino, Arturo and Nielsen, Mads Anker and Schewior, Kevin},
title = {{Combinatorial Perpetual Scheduling: Existence and Computation of Low-Height Schedules}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {142:1--142:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.142},
URN = {urn:nbn:de:0030-drops-265319},
doi = {10.4230/LIPIcs.ICALP.2026.142},
annote = {Keywords: Perpetual Scheduling, Matroids, Bamboo Garden Trimming, Pinwheel}
}
Document
Track A: Algorithms, Complexity and Games
From Worst-Case Hardness of NP to Quantum Cryptography via Quantum Indistinguishability Obfuscation
Abstract
Indistinguishability obfuscation (iO) has emerged as a powerful cryptographic primitive with many implications. While classical iO, combined with the infinitely-often worst-case hardness of NP, is known to imply one-way functions (OWFs) and a range of advanced cryptographic primitives, the cryptographic implications of quantum iO remain poorly understood. In this work, we initiate a study of the power of quantum iO. We define several natural variants of quantum iO, distinguished by whether the obfuscation algorithm, evaluation algorithm, and description of obfuscated program are classical or quantum. For each variant, we identify quantum cryptographic primitives that can be constructed under the assumption of quantum iO and the infinitely-often quantum worst-case hardness of NP (i.e., NP ⊈ i.o.BQP). In particular, we construct pseudorandom unitaries, QCCC quantum public-key encryption and (QCCC) quantum symmetric-key encryption, and several primitives implied by them such as one-way state generators, (efficiently-verifiable) one-way puzzles, and EFI pairs, etc. While our main focus is on quantum iO, even in the classical setting, our techniques yield a new and arguably simpler construction of OWFs from classical (imperfect) iO and the infinitely-often worst-case hardness of NP.
Cite as
Tomoyuki Morimae, Yuki Shirakawa, and Takashi Yamakawa. From Worst-Case Hardness of NP to Quantum Cryptography via Quantum Indistinguishability Obfuscation. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 143:1-143:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{morimae_et_al:LIPIcs.ICALP.2026.143,
author = {Morimae, Tomoyuki and Shirakawa, Yuki and Yamakawa, Takashi},
title = {{From Worst-Case Hardness of NP to Quantum Cryptography via Quantum Indistinguishability Obfuscation}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {143:1--143:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.143},
URN = {urn:nbn:de:0030-drops-265321},
doi = {10.4230/LIPIcs.ICALP.2026.143},
annote = {Keywords: Quantum cryptography, Indistinguishability Obfuscation}
}
Document
Track A: Algorithms, Complexity and Games
Undirected Replacement Paths: Dual Fault Reduces to Single Source
Abstract
Given a graph and two vertices s and t, the Replacement Path Problem (RP) is to compute for every edge e, the distance between s and t when e is removed. There are two natural extensions to RP:
- Single Source Replacement Paths (SSRP): Given a graph 𝐆 and a source node s, compute for every vertex v and every edge e the s-v distance in 𝐆⧵e. That is, we do not fix the target anymore.
- 2-Fault Replacement Paths (2-FRP): Given a graph 𝐆 and two nodes s and t, compute for every pair of edges e,e' the s-t distance in 𝐆⧵e,e'. That is, there are two failures instead of one.
Previously, there was no known formal reduction between SSRP and 2-FRP. It seemed plausible that 2-FRP would be computationally harder because there are no settings where 2-FRP admits a faster algorithm than SSRP. In directed unweighted graphs there is a provable gap in complexity, and in undirected graphs many of the known 2-FRP algorithms in a variety of settings are much slower than those for SSRP in the same setting.
The main contribution of this paper is a tight reduction from undirected 2-FRP to undirected SSRP, showing that contrary to prior intuition, 2-FRP is not harder than SSRP. As our reduction is weight-preserving, we get new algorithms for 2-FRP that match the best-known runtimes for SSRP:
(a) 𝒪̃(M n^ω) for weights in [1..M] [Grandoni and Vassilevska Williams, FOCS 2012 & TALG 2019], improving upon 𝒪(Mn^{2.87}) [Chechik, Zhang, ICALP 2024];
(b) n³/2^Ω(√{log n}) for weights in [1..poly(n)] [Grandoni and Vassilevska Williams, FOCS 2012 & TALG 2019], improving over the previous n³polylog(n) running time [Vassilevska W., Woldeghebriel and Xu, FOCS 2022];
(c) 𝒪̃(mn^{1/2} + n²) combinatorial time for unweighted graphs [Chechik and Cohen, SODA 2019], and more generally for rational weights in [1,2] [Chechik and Magen, ICALP 2020], improving upon 𝒪̃(n^{3-1/18}) [Chechik, Zhang ICALP 2024].
We complement these upper bounds with tight lower bounds under fine-grained hypotheses.
Cite as
Jakob Nogler and Virginia Vassilevska Williams. Undirected Replacement Paths: Dual Fault Reduces to Single Source. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 144:1-144:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nogler_et_al:LIPIcs.ICALP.2026.144,
author = {Nogler, Jakob and Vassilevska Williams, Virginia},
title = {{Undirected Replacement Paths: Dual Fault Reduces to Single Source}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {144:1--144:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.144},
URN = {urn:nbn:de:0030-drops-265332},
doi = {10.4230/LIPIcs.ICALP.2026.144},
annote = {Keywords: Single Source Replacement Paths, Dualt Fault Replacement Paths, Fine-Grained Complexity}
}
Document
Track A: Algorithms, Complexity and Games
Inapproximability of Counting Permutation Patterns
Abstract
Detecting and counting copies of permutation patterns are fundamental algorithmic problems, with applications in the analysis of rankings, nonparametric statistics, and property testing tasks such as independence and quasirandomness testing. From an algorithmic perspective, there is a sharp difference in complexity between detecting and counting the copies of a given length-k pattern in a length-n permutation. The former admits a 2^𝒪(k²) ⋅ n time algorithm (Guillemot and Marx, 2014) while the latter cannot be solved in time f(k) ⋅ n^o(k/log k) unless the Exponential Time Hypothesis (ETH) fails (Berendsohn, Kozma, and Marx, 2021). In fact already for patterns of length 4, exact counting is unlikely to admit near-linear time algorithms under standard fine-grained complexity assumptions (Dudek and Gawrychowski, 2020).
Recently, Ben-Eliezer, Mitrović and Srivastava (2026) showed that for patterns of length up to 5, a (1+ε)-approximation of the pattern count can be computed in near-linear time, yielding a separation between exact and approximate counting for small patterns, and conjectured that approximate counting is asymptotically easier than exact counting in general. We strongly refute their conjecture by showing that, under ETH, no algorithm running in time f(k)⋅ n^o(k/log k) can approximate the number of copies of a length-k pattern within a multiplicative factor n^(1/2-ε)k. The lower bound on runtime matches the conditional lower bound for exact pattern counting, and the obtained bound on the multiplicative error factor is essentially tight, as an n^{k/2}-approximation can be computed in 2^𝒪(k²) ⋅ n time using an algorithm for pattern detection.
Cite as
Michal Opler. Inapproximability of Counting Permutation Patterns. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 145:1-145:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{opler:LIPIcs.ICALP.2026.145,
author = {Opler, Michal},
title = {{Inapproximability of Counting Permutation Patterns}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {145:1--145:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.145},
URN = {urn:nbn:de:0030-drops-265340},
doi = {10.4230/LIPIcs.ICALP.2026.145},
annote = {Keywords: permutation patterns, approximate counting, inapproximability, exponential time hypothesis, parameterized complexity}
}
Document
Track A: Algorithms, Complexity and Games
Color Fault-Tolerant Distance Preservers: Õptimal Size in Conditionally Õptimal Time
Abstract
We revisit the problem of fault-tolerant (FT) distance preservers, when failure events in the network admit a form of correlation modeled as color faults.
FT distance preservers are sparse subgraphs that preserve distances between specified pairs of vertices, even after some edge or vertex failures occur. In the classical fault model, any set of at most k edges or vertices might fail (where k ≥ 1 is a given parameter). Despite extensive research, the classical model admits significant and tantalizing gaps, both in terms of sparsity bounds and of algorithmic efficiency.
In this work, we study the problem in the recently introduced color fault-tolerant (CFT) model: the given graph G = (V,E) has arbitrary colors on its edges/vertices where each color appears at most k times, and is susceptible to color faults, where the failure of color c causes all the c-colored elements to crash. Our main contribution is in the multi-source setting, where G has a source-set S ⊆ V, and the CFT preserver should preserve S × V distances under any single color fault. We show the following results (where n = |V|, |E| = m):
- There exists a CFT distance preserver H of G with Õ(n^{2 - 1/(k+1)} ⋅ |S|^{1/(k+1)}) edges.
- The above sparsity bound is worst-case optimal up to polylogarithmic terms.
- There is a combinatorial randomized algorithm that produces a preserver H whose size meets the above optimal sparsity bound, with running time of Õ(m ⋅ n^{1 - 1/(k+1)} ⋅ |S|^{1/(k+1)}).
- The above running time is conditionally optimal: a polynomial improvement would refute the combinatorial Boolean Matrix Multiplication (BMM) conjecture. Furthermore, the running time remains optimal even if we only require mild sparsification to m^{1-ε} edges. Our (conditionally) tight algorithm relies on a new approach for compressing fault-tolerant distance information in the presence of nearby faults, using the tool of sparse neighborhood covers. We believe that this technique may have further applications in the study of fault-tolerant graph sparsification.
Cite as
Merav Parter and Asaf Petruschka. Color Fault-Tolerant Distance Preservers: Õptimal Size in Conditionally Õptimal Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 146:1-146:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{parter_et_al:LIPIcs.ICALP.2026.146,
author = {Parter, Merav and Petruschka, Asaf},
title = {{Color Fault-Tolerant Distance Preservers: \~{O}ptimal Size in Conditionally \~{O}ptimal Time}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {146:1--146:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.146},
URN = {urn:nbn:de:0030-drops-265353},
doi = {10.4230/LIPIcs.ICALP.2026.146},
annote = {Keywords: fault tolerance, network design, distance preservers}
}
Document
Track A: Algorithms, Complexity and Games
Local Computation Algorithms for (Minimum) Spanning Trees on Expander Graphs
Abstract
We study local computation algorithms (LCAs) for constructing spanning trees. In this setting, the goal is to determine locally, for each edge e ∈ E, whether it belongs to a spanning tree T of the input graph G, where T is defined implicitly by G and the randomness of the algorithm. It is known that sublinear-probe LCAs for spanning trees do not exist in general graphs, even for simple graph families. We identify a natural and well-studied class of graphs - expander graphs - that do admit sublinear-time LCAs for spanning trees. This is perhaps surprising, as previous work on expanders only succeeded in designing LCAs for sparse spanning subgraphs, rather than full spanning trees. We design an LCA with probe complexity O(√n ((log²n)/ϕ² + d)) for graphs with conductance at least ϕ and maximum degree at most d (not necessarily constant), which is nearly optimal when ϕ and d are constants, since Ω(√n) probes are necessary even for expanders. Next, we show that for the natural class of Erdős-Rényi graphs G(n, p) with np = n^δ for any constant δ > 0 (which are expanders with high probability), the √n lower bound can be bypassed. Specifically, we give an average-case LCA for such graphs with probe complexity Õ(√{n^{1 - δ}}).
Finally, we extend our techniques to design LCAs for the minimum spanning tree (MST) problem on weighted expander graphs. Specifically, given a d-regular unweighted graph ̄{G} with sufficiently strong expansion, we consider the weighted graph G obtained by assigning to each edge an independent and uniform random weight from {1,…,W}, where W ≤ d/2 and W = o(log n). We show that there exists an LCA that is consistent with an exact MST of G, with probe complexity Õ(√nd²).
Cite as
Pan Peng and Yuyang Wang. Local Computation Algorithms for (Minimum) Spanning Trees on Expander Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 147:1-147:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{peng_et_al:LIPIcs.ICALP.2026.147,
author = {Peng, Pan and Wang, Yuyang},
title = {{Local Computation Algorithms for (Minimum) Spanning Trees on Expander Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {147:1--147:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.147},
URN = {urn:nbn:de:0030-drops-265361},
doi = {10.4230/LIPIcs.ICALP.2026.147},
annote = {Keywords: Local Computation Algorithms, (Minimum) Spanning Trees, Expander Graphs, Random Graphs}
}
Document
Track A: Algorithms, Complexity and Games
Alternation Depth of Threshold Decision Lists
Abstract
Linear decision lists are a computational model for Boolean functions. A linear decision list is built from a sequence of linear threshold function queries which are evaluated one by one: if a query returns true, the list outputs the value of the function, and if the answer is false, the process continues to the next query. The size of a linear decision list is the number of queries in it.
Linear decision lists form a natural and nontrivial subclass of depth-2 threshold circuits, the class of circuits that currently marks the frontier of explicit circuit lower bounds. Although some techniques for proving lower bounds against linear decision lists exist, they are quite limited, leaving important open problems unresolved. Moreover, for the related model of exact linear decision lists, no strong lower bounds are known.
We initiate the study of alternation depth of decision lists with linear threshold queries. The alternation depth is defined as the number of alternations in the sequence of output values of the decision list. We show that linear decision lists, both with bounded and unbounded weights in the threshold queries, form fine hierarchies with respect to alternation depth. A similar hierarchy exists for rectangle decision lists, the model closely related to communication complexity with NP oracles. We prove strong separations within these hierarchies and between them.
Next, we give a superpolynomial lower bound for an explicit function for exact linear decision lists of depth below n/log n. Such lower bounds were not previously known and do not follow directly from existing methods. We also establish a fine depth hierarchy for exact linear decision lists.
To prove these hierarchy separations, we use an iterative technique combined with existing techniques such as fooling sets and the analysis of blocky matrices. For the lower bound on exact linear decision lists, we combine the discrepancy method with an iterative analysis of blocky matrices.
Cite as
Vladimir Podolskii and Morgan Prior. Alternation Depth of Threshold Decision Lists. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 148:1-148:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{podolskii_et_al:LIPIcs.ICALP.2026.148,
author = {Podolskii, Vladimir and Prior, Morgan},
title = {{Alternation Depth of Threshold Decision Lists}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {148:1--148:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.148},
URN = {urn:nbn:de:0030-drops-265373},
doi = {10.4230/LIPIcs.ICALP.2026.148},
annote = {Keywords: linear decision lists, threshold decision lists, rectangle decision lists, threshold circuits}
}
Document
Track A: Algorithms, Complexity and Games
Colorful Minors
Abstract
We introduce the notion of colorful minors, which generalizes the classical concept of rooted minors in graphs. A q-colorful graph is defined as a pair (G, χ), where G is a graph and χ assigns to each vertex a (possibly empty) subset of at most q colors. The colorful minor relation enhances the classical minor relation by merging color sets at contracted edges and allowing the removal of colors from vertices. This framework naturally models algorithmic problems involving graphs with (possibly overlapping) annotated vertex sets. We develop a structural theory for colorful minors by establishing three core theorems characterizing ℋ-colorful minor-free graphs, where ℋ consists either of a clique or a grid with all vertices assigned all colors, or of grids with colors segregated and ordered on the outer face. Our results reveal that when exclusion is imposed not only on graphs but also to the way colors are distributed in them, a more refined structural landscape appears.
Leveraging our structural insights, we provide a complete classification - parameterized by the number q of colors - of all colorful graphs that exhibit the Erdős–Pósa property with respect to colorful minors. On the algorithmic side, we deduce that colorful minor testing is fixed-parameter tractable. Together with the fact that the colorful minor relation forms a well-quasi-order, this implies that every colorful minor-monotone parameter on colorful graphs admits a fixed-parameter algorithm. Furthermore, we derive two algorithmic meta-theorems (AMTs) whose structural conditions are linked to extensions of treewidth and Hadwiger number on colorful graphs. Our results suggest how known AMTs can be extended to incorporate not only the structure of the input graph but also the way the colored vertices are distributed in it.
Cite as
Evangelos Protopapas, Dimitrios M. Thilikos, and Sebastian Wiederrecht. Colorful Minors. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 149:1-149:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{protopapas_et_al:LIPIcs.ICALP.2026.149,
author = {Protopapas, Evangelos and Thilikos, Dimitrios M. and Wiederrecht, Sebastian},
title = {{Colorful Minors}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {149:1--149:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.149},
URN = {urn:nbn:de:0030-drops-265380},
doi = {10.4230/LIPIcs.ICALP.2026.149},
annote = {Keywords: Graph Minors, Colorful Minors, Annotated Graphs, Rooted Minors, Erd\H{o}s-P\'{o}sa property, Structural Graph Theory, Obstruction sets, Algorithmic Meta-Theorems, Bidimensionality}
}
Document
Track A: Algorithms, Complexity and Games
Optimal k-Secretary with Logarithmic Memory
Abstract
We study memory-bounded algorithms for the k-secretary problem. The algorithm of Kleinberg (SODA 2005) achieves an optimal competitive ratio of 1 - O(1/√k), yet a straightforward implementation requires Ω(k) memory. Our main result is a k-secretary algorithm that matches the optimal competitive ratio using O(log k) words of memory. We prove this result by establishing a general reduction from k-secretary to (random-order) quantile estimation, the problem of finding the k-th largest element in a stream. We show that a quantile estimation algorithm with an O(k^{α}) expected error (in terms of the rank) gives a (1 - O(1/k^{1-α}))-competitive k-secretary algorithm with O(1) extra words. We then introduce a new quantile estimation algorithm that achieves an O(√k) expected error bound using O(log k) memory. Of independent interest, we give a different algorithm that uses O(√k) words and finds the k-th largest element exactly with high probability, generalizing a result of Munro and Paterson (1980).
Cite as
Mingda Qiao and Wei Zhang. Optimal k-Secretary with Logarithmic Memory. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 150:1-150:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{qiao_et_al:LIPIcs.ICALP.2026.150,
author = {Qiao, Mingda and Zhang, Wei},
title = {{Optimal k-Secretary with Logarithmic Memory}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {150:1--150:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.150},
URN = {urn:nbn:de:0030-drops-265394},
doi = {10.4230/LIPIcs.ICALP.2026.150},
annote = {Keywords: Streaming algorithms, online algorithms, secretary problem}
}
Document
Track A: Algorithms, Complexity and Games
Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity
Abstract
We study the average-case hardness of establishing that a graph does not have a large clique in both proof and communication complexity. We show exponential lower bounds on the length of cutting planes and bounded-depth resolution over parities refutations of the binary encoding of clique formulas on randomly sampled dense graphs. Moreover, we show that the randomized communication complexity of finding a falsified clause in these formulas is polynomial.
Cite as
Susanna F. de Rezende, David Engström, Yassine Ghannane, Duri Andrea Janett, and Artur Riazanov. Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 151:1-151:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{derezende_et_al:LIPIcs.ICALP.2026.151,
author = {de Rezende, Susanna F. and Engstr\"{o}m, David and Ghannane, Yassine and Janett, Duri Andrea and Riazanov, Artur},
title = {{Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {151:1--151:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.151},
URN = {urn:nbn:de:0030-drops-265407},
doi = {10.4230/LIPIcs.ICALP.2026.151},
annote = {Keywords: proof complexity, communication complexity, cutting planes, bounded-depth resolution over parities, clique problem, average-case hardness, binary encoding}
}
Document
Track A: Algorithms, Complexity and Games
Partial Derivative Complexity of a Product of Linearly Independent Quadratics
Abstract
The partial derivative method is a central tool in algebraic complexity, underlying lower bounds for multilinear formulas, bounded depth circuits, and algebraic branching programs. A key feature of this measure is its subadditivity and submultiplicativity, which are usually used to upper bound the measure. However, proving lower bounds requires bounding the measure of explicit polynomials from below, and in some cases, a sharp estimate is required. For example, a frequently used fact is that the dimension of the space spanned by order k partial derivatives of a product of n linearly independent linear functions is binom(n,k).
Beyond the linear case, however, not much is known about the behavior of the (general) partial derivative measure under multiplication. In particular, it has been conjectured that for algebraically independent polynomials g₁,… ,g_r ∈ ℂ[𝐱], the partial derivative complexity of the product ∏_{i=1}^r g_i(𝐱) grows exponentially with r (see [Chaugule et al., 2023]), but prior to this work such bounds were only known when the g_i’s are linear polynomials, or satisfy additional restrictions.
In this paper, we show a lower bound of exp(Ω(r^{1/6})) for the measure of a product of r linearly independent quadratic polynomials. This is the first result to show such a lower bound on the partial derivative measure of a product of nonlinear polynomials, without any further restrictions. Interestingly, we only assume linear independence, which is weaker than algebraic independence. Our proof relies on algebraic-geometric and combinatorial techniques, combining the Jacobian approach of [Chaugule et al., 2023] together with the theory of wide algebras introduced in [Ananyan and Hochster, 2020; Oliveira and Sengupta, 2022; Garg et al., 2023]. To our knowledge, this is the first use of wide-algebra techniques for proving lower bounds on partial derivative complexity, and one of the first applications of these techniques outside the context of Sylvester-Gallai type problems.
Cite as
Nir Shalmon and Amir Shpilka. Partial Derivative Complexity of a Product of Linearly Independent Quadratics. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 152:1-152:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{shalmon_et_al:LIPIcs.ICALP.2026.152,
author = {Shalmon, Nir and Shpilka, Amir},
title = {{Partial Derivative Complexity of a Product of Linearly Independent Quadratics}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {152:1--152:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.152},
URN = {urn:nbn:de:0030-drops-265411},
doi = {10.4230/LIPIcs.ICALP.2026.152},
annote = {Keywords: algebraic complexity theory, partial derivatives, arithmetic circuits, quadratic polynomials}
}
Document
Track A: Algorithms, Complexity and Games
Dynamic Set Cover with Worst-Case Recourse
Abstract
In the dynamic set cover (SC) problem, the input is a dynamic universe of at most n elements and a fixed collection of m sets, where each element belongs to at most f sets and each set has a cost in [1/C,1]. The objective is to efficiently maintain an approximate minimum SC under element updates. Efficiency is primarily measured by the update time, but another important parameter is the recourse (the number of changes to solution per update). Ideally, one would like to achieve low worst-case bounds on both update time and recourse.
One can achieve an approximation of (1+ε)ln n (greedy-based) or (1+ε)f (primal–dual-based) with worst-case update time O(f log n) (ignoring ε-dependencies). However, despite a large body of work, no algorithm with low update time (even amortized) and nontrivial worst-case recourse is known even for unweighted instances (C = 1)!
We remedy this by providing a transformation that, given a SC algorithm with approximation α and update time T as a black-box, returns a set cover algorithm with approximation (2 + ε)α, update time O(T + α C) and worst-case recourse O(α C). Our main results are obtained by leveraging this transformation for constant C:
- For f = O(log n), applying the transformation on the best primal-dual-based algorithm yields worst-case recourse O(f). For constant f (e.g., vertex cover), we get near-optimal bounds on all parameters.
- For f = Ω(log n), applying the transformation on the best greedy-based algorithm yields worst-case recourse O(log n). As our main technical contribution, we show that by opening the black box and exploiting a certain robustness property of the greedy-based algorithm, the worst-case recourse can be reduced to O(1), without sacrificing the other parameters, yielding a ((2 + ε) ln n)-approximation with worst-case update time O(flog n) and O(1) worst-case recourse.
Cite as
Shay Solomon and Amitai Uzrad. Dynamic Set Cover with Worst-Case Recourse. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 153:1-153:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{solomon_et_al:LIPIcs.ICALP.2026.153,
author = {Solomon, Shay and Uzrad, Amitai},
title = {{Dynamic Set Cover with Worst-Case Recourse}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {153:1--153:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.153},
URN = {urn:nbn:de:0030-drops-265425},
doi = {10.4230/LIPIcs.ICALP.2026.153},
annote = {Keywords: Dynamic graphs, set cover, recourse}
}
Document
Track A: Algorithms, Complexity and Games
Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer
Abstract
We settle the problem of estimating the trace distance and (square root) fidelity between n-qubit pure quantum states to within additive error ε, given their independent samples, which was raised as an open question by Wang (IEEE Trans. Inf. Theory 2024). This is achieved by a quantum algorithm with optimal sample complexity Θ(1/ε²), improving the long-standing folklore with sample complexity O(1/ε⁴). At the heart of our algorithm is a samplized phase estimation of the product of two Householder reflections. This is realized by an improved (multi-)samplizer for pure states, through which any quantum query algorithm using Q queries to the reflection operator I - 2 |ψ⟩⟨ψ| can be converted to a δ-close (in the diamond norm distance) quantum sample algorithm using Θ(Q²/δ) samples of the state |ψ⟩. This samplizer for pure states is also shown to be optimal.
Cite as
Qisheng Wang and Zhicheng Zhang. Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 154:1-154:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wang_et_al:LIPIcs.ICALP.2026.154,
author = {Wang, Qisheng and Zhang, Zhicheng},
title = {{Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {154:1--154:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.154},
URN = {urn:nbn:de:0030-drops-265433},
doi = {10.4230/LIPIcs.ICALP.2026.154},
annote = {Keywords: Quantum algorithms, sample complexity, trace distance, fidelity, pure states, lower bounds, samplizer}
}
Document
Track A: Algorithms, Complexity and Games
A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity
Abstract
We investigates a model of hybrid classical-quantum communication complexity, in which two parties first exchange classical messages and subsequently communicate using quantum messages. We study the trade-off between the classical and quantum communication for composed functions of the form f∘Gⁿ, where f:{0,1}ⁿ → {±1} and G is an inner product function of Θ(log n) bits. To prove the trade-off, we establish a novel lifting theorem for hybrid communication complexity. This theorem unifies two previously separate lifting paradigms: the query-to-communication lifting framework for classical communication complexity and the approximate-degree-to-generalized-discrepancy lifting methods for quantum communication complexity. Our hybrid lifting theorem therefore offers a new framework for proving lower bounds in hybrid classical-quantum communication models.
As a corollary, we show that any hybrid protocol communicating c classical bits followed by q qubits to compute f∘Gⁿ must satisfy c+q² = Ω(max{deg(f), bs(f)}⋅log n), where deg(f) is the degree of f and {bs}(f) is the block sensitivity of f. For read-once formula f, this yields an almost tight trade-off: either they have to exchange Θ(n⋅log n) classical bits or Θ(√n ⋅ log n) qubits, showing that classical pre-processing cannot significantly reduce the quantum communication required. To the best of our knowledge, this is the first non-trivial trade-off between classical and quantum communication in hybrid two-way communication complexity.
Cite as
Xudong Wu, Guangxu Yang, and Penghui Yao. A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 155:1-155:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wu_et_al:LIPIcs.ICALP.2026.155,
author = {Wu, Xudong and Yang, Guangxu and Yao, Penghui},
title = {{A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {155:1--155:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.155},
URN = {urn:nbn:de:0030-drops-265449},
doi = {10.4230/LIPIcs.ICALP.2026.155},
annote = {Keywords: Hybrid communication, Generalized discrepancy bound, Dual polynomial, Query-to-communication lifting}
}
Document
Track A: Algorithms, Complexity and Games
Edge-Weighted Online Stochastic Matching Under Jaillet-Lu LP
Abstract
The online stochastic matching problem was introduced by [Feldman et al., 2009], together with the (1-1/e)-competitive Suggested Matching algorithm. In the most general edge-weighted setting, this ratio has not been improved for more than one decade, until recently [Yan, 2024] beat the 1-1/e bound and [Qiu et al., 2023] further improved it to 0.650. Both works measure the online competitiveness against the offline LP relaxation introduced by Jaillet and Lu [Jaillet and Lu, 2014]. The same LP has also played an important role in other settings as it is a natural choice for two-choice online algorithms.
In this paper, we prove an upper bound of 0.663 and a lower bound of 0.662 for edge-weighted online stochastic matching under Jaillet-Lu LP. We propose a simple hard instance and identify the optimal online algorithm for this specific instance which has a competitive ratio of < 0.663. Despite the simplicity of the instance, we then show that a near-optimal algorithm for it, which has a competitive ratio of > 0.662, can be generalized to work on all instances without any loss.
As our algorithm is generalized from a real near-optimal algorithm instead of manually combining trivial strategies, it has two natural advantages compared with previous works: (1) its matching strategy varies from time to time; (2) it utilizes global information about offline vertices. On the other hand, the upper bound suggests that more powerful LPs and multiple-choice strategies are needed if we want to further improve the ratio by > 0.001.
In addition to our main result, we also generalize the asymptotic equivalence between the Poisson arrival model and the original online stochastic matching established by [Huang and Shu, 2021], removing the requirement of approximate monotonicity for the online algorithm.
Cite as
Shuyi Yan. Edge-Weighted Online Stochastic Matching Under Jaillet-Lu LP. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 156:1-156:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{yan:LIPIcs.ICALP.2026.156,
author = {Yan, Shuyi},
title = {{Edge-Weighted Online Stochastic Matching Under Jaillet-Lu LP}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {156:1--156:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.156},
URN = {urn:nbn:de:0030-drops-265450},
doi = {10.4230/LIPIcs.ICALP.2026.156},
annote = {Keywords: Online stochastic matching}
}
Document
Track A: Algorithms, Complexity and Games
Fully Dynamic Spectral and Cut Sparsifiers for Directed Graphs
Abstract
Recent years have seen extensive research on directed graph sparsification. In this work, we initiate the study of fast fully dynamic spectral and cut sparsification algorithms for directed graphs.
We introduce a new notion of spectral sparsification called degree-balance preserving spectral approximation, which maintains the difference between the in-degree and out-degree of each vertex. The approximation error is measured with respect to the corresponding undirected Laplacian. This notion is equivalent to direct Eulerian spectral approximation when the input graph is Eulerian. Our algorithm achieves an amortized update time of O(ε^{-2} ⋅ polylog(n)) and produces a sparsifier of size O(ε^{-2} n ⋅ polylog(n)). Additionally, we present an algorithm that maintains a constant-factor approximation sparsifier of size O(n ⋅ polylog(n)) against an adaptive adversary for O(polylog(n))-partially symmetrized graphs, a notion introduced in [Kyng-Meierhans-Probst Gutenberg '22]. A β-partial symmetrization of a directed graph G is the union of G and β ⋅ G, where G is the corresponding undirected graph of G. This algorithm also achieves a polylogarithmic amortized update time.
Moreover, we develop a fully dynamic algorithm for maintaining a cut sparsifier for β-balanced directed graphs, where the ratio between weighted incoming and outgoing edges of any cut is at most β. This algorithm explicitly maintains a cut sparsifier of size O(ε^{-2}β n ⋅ polylog(n)) in worst-case update time O(ε^{-2}β ⋅ polylog(n)).
Cite as
Yibin Zhao. Fully Dynamic Spectral and Cut Sparsifiers for Directed Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 157:1-157:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{zhao:LIPIcs.ICALP.2026.157,
author = {Zhao, Yibin},
title = {{Fully Dynamic Spectral and Cut Sparsifiers for Directed Graphs}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {157:1--157:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.157},
URN = {urn:nbn:de:0030-drops-265468},
doi = {10.4230/LIPIcs.ICALP.2026.157},
annote = {Keywords: Dynamic graph algorithm, Spectral graph theory, Sparsifier}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Revisiting Finiteness of Matrix Monoids
Abstract
This paper concerns decision problems related to finite monoids of rational matrices. We show that determining finiteness of a given finitely presented monoid is in PSpace, improving the known coNExp^NP bound. We also show that the membership problem for finite matrix monoids is PSpace-complete, improving the known NExp-upper bound. Our two complexity results are corollaries of a new polynomial bit-size bound on matrix entries in finite monoids. This is obtained by reduction to the case of matrix groups, using the structure theory of noncommutative algebras and of matrix monoids. Our techniques also give us a polynomial-time algorithm for deciding whether a monoid of rational matrices is conjugate to a monoid of integer matrices.
Cite as
Rida Ait El Manssour, Roland Guttenberg, Nathan Lhote, Mahsa Shirmohammadi, and James Ben Worrell. Revisiting Finiteness of Matrix Monoids. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 158:1-158:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aitelmanssour_et_al:LIPIcs.ICALP.2026.158,
author = {Ait El Manssour, Rida and Guttenberg, Roland and Lhote, Nathan and Shirmohammadi, Mahsa and Worrell, James Ben},
title = {{Revisiting Finiteness of Matrix Monoids}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {158:1--158:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.158},
URN = {urn:nbn:de:0030-drops-265745},
doi = {10.4230/LIPIcs.ICALP.2026.158},
annote = {Keywords: Matrix Semigroups, Finiteness, Integrality, Bitsize Bound}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Unambiguisability and Register Minimisation of Min-Plus Models
Abstract
We study the unambiguisability problem for min-plus (tropical) weighted automata (WFAs), and the register-minimisation problem for tropical Cost Register Automata (CRAs), which are expressively-equivalent to WFAs. Both problems ask whether the “amount of nondeterminism’’ in the model can be reduced. We show that WFA unambiguisability is decidable for tropical WFAs. Our proof is via reduction to WFA determinisability, which was recently shown to be decidable. To obtain this reduction, we develop a characterisation of unambiguisability via gaps between runs. On the negative side, we show that CRA register minimisation is undecidable already for inputs with 7 registers, and hence also for any larger fixed number of registers.
Cite as
Shaull Almagor, Guy Arbel, and Sarai Sheinvald. Unambiguisability and Register Minimisation of Min-Plus Models. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 159:1-159:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{almagor_et_al:LIPIcs.ICALP.2026.159,
author = {Almagor, Shaull and Arbel, Guy and Sheinvald, Sarai},
title = {{Unambiguisability and Register Minimisation of Min-Plus Models}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {159:1--159:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.159},
URN = {urn:nbn:de:0030-drops-265479},
doi = {10.4230/LIPIcs.ICALP.2026.159},
annote = {Keywords: Automata, Weighted Automata, Determinisation, Unambiguous, Unambiguisation, Tropical, Min Plus}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Gray Codes with Constant Delay and Constant Auxiliary Space
Abstract
We give the first two algorithms to enumerate all binary words of {0,1}^𝓁 (like Gray codes) while ensuring that the delay and the auxiliary space is independent from 𝓁, i.e., constant time for each word, and constant memory in addition to the 𝓁 bits storing the current word. Our algorithms are given in two new computational models: tape machines and deque machines. We also study more restricted models, queue machines and stack machines, and show that they cannot enumerate all binary words with constant auxiliary space, even with unrestricted delay.
A tape machine is a Turing machine that stores the current binary word on a single working tape of length 𝓁 (which never increases), using no other tape. The machine has a single head and must edit its tape to reach all possible words of {0,1}^𝓁, and output them (in unit time, by entering special output states), with no duplicates. Hence a tape machine uses constant auxiliary space by definition (up to the head position). We construct a tape machine that achieves this task with constant delay between consecutive outputs, so that the machine implements a so-called skew-tolerant quasi-Gray code. We then construct a more involved tape machine that implements a Gray code.
A deque machine stores the current binary word on a double-ended queue of length 𝓁, and stores a constant-size internal state. It works as a tape machine, except that it modifies the content of the deque by performing push and pop operations on the endpoints. Hence again a deque machine uses constant auxiliary space by definition. We construct deque machines that enumerate all words of {0,1}^𝓁 with constant-delay. The main technical challenge in this model is to correctly detect when enumeration has finished.
Cite as
Antoine Amarilli, Claire David, Nadime Francis, Victor Marsault, Mikaël Monet, and Yann Strozecki. Gray Codes with Constant Delay and Constant Auxiliary Space. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 160:1-160:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{amarilli_et_al:LIPIcs.ICALP.2026.160,
author = {Amarilli, Antoine and David, Claire and Francis, Nadime and Marsault, Victor and Monet, Mika\"{e}l and Strozecki, Yann},
title = {{Gray Codes with Constant Delay and Constant Auxiliary Space}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {160:1--160:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.160},
URN = {urn:nbn:de:0030-drops-265485},
doi = {10.4230/LIPIcs.ICALP.2026.160},
annote = {Keywords: Gray code, Constant delay, Constant auxiliary space, Enumeration algorithms, Linear bounded automata, Tape machine, Deque machines, Counter implementation}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Out-Of-Order Membership in Regular Languages
Abstract
We introduce the task of out-of-order membership to a formal language L, where the letters of a word w are revealed one by one in an arbitrary order. The length |w| is known in advance, but the content of w is streamed as pairs (i, w[i]), received exactly once for each position i, in arbitrary order. We study efficient algorithms for this task when L is regular, seeking tight complexity bounds as a function of |w| for a fixed target language. Most of our results apply to an algebraically defined variant dubbed out-of-order evaluation: this problem is defined for a fixed finite monoid or semigroup S, and our goal is to compute the ordered product of the streamed elements of w.
We show that, for any fixed regular language or finite semigroup, both problems can be solved in constant time per streamed symbol and in linear space. However, the precise space complexity strongly depends on the algebraic structure of the target language or evaluation semigroup. Our main contributions are therefore to show (deterministic) space complexity characterizations, which we do for out-of-order evaluation of monoids and semigroups.
For monoids, we establish a trichotomy: the space complexity is either Θ(1), Θ(log n), or Θ(n), where n = |w|. More specifically, the problem admits a constant-space solution for commutative monoids, while all non-commutative monoids require Ω(log n) space. We further identify a class of monoids admitting an O(log n)-space algorithm, and show that all remaining monoids require Ω(n) space.
For general semigroups, the situation is more intricate. We characterize a class of semigroups admitting constant-space algorithms for out-of-order evaluation, and show that semigroups outside this class require at least Ω(log n) space. At the same time, we exhibit semigroups for which specialized techniques yield intermediate bounds such as an O(√n)-space algorithm, suggesting that the landscape may be richer and less well-behaved than for the monoid setting.
Cite as
Antoine Amarilli, Sebastien Labbe, and Charles Paperman. Out-Of-Order Membership in Regular Languages. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 161:1-161:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{amarilli_et_al:LIPIcs.ICALP.2026.161,
author = {Amarilli, Antoine and Labbe, Sebastien and Paperman, Charles},
title = {{Out-Of-Order Membership in Regular Languages}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {161:1--161:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.161},
URN = {urn:nbn:de:0030-drops-265498},
doi = {10.4230/LIPIcs.ICALP.2026.161},
annote = {Keywords: Automata, Complexity, Algebra}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Recursion and Proof Theoretical Characterizations of Small Circuit Classes with Modulo Counting via Discrete Differential Equations
Abstract
The paper proposes an implicit (i.e., machine-independent) complexity approach to studying computation by polynomial-size, constant-depth circuits with gates counting modulo a constant through the lens of discrete ordinary differential equations (ODEs). So far, recursion-theoretic characterizations have been provided for functions computed by circuits of constant depth, including gates counting modulo 2 and 6 only (i.e., for the classes FAC⁰[2] and FAC⁰[6], resp.). In this paper, it is shown that considering ODE schemas, rather than bounded recursion, allows for a more fine-grained analysis, leading to (uniform) characterizations for all classes FAC⁰[n] (n ∈ ℕ), i.e. functions computed by circuits including counting modulo n gates. Inspired by the syntactic form of the ODE schemas, we go further in this direction and present first-order bounded theories for capturing provably total functions in each of these classes.
Cite as
Melissa Antonelli, Arnaud Durand, and Rui Li. Recursion and Proof Theoretical Characterizations of Small Circuit Classes with Modulo Counting via Discrete Differential Equations. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 162:1-162:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{antonelli_et_al:LIPIcs.ICALP.2026.162,
author = {Antonelli, Melissa and Durand, Arnaud and Li, Rui},
title = {{Recursion and Proof Theoretical Characterizations of Small Circuit Classes with Modulo Counting via Discrete Differential Equations}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {162:1--162:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.162},
URN = {urn:nbn:de:0030-drops-265501},
doi = {10.4230/LIPIcs.ICALP.2026.162},
annote = {Keywords: Implicit complexity, circuit complexity, small circuit classes with counting, discrete ODEs, recursion theory, bounded arithmetic}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Shuffles of Context-Free Languages Along Regular Trajectories
Abstract
In single-core processors, concurrency requires that multiple processes be interleaved into a single thread of execution by a scheduler. The language-theoretic operation that corresponds to this is the shuffle of two languages: the set of words obtained by interleaving a word from each language in an arbitrary, letter-wise fashion. It is well known that regular languages are closed under shuffles, while context-free languages (CFLs) are not.
Following an established line of research, this paper considers shuffles according to regular "trajectories," that is, subject to scheduling constraints expressed by an automaton. Unsurprisingly, some trajectories allow for CFLs to be shuffled into CFLs (e.g., simple concatenation of the two words), while others do not. This paper provides a robust toolset to show that a given trajectory would always shuffle two nonregular CFLs into a nonCFL. In the case of deterministic CFLs (DCFLs), a salient trichotomy of trajectories depending on how they shuffle DCFLs is provided.
These results are based on lemmata of independent interest regarding how pushdown automata (PDA) must invoke the stack when accepting a nonregular CFL or DCFL. The latter case relies on a recent result of Jančar and Šíma (MFCS'2021); answering an open question therein, it is demonstrated that said result cannot be generalized to arbitrary CFLs, leading to dedicated machinery for both cases.
Cite as
Corentin Barloy, Michaël Cadilhac, and Kyle Ockerlund. Shuffles of Context-Free Languages Along Regular Trajectories. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 163:1-163:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{barloy_et_al:LIPIcs.ICALP.2026.163,
author = {Barloy, Corentin and Cadilhac, Micha\"{e}l and Ockerlund, Kyle},
title = {{Shuffles of Context-Free Languages Along Regular Trajectories}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {163:1--163:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.163},
URN = {urn:nbn:de:0030-drops-265513},
doi = {10.4230/LIPIcs.ICALP.2026.163},
annote = {Keywords: Context-free languages, shuffles, concurrency, non-regularity}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Everybody Wants to Be a Seed: Arbitrary Single-Tile Seeds in the Abstract Tile Assembly Model
Abstract
Winfree’s abstract Tile Assembly Model (aTAM) is one of the most popular abstract models for DNA nano-computing. This papers presents a seedless version of the aTAM. Instead of a designated seed assembly, any single isolated tile can initiate the assembly. This paper shows that such system can simulate any aTAM system at constant scale and it presents a tile set to do so. All systems can be simulated at scale at most 10 and most of them can be simulated at scale 5, provided that their seeds are large enough.
Removing the need for seed in self-assembly is a new way of solving the nucleation problem, that is, the problem of controlling the initialisation of any self-assembly process. It also allows for a convergence of the aTAM towards other classical tiling models, such as Wang’s.
The systems presented are based on independent modules made of tiles. The modules are carefully designed so they act kind of like stem cells. When a module self-assembles from an isolated tile, it grows a pseudo-seed which represents the seed of the simulated system with a Hamiltonian cycle along the borders of the corresponding macrotiles. The other modules, which appear later on during the simulation as components of the "regular" macrotiles, depend on each other to self-assemble. In such cases, the assembly is synchronised in a way so that the growth of any pseudo-seed is blocked. This ensures that a macrotile self-assembles only when interacting either with the pseudo-seed or with previously completed macrotiles.
Cite as
Florent Becker and Marie de Sainte Marie. Everybody Wants to Be a Seed: Arbitrary Single-Tile Seeds in the Abstract Tile Assembly Model. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 164:1-164:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{becker_et_al:LIPIcs.ICALP.2026.164,
author = {Becker, Florent and de Sainte Marie, Marie},
title = {{Everybody Wants to Be a Seed: Arbitrary Single-Tile Seeds in the Abstract Tile Assembly Model}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {164:1--164:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.164},
URN = {urn:nbn:de:0030-drops-265520},
doi = {10.4230/LIPIcs.ICALP.2026.164},
annote = {Keywords: DNA origami, self-assembly, kinetic modeling, computational modeling, cellular automata}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Automata on S-Adic Words
Abstract
A fundamental question in logic and verification is the following: for which unary predicates P_1, …, P_k is the monadic second-order theory of ⟨ℕ;<,P_1,…,P_k⟩ decidable? Equivalently, for which infinite words α can we decide whether a given Büchi automaton 𝒜 accepts α? Carton and Thomas showed decidability in the case that α is a fixed point of a letter-to-word substitution σ, i.e., σ(α) = α. However, abundantly more words, e.g., Sturmian words, are characterised by a broader notion of self-similarity that involves a set S of substitutions. A word α is said to be directed by a sequence s = (σ_n)_{n ∈ ℕ} over S if there is a sequence of words (α_n)_{n ∈ ℕ} such that α₀ = α and α_n = σ_n(α_{n+1}) for all n; such α are called S-adic. We study the automaton acceptance problem for such words and prove, among others, the following: given finite S and an automaton 𝒜, we can compute an automaton ℬ that accepts s ∈ S^ω if and only if s directs a word α accepted by 𝒜. Thus we can algorithmically answer questions of the form "Which S-adic words are accepted by a given automaton 𝒜?"
Cite as
Valérie Berthé, Toghrul Karimov, and Mihir Vahanwala. Automata on S-Adic Words. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 165:1-165:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{berthe_et_al:LIPIcs.ICALP.2026.165,
author = {Berth\'{e}, Val\'{e}rie and Karimov, Toghrul and Vahanwala, Mihir},
title = {{Automata on S-Adic Words}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {165:1--165:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.165},
URN = {urn:nbn:de:0030-drops-265534},
doi = {10.4230/LIPIcs.ICALP.2026.165},
annote = {Keywords: Sturmian words, S-adic words, automata theory, word combinatorics}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
The Complexity of Bisimilarity and Model Checking in Finitary Diagrams
Abstract
Inspired by the work of Dubut, Goubault, and Goubault-Larrecq (ICALP 2015) on natural homology, Dubut (RAMiCS 2020) introduces finitary diagrams and studies bisimilarity and diagrammatic path logics for them. To this aim, he defines a fragment of the existential theory of the reals, called the existential theory of invertible matrices (ETIM). Using a PSPACE upper bound for this fragment, he proves that for finitary diagrams, bisimilarity can be decided in EXPSPACE and model checking for diagrammatic path logic in PSPACE.
We significantly improve both these bounds and settle the complexity of model checking for finitary diagrams. As our first main result, we show that there is an efficient randomized algorithm for ETIM. Combining this with the previous work by Dubut, we obtain an NEXP upper bound for bisimilarity of finitary diagrams and an NP upper bound for diagrammatic path logic. We also provide a matching NP-hardness proof for the latter. The hardness proof introduces constrained layered poset problems, which may be of independent interest, and connects them to finitary diagrams using Gabriel’s theorem for representations of path quivers. For bisimilarity over finite fields, we further improve the upper bound to PSPACE. In ETIM, we quantify over invertible matrices. We finally ask what happens if we instead quantify over matrices from the special linear group, that is, of determinant one. We show that in this case, the resulting fragment is equivalent to the existential theory of the reals, under a mild generalization of the allowed linear constraints.
Cite as
Markus Bläser, Sagnik Dutta, and Samuel Okyay. The Complexity of Bisimilarity and Model Checking in Finitary Diagrams. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 166:1-166:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blaser_et_al:LIPIcs.ICALP.2026.166,
author = {Bl\"{a}ser, Markus and Dutta, Sagnik and Okyay, Samuel},
title = {{The Complexity of Bisimilarity and Model Checking in Finitary Diagrams}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {166:1--166:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.166},
URN = {urn:nbn:de:0030-drops-265548},
doi = {10.4230/LIPIcs.ICALP.2026.166},
annote = {Keywords: Bisimilarity, Finitary diagrams, Model checking, Existential Theory of the Reals}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Population Protocols over Ordered Agents
Abstract
Population protocols are a distributed computation model in which a collection of anonymous, finite-state agents interact in randomly chosen pairs and update their states according to a fixed transition function. The computation is defined by the eventual stabilization of the population to a consensus that represents the output. In practice, it is natural to allow each agent to carry a unique identifier and compare it with that of another agent before interacting. We model this extension by having agents be totally ordered and interactions between two agents to be fireable only if their pair of identifiers falls in some condition set. For instance, PP[<] allows for two agents to interact only if the first one appears before the second one.
We study population protocols over ordered agents PP[𝒩] where 𝒩 is a set of predicates available to restrict transition firing. We also study IO-PP[𝒩], the immediate observation fragment of PP[𝒩] where only one agent changes state per interaction. Our main result is that IO-PP[<] recognizes exactly the unambiguous star-free languages, which admits many other characterizations, such as two-variable first-order logic or two-way deterministic partially-ordered automata. We also provide a logic and an automaton model that fits in PP[<]. We further show that if the successor predicate appears in a set 𝒩 of NSPACE(n)-computable predicates, then IO-PP[𝒩] = PP[𝒩] = NSPACE(n). Finally, we investigate the problem of deciding whether a given population protocol always stabilizes to a consensus. While this problem is decidable for unordered population protocols, we show that this is undecidable already for PP[<] and IO-PP[+1], but conditionally decidable for IO-PP[<].
Cite as
Michael Blondin, Michaël Cadilhac, Benjamin Courchesne, Lucie Guillou, Corto Mascle, and Isa Vialard. Population Protocols over Ordered Agents. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 167:1-167:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blondin_et_al:LIPIcs.ICALP.2026.167,
author = {Blondin, Michael and Cadilhac, Micha\"{e}l and Courchesne, Benjamin and Guillou, Lucie and Mascle, Corto and Vialard, Isa},
title = {{Population Protocols over Ordered Agents}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {167:1--167:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.167},
URN = {urn:nbn:de:0030-drops-265557},
doi = {10.4230/LIPIcs.ICALP.2026.167},
annote = {Keywords: Population protocols, First-order logic, Partially-ordered automata, Unambiguous star-free languages}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
The Network Satisfaction Problem for Relation Algebras with at Most 4 Atoms
Abstract
Andréka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable [Hajnal Andréka and Roger D. Maddux, 1994]. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard [Matteo Cristiani and Robin Hirsch, 2004]. The literature contains many results on representations of relation algebras; in particular, some relation algebras with four atoms are not representable. We extend the result of Cristiani and Hirsch to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.
Cite as
Manuel Bodirsky, Moritz Jahn, Simon Knäuer, Matěj Konečný, and Paul Winkler. The Network Satisfaction Problem for Relation Algebras with at Most 4 Atoms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 168:1-168:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2026.168,
author = {Bodirsky, Manuel and Jahn, Moritz and Kn\"{a}uer, Simon and Kone\v{c}n\'{y}, Mat\v{e}j and Winkler, Paul},
title = {{The Network Satisfaction Problem for Relation Algebras with at Most 4 Atoms}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {168:1--168:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.168},
URN = {urn:nbn:de:0030-drops-265564},
doi = {10.4230/LIPIcs.ICALP.2026.168},
annote = {Keywords: Constraint Satisfaction, Computational Complexity, Relation Algebras, Network Satisfaction, Normal Representations, Polynomial-Time Algorithms}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
The Complexity of Finding Coset-Generating Polymorphisms and the Promise Metaproblem
Abstract
We show that the metaproblem for coset-generating polymorphisms is NP-complete, answering a question of Chen and Larose: given a finite structure, the computational question is whether this structure has a polymorphism of the form (x,y,z) ↦ x y^{-1} z with respect to some group; such operations are also called coset-generating, or heaps.
Furthermore, we introduce a promise version of the metaproblem, parametrised by two polymorphism conditions Σ₁ and Σ₂ and defined analogously to the promise constraint satisfaction problem. We give sufficient conditions under which the promise metaproblem for (Σ₁,Σ₂) is in 𝖯 and under which it is NP-hard. In particular, the promise metaproblem is in 𝖯 if Σ₁ states the existence of a Maltsev polymorphism and Σ₂ states the existence of an abelian heap polymorphism - despite the fact that neither the metaproblem for Σ₁ nor the metaproblem for Σ₂ is known to be in 𝖯. We also show that the creation-metaproblem for Maltsev polymorphisms, under the promise that a heap polymorphism exists, is in 𝖯 if and only if there is a uniform polynomial-time algorithm for CSPs with a heap polymorphism.
Cite as
Manuel Bodirsky and Armin Weiß. The Complexity of Finding Coset-Generating Polymorphisms and the Promise Metaproblem. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 169:1-169:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2026.169,
author = {Bodirsky, Manuel and Wei{\ss}, Armin},
title = {{The Complexity of Finding Coset-Generating Polymorphisms and the Promise Metaproblem}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {169:1--169:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.169},
URN = {urn:nbn:de:0030-drops-265574},
doi = {10.4230/LIPIcs.ICALP.2026.169},
annote = {Keywords: constraint satisfaction problem, coset-generating polymorphisms, metaproblem, heap, abelian heap, uniform polynomial-time algorithm, NP-hardness}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Transducers on Compressed Strings
Abstract
We study string-to-string functions which are compatible with compression in the following sense: given a compressed representation of an input string, one can compute in polynomial time a compressed representation of the output string. As the compression formalism, we use straight-line programs (i.e. context-free grammars that produce only one string). As the functions, we use finite state transducers, with a focus on the regular and polyregular functions. We show that all regular functions are compatible with compression, but this is no longer true for the polyregular functions. We identify a subclass of the polyregular functions - the so-called rectangular polyregular functions - which is compatible with compression, and we characterise this subclass in terms of a functional programming language.
Cite as
Mikołaj Bojańczyk and Markus Lohrey. Transducers on Compressed Strings. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 170:1-170:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bojanczyk_et_al:LIPIcs.ICALP.2026.170,
author = {Boja\'{n}czyk, Miko{\l}aj and Lohrey, Markus},
title = {{Transducers on Compressed Strings}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {170:1--170:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.170},
URN = {urn:nbn:de:0030-drops-265580},
doi = {10.4230/LIPIcs.ICALP.2026.170},
annote = {Keywords: polyregular functions, grammar compression}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Multi-Environment MDPs with Prior and Universal Semantics
Abstract
Multiple-environment Markov decision processes (MEMDPs) equip an MDP with several probabilistic transition functions (one per possible environment) so that the state is observable but the environment is not. Previous work studies two semantics: (i) the universal semantics, where an adversary picks the environment; and (ii) the prior semantics, where the environment is drawn once before execution from a fixed distribution. We clarify the relation between these semantics. For parity objectives, we show that the qualitative questions, i.e. value one, coincide, and we develop a new algorithm for the general value of MEMDP with prior semantics. In particular, we show that the prior value of an MEMDP with a parity objective can be approximated to any precision with a space efficient algorithm; equivalently, the associated gap problem is decidable in PSPACE when probabilities are given in unary (and in EXPSPACE otherwise). We then prove that the universal value equals the infimum of prior values over all beliefs. This yields a new algorithm for the universal gap problem with the same complexity (PSPACE for unary probabilities, EXPSPACE in general), improving on earlier doubly-exponential-space procedures. Finally, we observe that MEMDPs under the prior semantics form an important tractable subclass of POMDPs: our algorithms exploit the fact that belief entropy never increases, and we establish that any POMDP with this property reduces effectively to a prior-MEMDP, showing that prior-MEMDPs capture a broad and practically relevant subclass of POMDPs.
Cite as
Benjamin Bordais and Jean-François Raskin. Multi-Environment MDPs with Prior and Universal Semantics. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 171:1-171:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bordais_et_al:LIPIcs.ICALP.2026.171,
author = {Bordais, Benjamin and Raskin, Jean-Fran\c{c}ois},
title = {{Multi-Environment MDPs with Prior and Universal Semantics}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {171:1--171:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.171},
URN = {urn:nbn:de:0030-drops-265592},
doi = {10.4230/LIPIcs.ICALP.2026.171},
annote = {Keywords: Multi-Environement MDP, approximation algorithm, Partially-observable MDP}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Preservation Theorems in Semiring Semantics
Abstract
We study the status of classical model-theoretic preservation theorems such as the Łoś-Tarski theorem and the homomorphism preservation theorem in the context of semiring semantics.
Semiring semantics has its origins in the provenance analysis of database queries but has been extended to a systematic way of evaluating logical statements to values in a commutative semiring. Depending on the underlying semiring, this allows us to track descriptions of the atomic facts that are responsible for the truth of a statement or practical information about the evaluation such as costs or confidence. The systematic development of semiring semantics for first-order logic and other logical systems raises the question to what extent classical model-theoretic results can be generalised to this setting and how such results depend on the underlying semiring.
The definitions of semantic properties such as preservation under extensions, substructures, or homomorphisms naturally generalise to the setting of semiring semantics. However, the status of the corresponding preservation theorem strongly depends on the algebraic properties of the particular semirings. We prove that these preservation theorems do indeed hold for all lattice semirings (a quite large class, encompassing practically relevant semirings and in particular all min-max semirings). The proofs combine adaptations of the classical compactness and amalgamation methods with specific reduction methods for logical entailment that have been developed in semiring semantics. On the other side, variants of the existential preservation theorem fail for many other semirings, including the tropical semiring, the Viterbi semiring, the Łukasiewicz semiring, and the natural semirings ℕ and ℕ^∞. Surprisingly, the existential preservation theorem does hold for finite interpretations in a number of semirings, including the three-element min-max semiring, which extends the Boolean by just a single additional truth value. Thus, the situation for these semirings is in sharp contrast to the Boolean case, where the Łoś-Tarski theorem holds in general, but not in the finite.
Cite as
Sophie Brinke, Anuj Dawar, Erich Grädel, and Benedikt Pago. Preservation Theorems in Semiring Semantics. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 172:1-172:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brinke_et_al:LIPIcs.ICALP.2026.172,
author = {Brinke, Sophie and Dawar, Anuj and Gr\"{a}del, Erich and Pago, Benedikt},
title = {{Preservation Theorems in Semiring Semantics}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {172:1--172:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.172},
URN = {urn:nbn:de:0030-drops-265605},
doi = {10.4230/LIPIcs.ICALP.2026.172},
annote = {Keywords: Semiring semantics, preservation theorems, model theory, algebraic logics}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Stone Duality Proofs for Colorless Distributed Computability Theorems
Abstract
Twenty years ago, Herlihy/Shavit and Saks/Zaharoglou won the Gödel prize for the introduction of a simplicial semantics for distributed computing. This line of work culminated in a characterization of the distributed tasks which can be solved by asynchronous wait-free systems, resulting in the Asynchronous Computability Theorem (ACT). In this paper, we extend this semantics by identifying spectral topology as the natural generalization of the finite combinatorial topology they employed. In particular, we extend the topological approach of ACT to any round-based, content-neutral, full-information protocol. This family of protocols contains the Iterated Immediate Snapshot model (IIS), to which many distributed computation models can be reduced. In this sense, our work provides first steps towards a unified topological framework for distributed computing.
The main insight of this work is in considering global states obtained after finite executions of a distributed protocol not as abstract simplicial complexes as was previously done, but as finite spectral spaces, considering the Alexandrov topology on the associated face posets. Using this point-set topological approach, coupled with the interpretation of a distributed protocol as an endofunctor Π on the category of simplicial complexes, we show that any initial configuration ℐ can be associated to a projective limit system of finite complexes. The limit thereof is a spectral space Π^∞(ℐ) which precisely encodes the behavior of the protocol presented by Π. This leads us to derive a new general distributed computability theorem using Stone duality: a protocol Π solves a colorless task (ℐ,𝒪,Δ) if and only if there exists a spectral map f:Π^∞(ℐ) → 𝒪 compatible with Δ.
From this general characterization, we derive known colorless computability theorems, and provide new insights into the previously established connection between task-solvability and continuous maps between geometric realizations. This is achieved through Stone duality, a well established tool for such tight correspondences in computer science.
Cite as
Cameron Calk and Emmanuel Godard. Stone Duality Proofs for Colorless Distributed Computability Theorems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 173:1-173:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{calk_et_al:LIPIcs.ICALP.2026.173,
author = {Calk, Cameron and Godard, Emmanuel},
title = {{Stone Duality Proofs for Colorless Distributed Computability Theorems}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {173:1--173:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.173},
URN = {urn:nbn:de:0030-drops-265615},
doi = {10.4230/LIPIcs.ICALP.2026.173},
annote = {Keywords: simplicial complex, partial order, spectral spaces, Stone duality, categorical semantics}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
The Role of Counting Quantifiers in Laminar Set Systems
Abstract
Laminar set systems consist of non-crossing subsets of a universe with set inclusion essentially corresponding to the descendant relationship of a tree, the so-called laminar tree. Laminar set systems lie at the core of many graph decompositions such as modular decompositions, split decompositions, and bi-join decompositions. We show that from a laminar set system we can obtain the corresponding laminar tree by means of a monadic second order logic (MSO) transduction. This resolves an open question originally asked by Courcelle and is a satisfying resolution as MSO is the natural logic for set systems and is sufficient to define the property "laminar". Using results from Campbell et al. [STACS 2025], we can now obtain transductions for obtaining modular decompositions, co-trees, split decompositions and bi-join decompositions using MSO instead of CMSO. We further gain some insight into the expressive power of counting quantifiers and provide some results towards determining when counting quantifiers can be simulated in MSO in laminar set systems and when they cannot.
Cite as
Rutger Campbell and Noleen Köhler. The Role of Counting Quantifiers in Laminar Set Systems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 174:1-174:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{campbell_et_al:LIPIcs.ICALP.2026.174,
author = {Campbell, Rutger and K\"{o}hler, Noleen},
title = {{The Role of Counting Quantifiers in Laminar Set Systems}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {174:1--174:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.174},
URN = {urn:nbn:de:0030-drops-265625},
doi = {10.4230/LIPIcs.ICALP.2026.174},
annote = {Keywords: MSO-transductions, simulating counting quantifiers, laminar set systems, graph decompositions}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Loop Termination and Generalized Collatz Sequences
Abstract
Linear-constraint loops are programs whose transition relation is specified by a system of linear inequalities. The termination problem asks, given a loop, whether it admits an infinite computation. Decidability of termination remains open for linear-constraint loops over integers, rationals, and reals. We focus on loops over integers and show that they are tightly connected to generalized Collatz sequences - integer sequences generated by maps that are linear on each residue class modulo a fixed natural number. We prove that termination of one-variable linear-constraint loops is decidable in polynomial time, provided a long-standing conjecture about generalized Collatz sequences holds. Conversely, we show that any decision procedure for one-variable loops would prove or refute specific instances of this conjecture, which remain open. Moreover, we show that if a one-variable loop has a cyclic trace, then it also has a cyclic trace of length at most two.
Cite as
Mishel Carelli. Loop Termination and Generalized Collatz Sequences. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 175:1-175:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{carelli:LIPIcs.ICALP.2026.175,
author = {Carelli, Mishel},
title = {{Loop Termination and Generalized Collatz Sequences}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {175:1--175:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.175},
URN = {urn:nbn:de:0030-drops-265635},
doi = {10.4230/LIPIcs.ICALP.2026.175},
annote = {Keywords: Program Verification, Loop Termination, Generalized Collatz Sequences, Linear-Constraint Loops}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Expregular Functions
Abstract
Polyregular functions form a robust class of string-to-string functions with polynomial growth, as evidenced by Bojańczyk (2018). This class admits numerous descriptions and enjoys several closure properties. Most notably, polyregular functions are regularity reflecting (i.e. the inverse image of a regular language is regular).
In this work, we propose a robust class of string-to-string functions with exponential growth which we call expregular functions. We consider the following three models for describing them:
- MSO set interpretations, which extend MSO interpretations (one of the models capturing polyregular functions), by operating on monadic variables instead of tuples of first-order variables;
- yield-Hennie machines, which are branching one-tape Turing machines with bounded visit; and
- Ariadne transducers, a new model of 2-way pushdown machines with a bounded visit restriction. Our main contribution is a translation from MSO set interpretations to yield-Hennie machines, which are known to be regularity reflecting (Dartois, Nguy~ên, Peyrat 2026). In particular this establishes that MSO set interpretations are regularity reflecting, which in turn settles a major conjecture about automatic structures: every automatic ω-word has a decidable MSO theory.
Yield-Hennie machine directly translate to Ariadne transducers, and our second contribution is to prove that Ariadne transducers also translate to MSO set interpretations, thus establishing the equivalence of the three models. This is obtained by showing that that Ariadne automata - the automaton model corresponding to Ariadne transducers - recognise regular languages.
Cite as
Thomas Colcombet, Nathan Lhote, and Pierre Ohlmann. Expregular Functions. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 176:1-176:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{colcombet_et_al:LIPIcs.ICALP.2026.176,
author = {Colcombet, Thomas and Lhote, Nathan and Ohlmann, Pierre},
title = {{Expregular Functions}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {176:1--176:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.176},
URN = {urn:nbn:de:0030-drops-265647},
doi = {10.4230/LIPIcs.ICALP.2026.176},
annote = {Keywords: monadic second-order logic, exponential growth, automatic structures}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Exploring VASS Parameterised by Geometric Dimension
Abstract
The geometric dimension g of a Vector Addition System with States (VASS) is the dimension of the vector space generated by cycles in the VASS; this parameter refines the standard dimension d, the number of counters. Recently, it was discovered that the fastest-known algorithm for solving the reachability problem for VASS has the same complexity in terms of g as in terms of d. This suggests that the geometric dimension may in fact be a more adequate parameter for measuring the complexity of VASS reachability problems. We initiate a more systematic study of the geometric dimension. We discuss differences between two parameters: the geometric dimension and the SCC dimension. Our main technical result states that classical results about the coverability and boundedness problems can be improved from dimension d to geometric dimension g. Namely, coverability is witnessed by runs of length n^{2^𝒪(g)} instead of n^{2^𝒪(d)}, and unboundedness can be witnessed by runs of length n^{2^𝒪(g log g)} instead of n^{2^𝒪(d log d)}, where n is the size of the instance. We also study integer reachability and simultaneous unboundedness in VASS parameterised by the geometric dimension.
Cite as
Wojciech Czerwiński, Roland Guttenberg, Łukasz Orlikowski, Henry Sinclair-Banks, and Yangluo Zheng. Exploring VASS Parameterised by Geometric Dimension. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 177:1-177:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{czerwinski_et_al:LIPIcs.ICALP.2026.177,
author = {Czerwi\'{n}ski, Wojciech and Guttenberg, Roland and Orlikowski, {\L}ukasz and Sinclair-Banks, Henry and Zheng, Yangluo},
title = {{Exploring VASS Parameterised by Geometric Dimension}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {177:1--177:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.177},
URN = {urn:nbn:de:0030-drops-265655},
doi = {10.4230/LIPIcs.ICALP.2026.177},
annote = {Keywords: vector addition systems, Petri nets, geometric dimensions, coverability problem, integer reachability problem, simultaneous unboundedness, reachability problem}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Asymptotic Hausdorff and Language Similarity
Abstract
We introduce the Asymptotic Hausdorff lifting, denoted AH_d, a general method for lifting an element-level metric d to a (pseudo-) metric on sets, that captures asymptotic similarity in infinite domains equipped with a notion of size. The construction is designed to be insensitive to finite deviations and to avoid the limitations of classical Hausdorff-based approaches, which are often overly sensitive to outliers and fail to reflect asymptotic behavior.
Formal languages provide a central motivating instance of this framework, where elements are words and sets are languages. When applied to normalized edit distances, the Asymptotic Hausdorff lifting yields metric-valued distances between languages that reflect asymptotic edit behavior while preserving metric structure. We study the equivalence classes of regular languages induced by AH_d for normalized edit distances d, and characterize their asymptotic essence. Focusing in particular on the normalized edit distance of Marzal and Vidal, ned, we investigate the computation of AH_ned for regular languages and for bounded context-free languages.
Cite as
Dana Fisman and Gal Meirom. Asymptotic Hausdorff and Language Similarity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 178:1-178:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fisman_et_al:LIPIcs.ICALP.2026.178,
author = {Fisman, Dana and Meirom, Gal},
title = {{Asymptotic Hausdorff and Language Similarity}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {178:1--178:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.178},
URN = {urn:nbn:de:0030-drops-265660},
doi = {10.4230/LIPIcs.ICALP.2026.178},
annote = {Keywords: Automata theory, formal Languages, Metric Spaces, Language similarity, Edit Distance, asymptotic Analysis}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Transducing Linear Decompositions of Tournaments
Abstract
Bojańczyk, Pilipczuk, and Grohe [LICS '18] proved that for graphs of bounded linear clique-width, clique-width decompositions of small width can be produced by a CMSO transduction. We show that in the case of tournaments, a first-order transduction suffices. This implies that the logics CMSO and existential MSO are equivalent over bounded linear clique-width tournaments.
Cite as
Colin Geniet, Fatemeh Ghasemi, and Mamadou Moustapha Kanté. Transducing Linear Decompositions of Tournaments. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 179:1-179:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{geniet_et_al:LIPIcs.ICALP.2026.179,
author = {Geniet, Colin and Ghasemi, Fatemeh and Kant\'{e}, Mamadou Moustapha},
title = {{Transducing Linear Decompositions of Tournaments}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {179:1--179:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.179},
URN = {urn:nbn:de:0030-drops-265670},
doi = {10.4230/LIPIcs.ICALP.2026.179},
annote = {Keywords: Clique-width, definable decompositions, tournaments}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Optimally Controlling a Random Population
Abstract
The population control problem is a parameterised problem where a controller sends messages to a whole population of identical finite-state agents, aiming to eventually move them all into a target state. The decision problem asks whether this can be achieved for arbitrarily large finite populations. We focus on the randomised version of this problem, where every agent is a copy of the same finite Markov Decision Process and non-determinism in the global action chosen by the controller is resolved independently and uniformly at random. Colcombet, Fijalkow and Ohlmann [Thomas Colcombet et al., 2021] showed that this problem is decidable, but without any complexity upper bound. We show that the random population control problem is in fact ExpTime-complete.
Cite as
Hugo Gimbert, Corto Mascle, and Patrick Totzke. Optimally Controlling a Random Population. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 180:1-180:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gimbert_et_al:LIPIcs.ICALP.2026.180,
author = {Gimbert, Hugo and Mascle, Corto and Totzke, Patrick},
title = {{Optimally Controlling a Random Population}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {180:1--180:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.180},
URN = {urn:nbn:de:0030-drops-265685},
doi = {10.4230/LIPIcs.ICALP.2026.180},
annote = {Keywords: Controller synthesis, Parameterized verification}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Set Automata and Limits of Decidability of Two-Variable Logic on Data Words
Abstract
We extend the two-variable logic on data words [Bojańczyk et al., 2011] with guarded regular binary predicates of the form L̃(x,y) that is true if positions x and y have the same data value and the factor strictly between x and y is in the regular language L. We characterise the class of aperiodic monoids for which the extension of the two-variable logic with guarded predicates recognised by the monoid is decidable, namely the class of idempotent monoids whose two-sided ideals are linearly ordered, called linear bands. For this, we introduce an automata formalism, set automata, that is equivalent to the class automata of Bojańczyk and Lasota and thus has an undecidable emptiness problem. The set updates used in the automaton form a semigroup of relations. We identify a subclass of set automata called ordered quasi-normal set automata that has a decidable emptiness problem by reduction to the emptiness problem of ordered multicounter automata. We show that the two-variable logic extended with guarded regular predicates recognised by a monoid M is expressively equivalent to a quasi-normal set automaton with the monoid of relations M. In particular, if M is a linear band then the resulting automaton is ordered, and the decidability result follows.
Cite as
Shibashis Guha, Amaldev Manuel, and S P Rishal. Set Automata and Limits of Decidability of Two-Variable Logic on Data Words. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 181:1-181:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{guha_et_al:LIPIcs.ICALP.2026.181,
author = {Guha, Shibashis and Manuel, Amaldev and Rishal, S P},
title = {{Set Automata and Limits of Decidability of Two-Variable Logic on Data Words}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {181:1--181:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.181},
URN = {urn:nbn:de:0030-drops-265696},
doi = {10.4230/LIPIcs.ICALP.2026.181},
annote = {Keywords: Data words, Two-variable FO, Class automata, Finite semigroups, Decidability}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Witnesses for Fixpoint Games on Lattices
Abstract
We construct witnesses that can be used to derive strategies in fixpoint games and provide proof that the least fixpoint of a function is either above or not below some given bound. We rely on a lattice-theoretical approach, including a Galois connection that connects a lattice representing the "logic universe", where the witness lives, with another lattice representing the "behaviour universe", over which the function is defined. In fact we consider two types of games - primal and dual games - and in both cases show how to derive winning strategies in the game from witnesses and construct witnesses from strategies. The two games differ wrt. their rules and the choice of basis of the lattice.
The theory can be instantiated to well-known examples: in particular we compare with the construction of distinguishing formulas in standard bisimilarity and behavioural metrics for probabilistic systems. As a new case study we consider witnesses for certifying lower bounds for the termination probability for Markov chains.
Cite as
Barbara König and Karla Messing. Witnesses for Fixpoint Games on Lattices. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 182:1-182:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{konig_et_al:LIPIcs.ICALP.2026.182,
author = {K\"{o}nig, Barbara and Messing, Karla},
title = {{Witnesses for Fixpoint Games on Lattices}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {182:1--182:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.182},
URN = {urn:nbn:de:0030-drops-265709},
doi = {10.4230/LIPIcs.ICALP.2026.182},
annote = {Keywords: fixpoint games, lattice theory, witnesses, bisimilarity, Galois connections, Hennessy-Milner theorem, distinguishing formulas, behavioural metrics}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
On the Constructive Dimension Spectrum of Polynomials
Abstract
Recently, Stull [Stull, 2025], [Stull, 2022] resolved a long-standing open problem posed by Lutz, on whether the set of effective Hausdorff dimensions of points on a straight line in ℝ² - the effective dimension spectrum of the line - contains a unit interval. This question is related to problems in classical fractal geometry like the Kakeya conjecture and Furstenberg sets. Stull posed an open question on the dimension spectra of polynomial curves.
For the first result, with new techniques which adapt the theory of classical real root-finding of polynomials to the current setting, we show that the dimension spectra of every polynomial curve contains at least two points. This answers an open question posed by Stull [Stull, 2025], [Stull, 2022]. We use the main result to construct a class of polynomials which have width strictly greater than 1, answering a second problem stated in [Stull, 2025], [Stull, 2022].
Stull [Stull, 2025] resolved the dimension spectrum conjecture for planar lines, showing that it contains a unit interval. For the second result, we resolve the conjecture for a subfamily of polynomials whose coefficients form a "low" dimension point in ℝ^{d+1}.
Cite as
Prajval Koul and Satyadev Nandakumar. On the Constructive Dimension Spectrum of Polynomials. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 183:1-183:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{koul_et_al:LIPIcs.ICALP.2026.183,
author = {Koul, Prajval and Nandakumar, Satyadev},
title = {{On the Constructive Dimension Spectrum of Polynomials}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {183:1--183:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.183},
URN = {urn:nbn:de:0030-drops-265717},
doi = {10.4230/LIPIcs.ICALP.2026.183},
annote = {Keywords: Kolmogorov Complexity, Dimension, Polynomials}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Approximating 1-In-3 SAT by Linearly Ordered Hypergraph 3-Colouring Is NP-Hard
Abstract
1-in-3 SAT is a classical NP-hard constraint satisfaction problem (CSP). Given a satisfiable instance of 1-in-3 SAT, it is NP-hard to find a satisfying assignment for it, but it may be possible to efficiently find a solution subject to a weaker (not necessarily Boolean) predicate than "1-in-3". There is a conjecture, which we call the Approximate 1-in-3 SAT conjecture, made independently by several researchers, that predicts a dichotomy: for certain choices of weaker predicates the problem becomes tractable and for the remaining choices the task remains NP-hard. Such problems belong to the Promise CSP (PCSP) framework, which studies how one CSP can be approximated by another, in a specific qualitative sense. The Approximate 1-in-3 SAT conjecture is notable because there is no P versus NP-hard dichotomy conjecture for general PCSPs yet (due to insufficient evidence). One specific predicate, corresponding to the problem of linearly ordered 3-colouring of 3-uniform hypergraphs, has been mentioned in several recent papers as an obstacle to further progress in proving the Approximate 1-in-3 SAT conjecture. We prove that the problem for this predicate is NP-hard, as predicted by the conjecture. This completes the proof of the conjecture for predicates on a 3-element domain.
Cite as
Andrei Krokhin and Danny Vagnozzi. Approximating 1-In-3 SAT by Linearly Ordered Hypergraph 3-Colouring Is NP-Hard. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 184:1-184:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{krokhin_et_al:LIPIcs.ICALP.2026.184,
author = {Krokhin, Andrei and Vagnozzi, Danny},
title = {{Approximating 1-In-3 SAT by Linearly Ordered Hypergraph 3-Colouring Is NP-Hard}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {184:1--184:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.184},
URN = {urn:nbn:de:0030-drops-265729},
doi = {10.4230/LIPIcs.ICALP.2026.184},
annote = {Keywords: Constraint satisfaction, complexity theory}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Persistent Amortised Analysis, Operationally
Abstract
Amortised analysis is a technique for proving a combined time bound for a batch of operations on a data structure, even if some of those operations are expensive. But the traditional method of amortised analysis yields incorrect time bounds when the data structure is used persistently. Persistence allows operations to be performed on previous versions of the data structure, which prevents us from amortising expensive restructuring work. In his seminal book, Chris Okasaki showed how to extend amortised analysis to persistent usage. His method works by extending the data structure with thunks and performing the analysis with debits rather than credits. His argument, that credits are unsound for analysing persistent usage, has become folklore.
In this paper, we provide a new perspective on the role of debits in Okasaki’s work. First, we set up an operational semantics of call-by-value lambda calculus with thunks, and show formally that traditional amortised analysis does not work in a persistent setting. Then we show that, contrary to the folklore, amortised analysis in a persistent setting can be performed purely in terms of credits without using debits at all. Finally, we provide a formal semantics for Okasaki’s original debit-based approach.
Cite as
Anton Lorenzen. Persistent Amortised Analysis, Operationally. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 185:1-185:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lorenzen:LIPIcs.ICALP.2026.185,
author = {Lorenzen, Anton},
title = {{Persistent Amortised Analysis, Operationally}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {185:1--185:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.185},
URN = {urn:nbn:de:0030-drops-265736},
doi = {10.4230/LIPIcs.ICALP.2026.185},
annote = {Keywords: Lazy Data Structures, Amortised Analysis}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Edit Distance of Finite-Valued Transducers
Abstract
Transducers generalise automata by producing output word(s) for each input word, thereby defining a relation over words. A transducer is said to be finite-valued if, for every input word, it produces at most k output words, for some constant k. If k = 1, then the transducer is said to be functional. The edit distance between two transducers is the minimal number of edits required to transform every output of one transducer into some output of the other, for each input word. This notion has been studied for functional transducers, where it is shown to be computable. However, it is uncomputable for transducers in general. In this work, we show the computability of the edit distance of finite-valued transducers, a class that is strictly more expressive than functional transducers.
Cite as
Prince Mathew and Saina Sunny. Edit Distance of Finite-Valued Transducers. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 186:1-186:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mathew_et_al:LIPIcs.ICALP.2026.186,
author = {Mathew, Prince and Sunny, Saina},
title = {{Edit Distance of Finite-Valued Transducers}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {186:1--186:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.186},
URN = {urn:nbn:de:0030-drops-265752},
doi = {10.4230/LIPIcs.ICALP.2026.186},
annote = {Keywords: Edit distance, Finite state transducers, Rational relations, Finite-valued, Multi-sequential}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Optimal Lower Bounds for Symmetric Modular Circuits
Abstract
A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. Håstad’s celebrated switching lemma yields exponential lower bounds for the dual problem - realising modular arithmetic with Boolean gates - but, a similar lower bound for modular circuits computing the Boolean AND function has remained elusive for almost 30 years.
We solve this problem for the restricted model of symmetric circuits: We consider MOD_m-circuits of arbitrary depth, and for an arbitrary modulus m ∈ ℕ, and obtain subexponential lower bounds for computing the n-ary Boolean AND function, under the assumption that the circuits are syntactically symmetric under all permutations of their n input gates. This lower bound is matched precisely by a construction due to (Idziak, Kawałek, Krzaczkowski, LICS'22), leading to the surprising conclusion that the optimal symmetric circuit size is already achieved with depth 2.
Motivated by another construction from (LICS'22), which achieves smaller size at the cost of greater depth, we also prove tight size lower bounds for circuits with a more liberal notion of symmetry characterised by a nested block structure on the input variables.
Cite as
Benedikt Pago. Optimal Lower Bounds for Symmetric Modular Circuits. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 187:1-187:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{pago:LIPIcs.ICALP.2026.187,
author = {Pago, Benedikt},
title = {{Optimal Lower Bounds for Symmetric Modular Circuits}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {187:1--187:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.187},
URN = {urn:nbn:de:0030-drops-265767},
doi = {10.4230/LIPIcs.ICALP.2026.187},
annote = {Keywords: symmetric circuits, modular counting, lower bounds, CC⁰}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Scoped MSO, Register Automata, and Expressions: Equivalence over Data Words
Abstract
This paper establishes logical and expression-based characterizations of the class of languages recognized by nondeterministic register automata with guessing (NRA) over infinite alphabets. We introduce Scoped MSO, a logic featuring a novel segment modality and syntactic restrictions on data comparisons. We prove this logic is expressively equivalent to NRA over data domains where "strong guessing" can be eliminated. Furthermore, we define Data Regular Expressions, a minimalist regular expression calculus built from quantifier-free regions and equipped with k-contracting concatenation, and demonstrate its equivalence to NRA over arbitrary relational structures. Together, these formalisms and the translations between them establish a robust correspondence between register automata, logic, and expressions over data words.
Cite as
Radosław Piórkowski. Scoped MSO, Register Automata, and Expressions: Equivalence over Data Words. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 188:1-188:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{piorkowski:LIPIcs.ICALP.2026.188,
author = {Pi\'{o}rkowski, Rados{\l}aw},
title = {{Scoped MSO, Register Automata, and Expressions: Equivalence over Data Words}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {188:1--188:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.188},
URN = {urn:nbn:de:0030-drops-265775},
doi = {10.4230/LIPIcs.ICALP.2026.188},
annote = {Keywords: register automata, data words, monadic second-order logic, infinite alphabets, Kleene theorem, nominal sets}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
From Sets to Points: Simplifying MSO Interpretations via Reparameterizations
Abstract
We study the conditions under which monadic second-order (MSO) interpretations can be simplified by replacing representations of elements as tuples of arbitrary sets with representations as tuples of finite sets or points. Using reparameterizations of MSO formulas, we prove that for formulas with free finite-set variables, it is decidable whether a point reparameterization exists, and that such a reparameterization can be effectively constructed over countable chains. Moreover, over countable Dedekind-complete labeled chains, a formula with free arbitrary set variables admits a finite-set reparameterization if and only if it has at most countably many satisfying assignments. These results yield effective simplification procedures for MSO interpretations over broad classes of countable linear orders.
Cite as
Alexander Rabinovich. From Sets to Points: Simplifying MSO Interpretations via Reparameterizations. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 189:1-189:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rabinovich:LIPIcs.ICALP.2026.189,
author = {Rabinovich, Alexander},
title = {{From Sets to Points: Simplifying MSO Interpretations via Reparameterizations}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {189:1--189:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.189},
URN = {urn:nbn:de:0030-drops-265789},
doi = {10.4230/LIPIcs.ICALP.2026.189},
annote = {Keywords: Monadic Second-Order Logic, MSO interpretation, interpretation simplification, reparameterization, decidability}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
A Diagrammatic Axiomatisation of Behavioural Distance of Nondeterministic Processes
Abstract
Behavioural distances provide a quantitative approach to comparing the states of transition systems, moving beyond traditional Boolean notions of equivalence. In this paper, we develop a sound and complete axiomatisation of behavioural distance for nondeterministic processes using Milner’s charts, a model that generalises finite-state automata by incorporating variable outputs. Charts provide a compelling setting for studying behavioural distances because they shift the focus from language equivalence to bisimilarity. Their axiomatic study lays the groundwork for quantitative analysis of more expressive models, such as weighted transition systems.
To formalise this approach, we adopt string diagrams as our syntax of choice. String diagrams closely mirror the graphical structure of charts, while providing a rigorous formalism that supports inductive reasoning and compositional semantics. Unlike traditional algebraic syntaxes, which require additional mechanisms such as binders and substitution, string diagrams offer a variable-free representation where recursion naturally decomposes into simpler components. This makes them well-suited for reasoning about behavioural distances and aligns with broader efforts to axiomatise automata-theoretic equivalences through a unified diagrammatic framework.
Cite as
Wojciech Różowski, Robin Piedeleu, Alexandra Silva, and Fabio Zanasi. A Diagrammatic Axiomatisation of Behavioural Distance of Nondeterministic Processes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 190:1-190:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rozowski_et_al:LIPIcs.ICALP.2026.190,
author = {R\'{o}\.{z}owski, Wojciech and Piedeleu, Robin and Silva, Alexandra and Zanasi, Fabio},
title = {{A Diagrammatic Axiomatisation of Behavioural Distance of Nondeterministic Processes}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {190:1--190:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.190},
URN = {urn:nbn:de:0030-drops-265790},
doi = {10.4230/LIPIcs.ICALP.2026.190},
annote = {Keywords: behavioural distance, quantitative equational reasoning, string diagrams}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Infinite-State Games with Energy Objectives Beyond Counters
Abstract
In the theory of games on infinite-state arenas, there is a stark contrast between (i) recursion-based models such as pushdown systems and extensions on one hand, and (ii) counter-based models like vector addition systems with states (VASS) on the other. For pushdown systems and extensions, there is a rich variety of decidable and well-understood games, whereas on VASS arenas, even extremely simple games are undecidable. Here, a VASS is an automaton with counters that can be incremented and decremented, but not tested for zero. Crucially, the counters can only assume non-negative values.
However, certain VASS games become decidable when using energy semantics: An energy game is played on a system with counters, but the arena includes configurations with negative counters. The requirement that the counters stay non-negative is, instead, part of the winning condition of the existential player.
We study an analogue of energy semantics - legality of instructions as part of the winning condition rather than arena - on a broad class of infinite-state systems, where we call them viability games. Specifically, we study viability games in the framework of valence systems over graph monoids, where (undirected, loops allowed) graphs specify various infinite-state systems, such as pushdowns, VASS counters, integer counters, and combinations thereof.
In our main results, we provide a complete description of the decidability and complexity landscape of viability games across valence systems over graph monoids. Our results reveal encouraging decidability properties. For example, in certain combinations of pushdowns and counters, viability games are decidable, despite non-termination games being undecidable there. Moreover, viability games are even decidable for certain systems where (single-player) control-state reachability is undecidable.
Cite as
Irmak Sağlam and Georg Zetzsche. Infinite-State Games with Energy Objectives Beyond Counters. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 191:1-191:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{saglam_et_al:LIPIcs.ICALP.2026.191,
author = {Sa\u{g}lam, Irmak and Zetzsche, Georg},
title = {{Infinite-State Games with Energy Objectives Beyond Counters}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {191:1--191:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.191},
URN = {urn:nbn:de:0030-drops-265803},
doi = {10.4230/LIPIcs.ICALP.2026.191},
annote = {Keywords: Games on Graphs, Decidability, Complexity, Energy Games, Vector addition systems, Pushdown, Groups, Valence Systems}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Deciding DFA-Primality Is NP-Hard
Abstract
A DFA 𝒜 is composite if there exist DFAs 𝒜_1,…,𝒜_t with ℒ(𝒜) = ⋂_{i=1}^t ℒ(𝒜_i) such that each 𝒜_i has strictly less states than the minimal DFA deciding ℒ(𝒜). Otherwise, it is prime.
Prime-DFA is the problem of deciding primality for a given DFA. It was defined by Kupferman and Mosheiff in 2015 and it was shown to be NL-hard and in ExpSpace.
This paper proves the NP-hardness of Prime-DFA, thereby making the first progress in closing this doubly-exponential gap. It proves the NP-hardness by a reduction from the propositional logic satisfiability problem. The correctness of the reduction relies on an involved characterization of primality for a class of DFAs which contains those that can occur in the reduction.
Cite as
Daniel Alexander Spenner. Deciding DFA-Primality Is NP-Hard. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 192:1-192:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{spenner:LIPIcs.ICALP.2026.192,
author = {Spenner, Daniel Alexander},
title = {{Deciding DFA-Primality Is NP-Hard}},
booktitle = {53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
pages = {192:1--192:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-428-4},
ISSN = {1868-8969},
year = {2026},
volume = {374},
editor = {Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.192},
URN = {urn:nbn:de:0030-drops-265819},
doi = {10.4230/LIPIcs.ICALP.2026.192},
annote = {Keywords: Deterministic finite automaton (DFA), Regular languages, Finite languages, Decomposition, Primality, NP-Hardness}
}