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LIPIcs, Volume 364
43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)
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Leibniz International Proceedings in Informatics (LIPIcs)
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- published at: 2026-02-25
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-412-3
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Complete Volume
LIPIcs, Volume 364, STACS 2026, Complete Volume
Abstract
LIPIcs, Volume 364, STACS 2026, Complete Volume
Cite as
43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 1-1566, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{mahajan_et_al:LIPIcs.STACS.2026,
title = {{LIPIcs, Volume 364, STACS 2026, Complete Volume}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {1--1566},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026},
URN = {urn:nbn:de:0030-drops-255846},
doi = {10.4230/LIPIcs.STACS.2026},
annote = {Keywords: LIPIcs, Volume 364, STACS 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
43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 0:i-0:xxviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mahajan_et_al:LIPIcs.STACS.2026.0,
author = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {0:i--0:xxviii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.0},
URN = {urn:nbn:de:0030-drops-255836},
doi = {10.4230/LIPIcs.STACS.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Query Languages for Machine-Learning Models (Invited Talk)
Abstract
In my invited talk and this accompanying paper, I discuss two logics for weighted finite structures: first-order logic with summation (FO(SUM)) and its recursive extension IFP(SUM). These logics originate from foundational work by Grädel, Gurevich, and Meer in the 1990s. In recent joint work with Standke, Steegmans, and Van den Bussche, we have investigated these logics as query languages for machine learning models, specifically neural networks, which are naturally represented as weighted graphs. I present illustrative examples of queries to neural networks that can be expressed in these logics and discuss fundamental results on their expressiveness and computational complexity.
Cite as
Martin Grohe. Query Languages for Machine-Learning Models (Invited Talk). In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{grohe:LIPIcs.STACS.2026.1,
author = {Grohe, Martin},
title = {{Query Languages for Machine-Learning Models}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {1:1--1:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.1},
URN = {urn:nbn:de:0030-drops-254904},
doi = {10.4230/LIPIcs.STACS.2026.1},
annote = {Keywords: Expressive power of query languages, fixed-point logics, weighted structures, neural networks, explainable AI}
}
Document
Invited Talk
Moments in Time: Algebraic Analysis for Solvable Loops (Invited Talk)
Abstract
With substantial progress in automated reasoning, algebraic approaches emerged to automatically analyse program loops in an exact manner. In this invited talk, we discuss recent results in characterizing the functional behaviour of loops with polynomial arithmetic and probabilistic updates. This problem remains unsolved even when we restrict consideration to loops that are non-nested, without conditionals, and/or without exit conditions [Ehud Hrushovski et al., 2023; Julian Müllner and others, 2024].
We are motivated by applications of computer-aided verification, in particular to assess the safety, security, and sensitivity of computer systems [M. Z. Kwiatkowska et al., 2011; Gilles Barthe et al., 2012; Gilles Barthe and others, 2018; Marcel Moosbrugger et al., 2023; Alessandro Abate et al., 2023; Andrey Kofnov and others, 2024]. We are interested in modeling, deciding, and solving loop analysis. The key to our work are moment-computable loops [L. Kovács, 2008; Marcel Moosbrugger et al., 2022] which allow us to set limits on what is decidable and solvable in loop analysis. Our approach combines algebra, statistics, and automated reasoning to mechanize loop analysis. Various techniques, such as martingale theory and quantifier elimination, can be seen as examples of moment-computable loop analysis.
This talk is structured within three inter-connected parts. We first bring moment-based loop analysis into the landscape of {loop invariant synthesis} and extend moment-computable loops with {termination guarantees}. We next automate the reasoning about (probabilistic) loops by summarizing loop semantics as (probabilistic) algebraic recurrences, whose closed-form solutions capture (higher-order) moments, and hence invariants, among loop variables. These recurrences together with loop tests yield moment-based (super)martingales necessary to prove loop termination and compute probability bounds on termination. We finally describe moment-computable loops whose invariant synthesis {decidable} or as {hard} as open problems, such as the Skolem problem [Graham Everest et al., 2003; Terrence Tao, 2008].
Cite as
Laura Kovács. Moments in Time: Algebraic Analysis for Solvable Loops (Invited Talk). In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kovacs:LIPIcs.STACS.2026.2,
author = {Kov\'{a}cs, Laura},
title = {{Moments in Time: Algebraic Analysis for Solvable Loops}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {2:1--2:2},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.2},
URN = {urn:nbn:de:0030-drops-254910},
doi = {10.4230/LIPIcs.STACS.2026.2},
annote = {Keywords: program analysis, algebraic reasoning, symbolic computation, loop invariants}
}
Document
Invited Talk
Advancements in Online Edge Coloring Algorithms (Invited Talk)
Abstract
We study online edge coloring, where edges of an n-vertex graph arrive sequentially and must be colored irrevocably so that adjacent edges receive different colors. The goal is to use as few colors as possible as a function of the maximum degree Δ.
This talk surveys recent progress that achieves near-optimal guarantees by leveraging martingale concentration arguments. Specifically, we show that near-optimal colorings (using (1+o(1))Δ colors) exhibit sharp threshold phenomena that match long-standing lower bounds, resolving and strengthening a conjecture of Bar‑Noy, Motwani, and Naor [Bar-Noy et al., 1992].
First, while the conjecture posited the existence of a randomized algorithm achieving a (1+o(1))Δ-edge-coloring for maximum degree Δ = ω(log n), we present a deterministic online algorithm that achieves this guarantee in the same regime. This result matches the known impossibility result for deterministic algorithms when Δ = O(log n), establishing a sharp threshold.
Second, improving the conditions under which near-optimal coloring is known to be possible with randomness, we present a randomized online algorithm achieving a (1+o(1))Δ-edge-coloring already for graphs with maximum degree Δ = ω(√{log n}). This establishes a sharp threshold for randomized algorithms, matching the lower bound in [Bar-Noy et al., 1992] for the Δ = O(√{log n}) regime.
This is joint work with Joakim Blikstad, Radu Vintan, and David Wajc [Joakim Blikstad et al., 2024; Joakim Blikstad et al., 2025].
Cite as
Ola Svensson. Advancements in Online Edge Coloring Algorithms (Invited Talk). In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{svensson:LIPIcs.STACS.2026.3,
author = {Svensson, Ola},
title = {{Advancements in Online Edge Coloring Algorithms}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {3:1--3:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.3},
URN = {urn:nbn:de:0030-drops-254928},
doi = {10.4230/LIPIcs.STACS.2026.3},
annote = {Keywords: Edge coloring, Martingale, Online algorithms}
}
Document
Unit Interval Selection in Random Order Streams
Abstract
We consider the Unit Interval Selection problem in the one-pass random order streaming model. In this setting, an algorithm is presented with a sequence of n unit-length intervals on the line that arrive in uniform random order, one at a time, and the objective is to output (an approximation of) a largest set of disjoint intervals using space linear in the size of an optimal solution. Previous work only considered adversarially ordered streams and established that, within these space constraints, a (2/3)-approximation can be achieved in such streams, and this is best possible, in that going beyond such an approximation factor requires space Ω(n) [Emek et al., TALG'16].
In this work, we show that an improved expected approximation factor can be achieved if the input stream is in uniform random order, where the expectation is taken over the stream order. More specifically, we give a one-pass streaming algorithm with expected approximation factor 0.7401 that uses space O(|OPT|), where OPT denotes an optimal solution. We also show that random order algorithms with expected approximation factor above 8/9 require space Ω(n), and algorithms that compute a better than 2/3-approximation with probability above 2/3 also require Ω(n) space.
On a technical level, we design an algorithm for the restricted domain [0, Δ), for some constant Δ, and use standard techniques to obtain an algorithm for unrestricted domains. For the restricted domain [0, Δ), we run O(Δ) recursive instances of our algorithm, with each instance targeting the situation where a specific interval of an optimal solution arrives first. We establish the interesting property of our algorithm that it performs worst when the input stream consists solely of a set of independent intervals. It then remains to analyse the algorithm on these simple instances.
Our lower bound is proved via communication complexity arguments, similar in spirit to the robust communication lower bounds established by [Chakrabarti et al., Theory Comput. 2016].
Cite as
Cezar-Mihail Alexandru, Adithya Diddapur, Magnús M. Halldórsson, Christian Konrad, and Kheeran K. Naidu. Unit Interval Selection in Random Order Streams. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alexandru_et_al:LIPIcs.STACS.2026.4,
author = {Alexandru, Cezar-Mihail and Diddapur, Adithya and Halld\'{o}rsson, Magn\'{u}s M. and Konrad, Christian and Naidu, Kheeran K.},
title = {{Unit Interval Selection in Random Order Streams}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {4:1--4:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.4},
URN = {urn:nbn:de:0030-drops-254933},
doi = {10.4230/LIPIcs.STACS.2026.4},
annote = {Keywords: Random order streaming algorithms, unit interval selection}
}
Document
On the Complexity of Language Membership for Probabilistic Words
Abstract
We study the membership problem to context-free languages L (CFLs) on probabilistic words, that specify for each position a probability distribution on the letters (assuming independence across positions). Our task is to compute, given a probabilistic word, what is the probability that a word drawn according to the distribution belongs to L. This problem generalizes the problem of counting how many words of length n belong to L, or of counting how many completions of a partial word belong to L.
We show that this problem is in polynomial time for unambiguous context-free languages (uCFLs), but can be #P-hard already for unions of two linear uCFLs. More generally, we show that the problem is in polynomial time for so-called poly-slicewise-unambiguous languages, where given a length n we can tractably compute an uCFL for the words of length n in the language. This class includes some inherently ambiguous languages, and implies the tractability of bounded CFLs and of languages recognized by unambiguous polynomial-time counter automata; but we show that the problem can be #P-hard for nondeterministic counter automata, even for Parikh automata with a single counter. We then introduce classes of circuits from knowledge compilation which we use for tractable counting, and show that this covers the tractability of poly-slicewise-unambiguous languages and of some CFLs that are not poly-slicewise-unambiguous. Extending these circuits with negation further allows us to show tractability for the language of primitive words, and for the language of concatenations of two palindromes. We finally show the conditional undecidability of the meta-problem that asks, given a CFG, whether the probabilistic membership problem for that CFG is tractable or #P-hard.
Cite as
Antoine Amarilli, Mikaël Monet, Paul Raphaël, and Sylvain Salvati. On the Complexity of Language Membership for Probabilistic Words. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{amarilli_et_al:LIPIcs.STACS.2026.5,
author = {Amarilli, Antoine and Monet, Mika\"{e}l and Rapha\"{e}l, Paul and Salvati, Sylvain},
title = {{On the Complexity of Language Membership for Probabilistic Words}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {5:1--5:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.5},
URN = {urn:nbn:de:0030-drops-254943},
doi = {10.4230/LIPIcs.STACS.2026.5},
annote = {Keywords: Automaton, probabilistic words, context-free grammar, membership problem}
}
Document
Threshold-Driven Streaming Graph: Expansion and Rumor Spreading
Abstract
A randomized distributed algorithm called RAES was introduced in [Becchetti et al., 2020] to extract a bounded-degree expander from a dense n-vertex expander graph G = (V, E). The algorithm relies on a simple threshold-based procedure. A key assumption in [Becchetti et al., 2020] is that the input graph G is static - i.e., both its vertex set V and edge set E remain unchanged throughout the process - while the analysis of raes in dynamic models is left as a major open question.
In this work, we investigate the behavior of RAES under a dynamic graph model induced by a streaming node-churn process (also known as the sliding window model), where, at each discrete round, a new node joins the graph and the oldest node departs. This process yields a bounded-degree dynamic graph 𝒢 = {G_t = (V_t, E_t) : t ∈ ℕ} that captures essential characteristics of peer-to-peer networks - specifically, node churn and threshold on the number of connections each node can manage. We prove that every snapshot G_t in the dynamic graph sequence has good expansion properties with high probability. Furthermore, we leverage this property to establish a logarithmic upper bound on the completion time of the well-known PUSH and PULL rumor spreading protocols over the dynamic graph 𝒢.
Cite as
Flora Angileri, Andrea Clementi, Emanuele Natale, Michele Salvi, and Isabella Ziccardi. Threshold-Driven Streaming Graph: Expansion and Rumor Spreading. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{angileri_et_al:LIPIcs.STACS.2026.6,
author = {Angileri, Flora and Clementi, Andrea and Natale, Emanuele and Salvi, Michele and Ziccardi, Isabella},
title = {{Threshold-Driven Streaming Graph: Expansion and Rumor Spreading}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {6:1--6:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.6},
URN = {urn:nbn:de:0030-drops-254957},
doi = {10.4230/LIPIcs.STACS.2026.6},
annote = {Keywords: Distributed Algorithms, Randomized Algorithms, Dynamic Random Graphs, Graph Expansion, Rumor Spreading}
}
Document
Refining the Complexity Landscape of Speed Scaling: Hardness and Algorithms
Abstract
We study the computational complexity of scheduling jobs on a single speed-scalable processor with the objective of capturing the trade-off between the (weighted) flow time and the energy consumption. This trade-off has been extensively explored in the literature through a number of problem formulations that differ in the specific job characteristics and the precise objective function. Nevertheless, the computational complexity of four important problem variants has remained unresolved and was explicitly identified as an open question in prior work (see [Barcelo et al., 2015]). In this paper, we settle the complexity of these variants.
More specifically, we prove that the problem of minimizing the objective of total (weighted) flow time plus energy is NP-hard for the cases of (i) unit-weight jobs with arbitrary sizes, and (ii) arbitrary-weight jobs with unit sizes. These results extend to the objective of minimizing the total (weighted) flow time subject to an energy budget and hold even when the schedule is required to adhere to a given priority ordering.
In contrast, we show that when a completion-time ordering is provided, the same problem variants become polynomial-time solvable. The latter result highlights the subtle differences between priority and completion orderings for the problem.
Cite as
Antonios Antoniadis, Denise Graafsma, Ruben Hoeksma, and Maria Vlasiou. Refining the Complexity Landscape of Speed Scaling: Hardness and Algorithms. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{antoniadis_et_al:LIPIcs.STACS.2026.7,
author = {Antoniadis, Antonios and Graafsma, Denise and Hoeksma, Ruben and Vlasiou, Maria},
title = {{Refining the Complexity Landscape of Speed Scaling: Hardness and Algorithms}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {7:1--7:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.7},
URN = {urn:nbn:de:0030-drops-254967},
doi = {10.4230/LIPIcs.STACS.2026.7},
annote = {Keywords: energy-efficient algorithms, scheduling, flow-time minimization, linear program, NP-hard, speed scaling}
}
Document
On the p-adic Skolem Problem
Abstract
The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) has a zero term. Showing decidability of this problem is equivalent to giving an effective proof of the Skolem-Mahler-Lech Theorem, which asserts that a non-degenerate LRS has finitely many zeros. The latter result was proven over 90 years ago via an ineffective method showing that such an LRS has only finitely many p-adic zeros. In this paper we consider the problem of determining whether a given LRS has a p-adic zero, as well as the corresponding function problem of computing exact representations of all p-adic zeros. We present algorithms for both problems and report on their implementation. The output of the algorithms is unconditionally correct, and termination is guaranteed subject to the p-adic Schanuel Conjecture (a standard number-theoretic hypothesis concerning the p-adic exponential function). While these algorithms do not solve the Skolem Problem, they can be exploited to find natural-number and rational zeros under additional hypotheses. To illustrate this, we apply our results to show decidability of the Simultaneous Skolem Problem (determine whether two coprime linear recurrences have a common natural-number zero), again subject to the p-adic Schanuel Conjecture.
Cite as
Piotr Bacik, Joël Ouaknine, David Purser, and James Worrell. On the p-adic Skolem Problem. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bacik_et_al:LIPIcs.STACS.2026.8,
author = {Bacik, Piotr and Ouaknine, Jo\"{e}l and Purser, David and Worrell, James},
title = {{On the p-adic Skolem Problem}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {8:1--8:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.8},
URN = {urn:nbn:de:0030-drops-254979},
doi = {10.4230/LIPIcs.STACS.2026.8},
annote = {Keywords: Skolem Problem, p-adic Schanuel Conjecture, Skolem Conjecture, Exponential Local-Global Principle, exponential polynomial}
}
Document
Algebraic Characterizations of Classes of Regular Languages in DynFO
Abstract
This paper explores the fine-grained structure of classes of regular languages maintainable in fragments of first-order logic within the dynamic descriptive complexity framework of Patnaik and Immerman. A result by Hesse states that the class of regular languages is maintainable by first-order formulas even if only unary auxiliary relations can be used. Another result by Gelade, Marquardt, and Schwentick states that the class of regular languages coincides with the class of languages maintainable by quantifier-free formulas with binary auxiliary relations.
We refine Hesse’s result and show that with unary auxiliary data ∃^*∀^*-formulas can maintain all regular languages. We then obtain precise algebraic characterizations of the classes of languages maintainable with quantifier-free formulas and positive ∃^*-formulas in the presence of unary auxiliary relations.
Cite as
Corentin Barloy, Felix Tschirbs, Nils Vortmeier, and Thomas Zeume. Algebraic Characterizations of Classes of Regular Languages in DynFO. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{barloy_et_al:LIPIcs.STACS.2026.9,
author = {Barloy, Corentin and Tschirbs, Felix and Vortmeier, Nils and Zeume, Thomas},
title = {{Algebraic Characterizations of Classes of Regular Languages in DynFO}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {9:1--9:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.9},
URN = {urn:nbn:de:0030-drops-254986},
doi = {10.4230/LIPIcs.STACS.2026.9},
annote = {Keywords: Dynamic descriptive complexity, formal languages, monoid theory}
}
Document
A Polynomial Bound on the Pathwidth of Graphs Edge-Coverable by k Shortest Paths
Abstract
Dumas, Foucaud, Perez and Todinca [SIAM J. Disc. Math., 2024] recently proved that every graph whose edge set can be covered by k shortest paths has pathwidth at most 3^k. In this paper, we improve this upper bound on the pathwidth to a polynomial bound; namely, we show that every graph whose edge set can be covered by k shortest paths has pathwidth O(k⁴), answering a question from the same paper. Moreover, we also prove that when k ≤ 3, every such graph has pathwidth at most k (and this bound is tight). Eventually, we show that even though there exist graphs with arbitrary large treewidth whose vertex set can be covered by 2 isometric trees, every graph whose set of edges can be covered by 2 isometric trees has treewidth at most 2.
Cite as
Julien Baste, Lucas De Meyer, Ugo Giocanti, Etienne Objois, and Timothé Picavet. A Polynomial Bound on the Pathwidth of Graphs Edge-Coverable by k Shortest Paths. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{baste_et_al:LIPIcs.STACS.2026.10,
author = {Baste, Julien and De Meyer, Lucas and Giocanti, Ugo and Objois, Etienne and Picavet, Timoth\'{e}},
title = {{A Polynomial Bound on the Pathwidth of Graphs Edge-Coverable by k Shortest Paths}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {10:1--10:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.10},
URN = {urn:nbn:de:0030-drops-254999},
doi = {10.4230/LIPIcs.STACS.2026.10},
annote = {Keywords: Structural Graph Theory, Coverings, Metrics, Pathwidth, Treewdidth, Parameterized Algorithms, Layerings}
}
Document
The Diameter of (Threshold) Geometric Inhomogeneous Random Graphs
Abstract
We prove that the diameter of threshold (zero temperature) Geometric Inhomogeneous Random Graphs (GIRG) is asymptotically almost surely Θ(log n). This has strong implications for the runtime of many distributed protocols on those graphs, which often have runtimes bounded as a function of the diameter.
The GIRG model exhibits many properties empirically found in real-world networks, and the runtime of various practical algorithms has empirically been found to scale in the same way for GIRG and for real-world networks, in particular related to computing distances, diameter, clustering, cliques and chromatic numbers. Thus the GIRG model is a promising candidate for deriving insight about the performance of algorithms in real-world instances.
The diameter was previously only known in the one-dimensional case, and the proof relied very heavily on dimension one. Our proof employs a similar Peierls-type argument alongside a novel renormalization scheme. Moreover, instead of using topological arguments (which become complicated in high dimensions) in establishing the connectivity of certain boundaries, we employ some comparatively recent and clearer graph-theoretic machinery. The lower bound is proven via a simple ad-hoc construction.
Cite as
Zylan Benjert, Kostas Lakis, Johannes Lengler, and Raghu Raman Ravi. The Diameter of (Threshold) Geometric Inhomogeneous Random Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 11:1-11:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{benjert_et_al:LIPIcs.STACS.2026.11,
author = {Benjert, Zylan and Lakis, Kostas and Lengler, Johannes and Ravi, Raghu Raman},
title = {{The Diameter of (Threshold) Geometric Inhomogeneous Random Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {11:1--11:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.11},
URN = {urn:nbn:de:0030-drops-255009},
doi = {10.4230/LIPIcs.STACS.2026.11},
annote = {Keywords: GIRG, Diameter, Distributed Algorithms, Complex Networks}
}
Document
Density Matters: A Complexity Dichotomy of Deleting Edges to Bound Subgraph Density
Abstract
We study τ-Bounded-Density Edge Deletion (τ-BDED), where given an undirected graph G, the task is to remove as few edges as possible to obtain a graph G' where no subgraph of G' has density more than τ. The density of a (sub)graph is the number of edges divided by the number of vertices. This problem was recently introduced and shown to be NP-hard for τ ∈ {2/3, 3/4, 1 + 1/25}, but polynomial-time solvable for τ ∈ {0,1/2,1} [Bazgan et al., JCSS 2025]. We provide a complete dichotomy with respect to the target density τ:
1) If 2τ ∈ ℕ (half-integral target density) or τ < 2/3, then τ-BDED is polynomial-time solvable.
2) Otherwise, τ-BDED is NP-hard.
We complement the NP-hardness with fixed-parameter tractability with respect to the treewidth of G. Moreover, for integral target density τ ∈ ℕ, we show τ-BDED to be solvable in randomized O(m^{1 + o(1)}) time. Our algorithmic results are based on a reduction to a new general flow problem on restricted networks that, depending on τ, can be solved via Maximum s-t-Flow or General Factors. We believe this connection between these variants of flow and matching to be of independent interest.
Cite as
Matthias Bentert, Tom-Lukas Breitkopf, Vincent Froese, Anton Herrmann, and André Nichterlein. Density Matters: A Complexity Dichotomy of Deleting Edges to Bound Subgraph Density. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bentert_et_al:LIPIcs.STACS.2026.12,
author = {Bentert, Matthias and Breitkopf, Tom-Lukas and Froese, Vincent and Herrmann, Anton and Nichterlein, Andr\'{e}},
title = {{Density Matters: A Complexity Dichotomy of Deleting Edges to Bound Subgraph Density}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {12:1--12:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.12},
URN = {urn:nbn:de:0030-drops-255012},
doi = {10.4230/LIPIcs.STACS.2026.12},
annote = {Keywords: Transshipment, Maximum Flow, General Factors, Matching, Graph Modification Problem}
}
Document
Line Cover and Related Problems
Abstract
We study several extensions of the classic Line Cover problem of covering a set of n points in the plane with k lines. Line Cover is known to be NP-hard and our focus is on two natural generalizations: (1) Line Clustering, where the objective is to find k lines in the plane that minimize the sum of squares of distances of a given set of input points to the closest line, and (2) Hyperplane Cover, where the goal is to cover n points in ℝ^d by k hyperplanes. We also consider the more general Projective Clustering problem, which unifies both of these and has numerous applications in machine learning, data mining, and computational geometry. In this problem one seeks k affine subspaces of dimension r minimizing the sum of squares of distances of a given set of n points in ℝ^d to the closest point within one of the k affine subspaces.
Our main contributions reveal interesting differences in the parameterized complexity of these problems. While Line Cover is fixed-parameter tractable parameterized by the number k of lines in the solution, we show that Line Clustering is W[1]-hard when parameterized by k and rule out algorithms of running time n^{o(k)} under the Exponential Time Hypothesis. Hyperplane Cover is known to be NP-hard even when d = 2 and by the work of Langerman and Morin [Discrete & Computational Geometry, 2005], it is FPT parameterized by k and d. We complement this result by establishing that Hyperplane Cover is W[2]-hard when parameterized by only k.
We complement our hardness results by presenting an algorithm for Projective Clustering. We show that this problem is solvable in n^{𝒪(dk(r+1))} time. Not only does this yield an upper bound for Line Clustering that asymptotically matches our lower bound, but it also significantly extends the seminal work on k-Means Clustering (the special case r = 0) by Inaba, Katoh, and Imai [SoCG 1994].
Cite as
Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana. Line Cover and Related Problems. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bentert_et_al:LIPIcs.STACS.2026.13,
author = {Bentert, Matthias and Fomin, Fedor V. and Golovach, Petr A. and Saha, Souvik and Seetharaman, Sanjay and Upasana, Anannya},
title = {{Line Cover and Related Problems}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {13:1--13:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.13},
URN = {urn:nbn:de:0030-drops-255023},
doi = {10.4230/LIPIcs.STACS.2026.13},
annote = {Keywords: Point Line Cover, Projective Clustering, W-hardness, XP algorithm}
}
Document
Computing Tarski Fixed Points in Financial Networks
Abstract
Modern financial networks are highly connected and result in complex interdependencies of the involved institutions. In the prominent Eisenberg-Noe model [Eisenberg and Noe, 2001], a fundamental aspect is clearing - to determine the amount of assets available to each financial institution in the presence of potential defaults and bankruptcy. A clearing state represents a fixed point that satisfies a set of natural axioms. Existence can be established (even in broad generalizations of the model) using Tarski’s theorem. While a maximal fixed point can be computed in polynomial time, the complexity of computing other fixed points is open. In this paper, we provide an efficient algorithm to compute a minimal fixed point. Our algorithm applies in a broad generalization of the Eisenberg-Noe model with any monotone, piecewise-linear payment functions and default costs. We also study claims trading, a local network adjustment to improve clearing, when networks are evaluated with minimal clearing. We provide an efficient algorithm to decide existence of Pareto-improving trades and compute optimal ones if they exist.
Cite as
Leander Besting, Martin Hoefer, and Lars Huth. Computing Tarski Fixed Points in Financial Networks. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{besting_et_al:LIPIcs.STACS.2026.14,
author = {Besting, Leander and Hoefer, Martin and Huth, Lars},
title = {{Computing Tarski Fixed Points in Financial Networks}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {14:1--14:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.14},
URN = {urn:nbn:de:0030-drops-255038},
doi = {10.4230/LIPIcs.STACS.2026.14},
annote = {Keywords: Tarski Fixed Points, Financial Networks, Minimal Clearing, Claims Trade}
}
Document
The Complexity of Resilience for Digraph Queries
Abstract
We prove a complexity dichotomy for the resilience problem for unions of conjunctive digraph queries (i.e., for existential positive sentences over the signature {R} of directed graphs). Specifically, for every union μ of conjunctive digraph queries, the following problem is in P or NP-complete: given a directed multigraph G and a natural number u, can we remove u edges from G so that G ⊧ ¬ μ? In fact, we verify a more general dichotomy conjecture from [Bodirsky et al., 2024] for all resilience problems in the special case of directed graphs, and show that for such unions of queries μ there exists a countably infinite (`dual') valued structure Δ_μ which either primitively positively constructs 1-in-3-3-SAT, and hence the resilience problem for μ is NP-complete by general principles, or has a pseudo cyclic canonical fractional polymorphism, and the resilience problem for μ is in P.
Cite as
Manuel Bodirsky and Žaneta Semanišinová. The Complexity of Resilience for Digraph Queries. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 15:1-15:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bodirsky_et_al:LIPIcs.STACS.2026.15,
author = {Bodirsky, Manuel and Semani\v{s}inov\'{a}, \v{Z}aneta},
title = {{The Complexity of Resilience for Digraph Queries}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {15:1--15:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.15},
URN = {urn:nbn:de:0030-drops-255045},
doi = {10.4230/LIPIcs.STACS.2026.15},
annote = {Keywords: valued constraints, unions of conjunctive queries, resilience, computational complexity, pp-constructions}
}
Document
To Buy or Not to Buy: Online Rent-Or-Buy on Node-Weighted Graphs
Abstract
We study the rent-or-buy variant of the online Steiner forest problem on node- and edge-weighted graphs. For n-node graphs with at most ̄{n} nodes of non-zero weight, and at most k̃ different arriving terminal pairs, we obtain the following:
- A deterministic, O(log n log ̄{n})-competitive algorithm against adaptive adversaries. This improves on the previous best, O(log⁴ n)-competitive algorithm obtained by the black-box reduction from [Yair Bartal et al., 2001] combined with the previously best deterministic algorithms for the simpler "buy-only" setting.
- A deterministic, O(̄{n}log k̃)-competitive algorithm against adaptive adversaries. This generalizes the O(log k̃)-competitive algorithm for the purely edge-weighted setting from [Seeun Umboh, 2015].
- A randomized, O(log k̃ log ̄{n})-competitive algorithm against oblivious adversaries. All previous approaches were based on the randomized, black-box reduction from [Awerbuch et al., 2004] that achieves a O(log k̃ log n)-competitive ratio when combined with an algorithm for the "buy-only" setting. Our key technical ingredient is a novel charging scheme to an instance of online prize-collecting set cover. This allows us to extend the witness-technique of [Seeun Umboh, 2015] to the node-weighted setting and obtain refined guarantees with respect to ̄{n}, already in the much simpler "buy-only" setting.
Cite as
Sander Borst and Moritz Venzin. To Buy or Not to Buy: Online Rent-Or-Buy on Node-Weighted Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{borst_et_al:LIPIcs.STACS.2026.16,
author = {Borst, Sander and Venzin, Moritz},
title = {{To Buy or Not to Buy: Online Rent-Or-Buy on Node-Weighted Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {16:1--16:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.16},
URN = {urn:nbn:de:0030-drops-255054},
doi = {10.4230/LIPIcs.STACS.2026.16},
annote = {Keywords: online network design, node-weighted Steiner forest, derandomization}
}
Document
Kernelization Dichotomies for Hitting Minors Under Structural Parameterizations
Abstract
For a finite collection of connected graphs ℱ, the ℱ-Minor Deletion problem consists in, given a graph G and an integer 𝓁, deciding whether G contains a vertex set of size at most 𝓁 whose removal results in an ℱ-minor-free graph. We lift the existence of (approximate) polynomial kernels for ℱ-Minor Deletion by the solution size to (approximate) polynomial kernels parameterized by the vertex-deletion distance to graphs of bounded elimination distance to ℱ-minor-free graphs. This results in exact polynomial kernels for every family ℱ that contains a planar graph, and an approximate polynomial kernel for Planar Vertex Deletion. Moreover, combining our result with a previous lower bound, we obtain the following infinite set of dichotomies, assuming NP ̸ ⊆ coNP/poly: for any finite set ℱ of biconnected graphs on at least three vertices containing a planar graph, and any minor-closed class of graphs {C}, ℱ-Minor Deletion admits a polynomial kernel parameterized by the vertex-deletion distance to {C} if and only if {C} has bounded elimination distance to ℱ-minor-free graphs. For instance, this yields dichotomies for Cactus Vertex Deletion, Outerplanar Vertex Deletion, and Treewidth-t Vertex Deletion for every integer t ≥ 0. Prior to our work, such dichotomies were only known for the particular cases of Vertex Cover and Feedback Vertex Set. Our approach builds on the techniques developed by Jansen and Pieterse [Theor. Comput. Sci. 2020] and also uses adaptations of some of the results by Jansen, de Kroon, and Włodarczyk [STOC 2021].
Cite as
Marin Bougeret, Eric Brandwein, and Ignasi Sau. Kernelization Dichotomies for Hitting Minors Under Structural Parameterizations. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 17:1-17:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bougeret_et_al:LIPIcs.STACS.2026.17,
author = {Bougeret, Marin and Brandwein, Eric and Sau, Ignasi},
title = {{Kernelization Dichotomies for Hitting Minors Under Structural Parameterizations}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {17:1--17:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.17},
URN = {urn:nbn:de:0030-drops-255067},
doi = {10.4230/LIPIcs.STACS.2026.17},
annote = {Keywords: hitting forbidden minors, parameterized complexity, polynomial kernel, structural parameterization, elimination distance, kernelization lower bound}
}
Document
Optimal Deterministic Rendezvous in Labeled Lines
Abstract
In a rendezvous task, a set of mobile agents initially dispersed in a network have to gather at an arbitrary common site. We consider the rendezvous problem on the infinite labeled line, with 2 initially asleep agents, without communication, and a synchronous notion of time. Each node on the line is labeled with a unique positive integer. The initial distance between the two agents is denoted by D. Time is divided into rounds and measured from the moment an agent first wakes up. We denote by τ the delay between the two agents' wake up times. If awake in a given round T, an agent at a node v has three options: stay at the node v, take port 0, or take port 1. If it decides to stay, the agent will still be at node v in round T+1. Otherwise, it will be at one of the two neighbors of v on the infinite line, depending on the port it chose. The agents achieve rendezvous in T rounds if they are at the same node in round T. We aim for a deterministic algorithm for this problem.
The problem was recently considered by Miller and Pelc [Distributed Computing, 2025]. With 𝓁_{max} the largest label of the two starting nodes, they showed that no algorithm can guarantee rendezvous in o(D log^* 𝓁_{max}) rounds. The lower bound follows from a connection with the LOCAL model of distributed computing, and holds even if the agents are guaranteed simultaneous wake-up (τ = 0) and are told their initial distance D. Miller and Pelc also gave an algorithm of optimal matching complexity O(D log^* 𝓁_{max}) when the agents know D, but only obtained the higher bound of O(D² (log^* 𝓁_{max})³) when D is unknown to the agents.
In this paper, we improve this second complexity to a tight O(D log^* 𝓁_{max}), closing the gap between the best known lower and upper bounds. In fact, our algorithm achieves rendezvous in O(D log^* 𝓁_{min}) rounds, where 𝓁_{min} is the smallest label within distance O(D) of the two starting positions. We obtain this result by having the agents compute sparse subsets of the nodes to gather at (formally, ruling sets over the line), as well as some general observations about the setting of rendezvous on labeled graphs.
Cite as
Yann Bourreau, Ananth Narayanan, and Alexandre Nolin. Optimal Deterministic Rendezvous in Labeled Lines. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bourreau_et_al:LIPIcs.STACS.2026.18,
author = {Bourreau, Yann and Narayanan, Ananth and Nolin, Alexandre},
title = {{Optimal Deterministic Rendezvous in Labeled Lines}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {18:1--18:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.18},
URN = {urn:nbn:de:0030-drops-255071},
doi = {10.4230/LIPIcs.STACS.2026.18},
annote = {Keywords: mobile agents, rendezvous, ruling set, deterministic algorithms, labeled line}
}
Document
A Linear Kernel for Independent Set Reconfiguration in Planar Graphs
Abstract
Fix a positive integer r, and a graph G that is K_{3,r}-minor-free. Let I_s and I_t be two independent sets in G, each of size k. We begin with a "token" on each vertex of I_s and seek to move all tokens to I_t, by repeated "token jumping", removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size k. Given G, I_s, and I_t, we ask whether there exists a sequence of token jumps that transforms I_s into I_t. When k is part of the input, this problem is known to be PSPACE-complete. But it was shown by Ito, Kamiński, and Ono [Ito et al., 2014] to be fixed-parameter tractable. That is, the problem can be solved in time f(k)⋅ P(n), for some function f and polynomial P, where n denotes the order of G.
Here we strengthen the upper bound on the running time in terms of k by showing that the problem has a kernel of size linear in k. More precisely, we transform an arbitrary input problem on a K_{3,r}-minor-free graph (for some fixed positive integer r) into an equivalent problem on a (K_{3,r}-minor-free) graph with order O(k). This answers positively a question of Bousquet, Mouawad, Nishimura, and Siebertz [Nicolas Bousquet et al., 2022] and improves the recent quadratic kernel of Cranston, Mühlenthaler, and Peyrille [Daniel W. Cranston et al., 2024]. For planar graphs, we further strengthen this upper bound to get a kernel of size at most 42k.
Cite as
Nicolas Bousquet and Daniel W. Cranston. A Linear Kernel for Independent Set Reconfiguration in Planar Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bousquet_et_al:LIPIcs.STACS.2026.19,
author = {Bousquet, Nicolas and Cranston, Daniel W.},
title = {{A Linear Kernel for Independent Set Reconfiguration in Planar Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {19:1--19:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.19},
URN = {urn:nbn:de:0030-drops-255081},
doi = {10.4230/LIPIcs.STACS.2026.19},
annote = {Keywords: Reconfiguration, Independent Set, Kernel, Planar graphs}
}
Document
Approximation Algorithms for Integer Programming with Resource Augmentation
Abstract
Solving a general integer program (IP) is NP-hard. The classic algorithm [Papadimitriou, J.ACM '81] for IPs has a running time n^{{𝒪}(m)}(m⋅max{Δ,‖b‖_{∞}})^{{𝒪}(m²)}, where m is the number of constraints, n is the number of variables, and Δ and ‖b‖_{∞} are, respectively, the largest absolute values among the entries in the constraint matrix and the right-hand side vector of the constraint. The running time is exponential in m, and becomes pseudo-polynomial if m is a constant. In recent years, there has been extensive research on FPT (fixed parameter tractable) algorithms for the so-called n-fold IPs, which may possess a large number of constraints, but the constraint matrix satisfies a specific block structure. It is remarkable that these FPT algorithms take as parameters Δ and the number of rows and columns of some small submatrices. If Δ is not treated as a parameter, then the running time becomes pseudo-polynomial even if all the other parameters are taken as constants.
This paper explores the trade-off between time and accuracy in solving an IP. We show that, for arbitrary small ε > 0, there exists an algorithm for IPs with m constraints that runs in {f(m,ε)}⋅poly(|I|) time, and returns a near-feasible solution that violates the constraints by at most εΔ. Furthermore, for n-fold IPs, we establish a similar result - our algorithm runs in time that depends on the number of rows and columns of small submatrices together with 1/ε, and returns a solution that slightly violates the constraints. Meanwhile, both solutions guarantee that their objective values are no worse than the corresponding optimal objective values satisfying the constraints. As applications, our results can be used to obtain additive approximation schemes for multidimensional knapsack as well as scheduling.
Cite as
Hauke Brinkop, Hua Chen, Lin Chen, Klaus Jansen, and Guochuan Zhang. Approximation Algorithms for Integer Programming with Resource Augmentation. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brinkop_et_al:LIPIcs.STACS.2026.20,
author = {Brinkop, Hauke and Chen, Hua and Chen, Lin and Jansen, Klaus and Zhang, Guochuan},
title = {{Approximation Algorithms for Integer Programming with Resource Augmentation}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {20:1--20:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.20},
URN = {urn:nbn:de:0030-drops-255090},
doi = {10.4230/LIPIcs.STACS.2026.20},
annote = {Keywords: Approximation algorithms, Resource augmentation, Integer programs, n-fold IPs}
}
Document
Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions
Abstract
We study the extension of Presburger arithmetic by the class of sub-polynomial Hardy field functions, and show the majority of these extensions to be undecidable. More precisely, we show that the theory Th(ℤ; < , +, ⌊f⌉), where f is a Hardy field function and ⌊⋅⌉ the nearest integer operator, is undecidable when f grows polynomially faster than x. Further, we show that when f grows sub-linearly quickly, but still as fast as some polynomial, the theory Th(ℤ; < , +, ⌊f⌉) is undecidable.
Cite as
Hera Brown and Jakub Konieczny. Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brown_et_al:LIPIcs.STACS.2026.21,
author = {Brown, Hera and Konieczny, Jakub},
title = {{Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {21:1--21:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.21},
URN = {urn:nbn:de:0030-drops-255103},
doi = {10.4230/LIPIcs.STACS.2026.21},
annote = {Keywords: Arithmetic theories, Hardy fields, Undecidability}
}
Document
Modular Counting over 3-Element and Conservative Domains
Abstract
In the Constraint Satisfaction Problem (CSP for short) the goal is to decide the existence of a homomorphism from a given relational structure {G} to a given relational structure {H}. If the structure {H} is fixed and {G} is the only input, the problem is denoted CSP({H}). In its counting version, #CSP({H}), the task is to find the number of such homomorphisms. The CSP and #CSP have been used to model a wide variety of combinatorial problems and have received a tremendous amount of attention from researchers from multiple disciplines.
In this paper we consider the modular version of the counting CSPs, that is, problems of the form #_pCSP({H}) of counting the number of homomorphisms to {H} modulo a fixed prime number p. Modular counting has been intensively studied during the last decade, although mainly in the case of graph homomorphisms. Here we continue the program of systematic research of modular counting of homomorphisms to general relational structures. The main results of the paper include a new way of reducing modular counting problems to smaller domains and a study of the complexity of such problems over 3-element domains and over conservative domains, that is, relational structures that allow to express (in a certain exact way) every possible unary predicate.
Cite as
Andrei A. Bulatov and Amirhossein Kazeminia. Modular Counting over 3-Element and Conservative Domains. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bulatov_et_al:LIPIcs.STACS.2026.22,
author = {Bulatov, Andrei A. and Kazeminia, Amirhossein},
title = {{Modular Counting over 3-Element and Conservative Domains}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {22:1--22:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.22},
URN = {urn:nbn:de:0030-drops-255114},
doi = {10.4230/LIPIcs.STACS.2026.22},
annote = {Keywords: Constraint Satisfaction Problem, Modular Counting}
}
Document
Simple Circuit Extensions for XOR in PTIME
Abstract
The Minimum Circuit Size Problem for Partial Functions (MCSP^*) is hard assuming the Exponential Time Hypothesis (ETH) (Ilango, 2020). This breakthrough result leveraged a characterization of the optimal {∧, ∨, ¬} circuits for n-bit OR (OR_n) and a reduction from the partial f-Simple Extension Problem where f = OR_n. It remains open to extend that reduction to show ETH-hardness of total MCSP. However, Ilango observed that the total f-Simple Extension Problem is easy whenever f is computed by read-once formulas (like OR_n). Therefore, extending Ilango’s proof to total MCSP would require replacing OR_n with a more complex but similarly well-understood Boolean function.
This work shows that the f-Simple Extension problem remains easy when f is the next natural candidate: XOR_n. We first develop a fixed-parameter tractable algorithm for the f-Simple Extension Problem that is efficient whenever the optimal circuits for f are (1) linear in size, (2) polynomially "few" and efficiently enumerable in the truth-table size (up to isomorphism and permutation of inputs), and (3) all have constant bounded fan-out. XOR_n satisfies all three of these conditions. When ¬ gates count towards circuit size, optimal XOR_n circuits are binary trees of n-1 subcircuits computing (¬)XOR₂ (Kombarov, 2011). We extend this characterization when ¬ gates do not contribute the circuit size. Thus, the XOR-Simple Extension Problem is in polynomial time under both measures of circuit complexity.
We conclude by discussing conjectures about the complexity of the f-Simple Extension problem for each explicit function f with known and unrestricted circuit lower bounds over the DeMorgan basis. Examining the conditions under which our Simple Extension Solver is efficient, we argue that multiplexer functions (MUX) are the most promising candidate for ETH-hardness of a Simple Extension Problem, towards proving ETH-hardness of total MCSP.
Cite as
Marco Carmosino, Ngu Dang, and Tim Jackman. Simple Circuit Extensions for XOR in PTIME. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{carmosino_et_al:LIPIcs.STACS.2026.23,
author = {Carmosino, Marco and Dang, Ngu and Jackman, Tim},
title = {{Simple Circuit Extensions for XOR in PTIME}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {23:1--23:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.23},
URN = {urn:nbn:de:0030-drops-255127},
doi = {10.4230/LIPIcs.STACS.2026.23},
annote = {Keywords: Minimum Circuit Size Problem, Circuit Lower Bounds, Exponential Time Hypothesis}
}
Document
Foremost, Fastest, Shortest: Temporal Graph Realization Under Various Path Metrics
Abstract
In this work, we follow the current trend on temporal graph realization, where one is given a property P and the goal is to determine whether there is a temporal graph, that is, a graph where the edge set changes over time, with property P. We consider the problems where the given property P is a prescribed matrix for the duration, length, or earliest arrival time of pairwise temporal paths. This means that we are given a matrix D and ask whether there is a temporal graph such that for any ordered pair of vertices (s,t), D_{s,t} equals the duration (length, or earliest arrival time, respectively) of any temporal path from s to t minimizing that specific temporal path metric. For shortest and earliest arrival temporal paths, we are the first to consider these problems as far as we know. We analyze these problems for many settings such as: strict and non-strict paths, periodic and non-periodic temporal graphs, and limited number of labels per edge (limited number of occurrences per edge over time). In contrast to all other path metrics, we show that for the earliest arrival times, we can achieve polynomial-time algorithms in periodic and non-periodic temporal graphs and for strict and and non-strict paths. However, the problem becomes NP-hard when the matrix does not contain a single integer but a set or range of possible allowed values. As we show, the problem can still be solved efficiently in this scenario, when the number of entries with more than one value is small, that is, we develop an FPT-algorithm for the number of such entries. For the setting of fastest paths, we achieve new hardness results that answers an open question by Klobas, Mertzios, Molter, and Spirakis [Theor. Comput. Sci. '25] about the parameterized complexity of the problem with respect to the vertex cover number and significantly improves over a previous hardness result for the feedback vertex set number. When considering shortest paths, we show that the periodic versions are polynomial-time solvable whereas the non-periodic versions become NP-hard.
Cite as
Justine Cauvi, Nils Morawietz, and Laurent Viennot. Foremost, Fastest, Shortest: Temporal Graph Realization Under Various Path Metrics. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cauvi_et_al:LIPIcs.STACS.2026.24,
author = {Cauvi, Justine and Morawietz, Nils and Viennot, Laurent},
title = {{Foremost, Fastest, Shortest: Temporal Graph Realization Under Various Path Metrics}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {24:1--24:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.24},
URN = {urn:nbn:de:0030-drops-255139},
doi = {10.4230/LIPIcs.STACS.2026.24},
annote = {Keywords: network design, temporal paths, foremost paths, fastest paths, shortest paths, non-strict paths, periodic temporal graphs}
}
Document
Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing
Abstract
Two graphs G and H are homomorphism indistinguishable over a graph class ℱ if they admit the same number of homomorphisms from every graph F ∈ ℱ. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems HomInd(ℱ) of deciding homomorphism indistinguishability over ℱ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of HomInd(ℱ), Seppelt (MFCS 2024) showed that HomInd(ℱ) is in randomised polynomial time for every graph class ℱ of bounded treewidth that can be defined in counting monadic second-order logic CMSO₂.
We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in P. For CMSO₂-definable graph classes ℱ of bounded pathwidth, we improve the previous complexity upper bound for HomInd(ℱ) from P to C_ = L and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as C_ = L-complete.
Cite as
Marek Černý and Tim Seppelt. Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cerny_et_al:LIPIcs.STACS.2026.25,
author = {\v{C}ern\'{y}, Marek and Seppelt, Tim},
title = {{Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {25:1--25:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.25},
URN = {urn:nbn:de:0030-drops-255144},
doi = {10.4230/LIPIcs.STACS.2026.25},
annote = {Keywords: treewidth, Courcelle’s theorem, logspace, multiplicity automata, polynomial identity testing}
}
Document
Approximate Cartesian Tree Matching with Substitutions
Abstract
The Cartesian tree of a sequence captures the relative order of the sequence’s elements. In recent years, Cartesian tree matching has attracted considerable attention, particularly due to its applications in time series analysis. Consider a text T of length n and a pattern P of length m. In the exact Cartesian tree matching problem, the task is to find all length-m fragments of T whose Cartesian tree coincides with the Cartesian tree CT(P) of the pattern. Although the exact version of the problem can be solved in linear time [Park et al., TCS 2020], it remains rather restrictive; for example, it is not robust to outliers in the pattern.
To overcome this limitation, we consider the approximate setting, where the goal is to identify all fragments of T that are close to some string whose Cartesian tree matches CT(P). In this work, we quantify closeness via the widely used Hamming distance metric. For a given integer parameter k > 0, we present an algorithm that computes all fragments of T that are at Hamming distance at most k from a string whose Cartesian tree matches CT(P). Our algorithm runs in time 𝒪(n √m ⋅ k^{2.5}) for k ≤ m^{1/5} and in time 𝒪(nk⁵) for k ≥ m^{1/5}, thereby improving upon the state-of-the-art 𝒪(nmk)-time algorithm of Kim and Han [TCS 2025] in the regime k = o(m^{1/4}).
On the way to our solution, we develop a toolbox of independent interest. First, we introduce a new notion of periodicity in Cartesian trees. Then, we lift multiple well-known combinatorial and algorithmic results for string matching and periodicity in strings to Cartesian tree matching and periodicity in Cartesian trees.
Cite as
Panagiotis Charalampopoulos, Jonas Ellert, and Manal Mohamed. Approximate Cartesian Tree Matching with Substitutions. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 26:1-26:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{charalampopoulos_et_al:LIPIcs.STACS.2026.26,
author = {Charalampopoulos, Panagiotis and Ellert, Jonas and Mohamed, Manal},
title = {{Approximate Cartesian Tree Matching with Substitutions}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {26:1--26:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.26},
URN = {urn:nbn:de:0030-drops-255151},
doi = {10.4230/LIPIcs.STACS.2026.26},
annote = {Keywords: Cartesian tree, Hamming distance, approximate pattern matching}
}
Document
The Communication Complexity of Combinatorial Auctions in Graphs
Abstract
We study truthful and non-truthful protocols for combinatorial auctions in which every item can be allocated to one of two agents (multigraphs), or more generally to a fixed number of agents (hypergraphs). We show some tight - both positive and impossibility - results for the communication complexity of approximating the optimal social welfare for general monotone, subadditive, or XOS valuations.
Cite as
George Christodoulou, Elias Koutsoupias, Annamária Kovács, and Ioannis Vlachos. The Communication Complexity of Combinatorial Auctions in Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{christodoulou_et_al:LIPIcs.STACS.2026.27,
author = {Christodoulou, George and Koutsoupias, Elias and Kov\'{a}cs, Annam\'{a}ria and Vlachos, Ioannis},
title = {{The Communication Complexity of Combinatorial Auctions in Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {27:1--27:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.27},
URN = {urn:nbn:de:0030-drops-255163},
doi = {10.4230/LIPIcs.STACS.2026.27},
annote = {Keywords: Auctions, Communication Complexity, Mechanism Design, Graphs}
}
Document
Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank
Abstract
Proving complexity lower bounds remains a challenging task: currently, we only know how to prove conditional uniform (algorithm) lower bounds and nonuniform (circuit) lower bounds in restricted circuit models. About a decade ago, Williams (STOC 2010) showed how to derive nonuniform lower bounds from uniform upper bounds: roughly, by designing a fast algorithm for checking satisfiability of circuits, one gets a lower bound for this circuit class. Since then, a number of results of this kind have been proved. For example, Jahanjou et al. (ICALP 2015) and Carmosino et al. (ITCS 2016) proved that if NSETH fails, then E^{NP} has series-parallel circuit size ω(n).
One can also derive nonuniform lower bounds from nondeterministic uniform lower bounds. Perhaps the most well-known example is the Karp-Lipton theorem (STOC 1980): if Σ₂ ≠ Π₂, then NP ⊄ P/poly. Some recent examples include the following. Nederlof (STOC 2020) proved a lower bound on the matrix multiplication tensor rank under an assumption that TSP cannot be solved faster than in 2ⁿ time. Belova et al. (SODA 2024) proved that there exists an explicit polynomial family of arithmetic circuit size Ω(n^{δ}), for any δ > 0, assuming that MAX-3-SAT cannot be solved faster than in 2ⁿ nondeterministic time. Williams (FOCS 2024) proved an exponential lower bound for ETHR ∘ ETHR circuits under the Orthogonal Vectors conjecture. Whereas all the lower bounds above are proved under strong assumptions that might eventually be refuted, the revealed connections are of great interest and may still give further insights: one may be able to weaken the used assumptions or to construct generators from other fine-grained reductions.
In this paper, we continue developing this line of research and show how uniform nondeterministic lower bounds can be used to construct generators of various types of combinatorial objects that are notoriously hard to analyze: Boolean functions of high circuit size, matrices of high rigidity, and tensors of high rank. Specifically, we prove the following.
- If, for some ε and k, k-SAT cannot be solved in input-oblivious co-nondeterministic time O(2^{(1/2+ε)n}), then there exists a monotone Boolean function family in coNP of monotone circuit size 2^{Ω(n / log n)}. Combining this with the result above, we get win-win circuit lower bounds: either E^{NP{}} requires series-parallel circuits of size ω(n) or coNP requires monotone circuits of size 2^{Ω(n / log n)}.
- If, for all ε > 0, MAX-3-SAT cannot be solved in co-nondeterministic time O(2^{(1 - ε)n}), then there exist small families of matrices with rigidity exceeding the best known constructions as well as small families of three-dimensional tensors of rank n^{1+Δ}, for some Δ > 0.
Cite as
Nikolai Chukhin, Alexander S. Kulikov, Ivan Mihajlin, and Arina Smirnova. Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chukhin_et_al:LIPIcs.STACS.2026.28,
author = {Chukhin, Nikolai and Kulikov, Alexander S. and Mihajlin, Ivan and Smirnova, Arina},
title = {{Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {28:1--28:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.28},
URN = {urn:nbn:de:0030-drops-255177},
doi = {10.4230/LIPIcs.STACS.2026.28},
annote = {Keywords: computational complexity, circuit complexity, lower bounds, conditional lower bounds, monotone circuits, matrix rigidity, tensor rank, arithmetic circuits, fine-grained complexity}
}
Document
A Pumping-Like Lemma for Languages over Infinite Alphabets
Abstract
We prove a kind of a pumping lemma for languages accepted by one-register alternating finite-memory automata. As a corollary, we obtain that the set of lengths of words in such languages is semi-linear.
Cite as
Yoav Danieli. A Pumping-Like Lemma for Languages over Infinite Alphabets. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{danieli:LIPIcs.STACS.2026.29,
author = {Danieli, Yoav},
title = {{A Pumping-Like Lemma for Languages over Infinite Alphabets}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {29:1--29:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.29},
URN = {urn:nbn:de:0030-drops-255185},
doi = {10.4230/LIPIcs.STACS.2026.29},
annote = {Keywords: infinite alphabets, pumping lemma, alternation, semi-linearity}
}
Document
Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions
Abstract
Given Boolean functions f, g : 𝔽₂ⁿ → {-1,+1}, we say they are linearly isomorphic if there exists A ∈ GL_n(𝔽₂) such that f(x) = g(Ax) for all x. We study this problem in the tolerant property testing framework under the known-unknown model, where g is given explicitly and f is accessible only via oracle queries, meaning the algorithm may adaptively request the value of f(x) for inputs x ∈ 𝔽₂ⁿ of its choice. Given parameters ε ≥ 0 and ω > 0, the goal is to distinguish whether there exists A ∈ GL_n(𝔽₂) such that the normalized Hamming distance between f and g(Ax) is at most ε, or whether for every A ∈ GL_n(𝔽₂) the distance is at least ε+ω.
Our main result is a tolerant tester making Õ ((m/ω) ⁴) queries to f, where m is an upper bound on the spectral norm of g, improving the previous Õ ((m/ω) ^{24}) bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of Ω(m²) for constant ω (for example, ω = 1/4), improving the prior Ω(log m) lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana-McFarland functions from symmetric-key cryptography.
Cite as
Swarnalipa Datta, Arijit Ghosh, Chandrima Kayal, Manaswi Paraashar, and Manmatha Roy. Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 30:1-30:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{datta_et_al:LIPIcs.STACS.2026.30,
author = {Datta, Swarnalipa and Ghosh, Arijit and Kayal, Chandrima and Paraashar, Manaswi and Roy, Manmatha},
title = {{Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {30:1--30:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.30},
URN = {urn:nbn:de:0030-drops-255194},
doi = {10.4230/LIPIcs.STACS.2026.30},
annote = {Keywords: Boolean Function, Isomorphism of Boolean Function, Fourier Analysis, Sublinear Algorithm, Property Testing}
}
Document
Lower Bounds for Ranking-Based Pivot Rules
Abstract
The existence of a polynomial pivot rule for the simplex method for linear programming, policy iteration for Markov decision processes, and strategy improvement for parity games each are prominent open problems in their respective fields. While numerous natural candidates for efficient rules have been eliminated, all existing lower bound constructions are tailored to individual or small sets of pivot rules. We introduce a unified framework for formalizing classes of rules according to the information about the input that they rely on. Within this framework, we show lower bounds for ranking-based classes of rules that base their decisions on orderings of the improving pivot steps induced by the underlying data. Our first result is a superpolynomial lower bound for strategy improvement, obtained via a family of sink parity games, which applies to memory-based generalizations of Bland’s rule that only access the input by comparing the ranks of improving edges in some global order. Our second result is a subexponential lower bound for policy iteration, obtained via a family of Markov decision processes, which applies to memoryless rules that only access the input by comparing improving actions according to their ranks in a global order, their reduced costs, and the associated improvements in objective value. Both results carry over to the simplex method for linear programming.
Cite as
Yann Disser, Georg Loho, Matthew Maat, and Nils Mosis. Lower Bounds for Ranking-Based Pivot Rules. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{disser_et_al:LIPIcs.STACS.2026.31,
author = {Disser, Yann and Loho, Georg and Maat, Matthew and Mosis, Nils},
title = {{Lower Bounds for Ranking-Based Pivot Rules}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {31:1--31:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.31},
URN = {urn:nbn:de:0030-drops-255207},
doi = {10.4230/LIPIcs.STACS.2026.31},
annote = {Keywords: lower bounds, Markov decision processes, parity games, pivot rules, policy iteration, simplex method}
}
Document
Optimal Verification of a Minimum-Weight Basis in an Uncertainty Matroid
Abstract
Research in explorable uncertainty addresses combinatorial optimization problems where there is partial information about the values of numeric input parameters, and exact values of these parameters can be determined by performing costly queries. The goal is to design an adaptive query strategy that minimizes the query cost incurred in computing an optimal solution. Solving such problems generally requires that we be able to solve the associated verification problem: given the answers to all queries in advance, find a minimum-cost set of queries that certifies an optimal solution to the combinatorial optimization problem. We present a polynomial-time algorithm for verifying a minimum-weight basis of a matroid, where each weight lies in a given uncertainty area. These areas may be finite sets, real intervals, or unions of open and closed intervals, strictly generalizing previous work by Erlebach and Hoffman which only handled the special case of open intervals. Our algorithm introduces new techniques to address the resulting challenges.
Verification problems are of particular importance in the area of explorable uncertainty, as the structural insights and techniques used to solve the verification problem often heavily influence work on the corresponding online problem and its stochastic variant. In our case, we use structural results from the verification problem to give a best-possible algorithm for a promise variant of the corresponding adaptive online problem. Finally, we show that our algorithms can be applied to two learning-augmented variants of the minimum-weight basis problem under explorable uncertainty.
Cite as
Haya Diwan, Lisa Hellerstein, Nicole Megow, and Jens Schlöter. Optimal Verification of a Minimum-Weight Basis in an Uncertainty Matroid. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{diwan_et_al:LIPIcs.STACS.2026.32,
author = {Diwan, Haya and Hellerstein, Lisa and Megow, Nicole and Schl\"{o}ter, Jens},
title = {{Optimal Verification of a Minimum-Weight Basis in an Uncertainty Matroid}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {32:1--32:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.32},
URN = {urn:nbn:de:0030-drops-255216},
doi = {10.4230/LIPIcs.STACS.2026.32},
annote = {Keywords: Matroid verification, minimum-weight basis, query strategy, uncertainty matroid, explorable uncertainty}
}
Document
Higher Hardness Results for the Reconfiguration of Odd Matchings
Abstract
We study the reconfiguration of odd matchings of combinatorial graphs. Odd matchings are matchings that cover all but one vertex of a graph. A reconfiguration step, or flip, is an operation that matches the isolated vertex and, consequently, isolates another vertex. The flip graph of odd matchings is a graph that has all odd matchings of a graph as vertices and an edge between two vertices if their corresponding matchings can be transformed into one another via a single flip.
We show that computing the diameter of the flip graph of odd matchings is Π₂^p-hard. This complements a recent result by Wulf [FOCS25] that it is Π₂^p-hard to compute the diameter of the flip graph of perfect matchings where a flip swaps matching edges along a single cycle of unbounded size.
Further, we show that computing the radius of the flip graph of odd matchings is Σ₃^p-hard. The respective decision problems for the diameter and the radius are also complete in the respective level of the polynomial hierarchy. This shows that computing the radius of the flip graph of odd matchings is provably harder than computing its diameter, unless the polynomial hierarchy collapses.
Finally, we reduce set cover to the problem of finding shortest flip sequences. As a consequence, we show APX-hardness and that the problem cannot be approximated by a sublogarithmic factor. By doing so, we answer a question asked by Aichholzer, Brenner, Dorfer, Hoang, Perz, Rieck, and Verciani [GD25].
Cite as
Joseph Dorfer. Higher Hardness Results for the Reconfiguration of Odd Matchings. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dorfer:LIPIcs.STACS.2026.33,
author = {Dorfer, Joseph},
title = {{Higher Hardness Results for the Reconfiguration of Odd Matchings}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {33:1--33:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.33},
URN = {urn:nbn:de:0030-drops-255222},
doi = {10.4230/LIPIcs.STACS.2026.33},
annote = {Keywords: Graph Reconfiguration Problems, Flip Graphs, Polynomial Hierarchy, APX-hardness}
}
Document
Algorithm and Strategy Construction for Sure-Almost-Sure Stochastic Parity Games
Abstract
We consider turn-based stochastic two-player games with a combination of a parity condition that must hold surely, that is in all possible outcomes, and of a parity condition that must hold almost-surely, that is with probability 1. The problem of deciding the existence of a winning strategy in such games is central in the framework of synthesis beyond worst-case where a hard requirement that must hold surely is combined with a softer requirement. Recent works showed that the problem is coNP-complete, and infinite-memory strategies are necessary in general, even in one-player games (i.e., Markov decision processes). However, memoryless strategies are sufficient for the opponent player. Despite these comprehensive results, the known algorithmic solution enumerates all memoryless strategies of the opponent, which is exponential in all cases, and does not construct a winning strategy when one exists.
We present a recursive algorithm, based on a characterisation of the winning region, that gives a deeper insight into the problem. In particular, we show how to construct a winning strategy to achieve the combination of sure and almost-sure parity, and we derive new complexity and memory bounds for special classes of the problem, defined by fixing the index of either of the two parity conditions.
Cite as
Laurent Doyen and Shibashis Guha. Algorithm and Strategy Construction for Sure-Almost-Sure Stochastic Parity Games. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 34:1-34:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{doyen_et_al:LIPIcs.STACS.2026.34,
author = {Doyen, Laurent and Guha, Shibashis},
title = {{Algorithm and Strategy Construction for Sure-Almost-Sure Stochastic Parity Games}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {34:1--34:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.34},
URN = {urn:nbn:de:0030-drops-255230},
doi = {10.4230/LIPIcs.STACS.2026.34},
annote = {Keywords: stochastic games, parity objectives, reactive synthesis}
}
Document
A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity
Abstract
We are interested in what happens when we take a Π₁ combinatorial statement, write its negation as a homogeneous quadratic feasibility problem (HQFP), and relax the problem into a positive semidefinite feasibility problem. This question is particularly interesting owing to the fact that any statement written as a PSD feasibility problem can be proven or disproven using a short proof. We investigate this for one very simple and one very complicated statement.
The simple statement we look at is the pigeonhole principle. We prove that the relaxed negation of the PHP remains unsatisfiable and we thus obtain a new "quantum" pigeonhole principle (QPHP) which is a stronger statement than the vanilla PHP. It states that if we take n copies of the same state, and measure each copy using a measurement with only n-1 outcomes (the measurement can be different for different copies), then there will be an outcome j and two copies i₁, i₂ where the resulting states, obtained when the outcome is j for both copies, are not orthogonal.
We then look at the statement "the deterministic communication complexity of f is ≤ k", where f could be either a function or a relation. We write this statement in two equivalent ways, using two different HQFPs. By relaxing to PSD feasibility, we increase the set of available protocols, and thus we always get a communication model which is stronger than deterministic communication complexity. An argument from proof complexity shows that any model obtained in this way will solve all Karchmer-Wigderson games efficiently. However, the argument is very indirect and does not give us an explicit protocol that solves the Karchmer-Wigderson games. We then work to find such protocols in the two communication models obtained by relaxing our two formulations.
When relaxing the first of the two formulations we obtain a structured variant of the γ₂ norm. This communication model is to subunit γ₂ norm matrices like deterministic protocols are to rectangles, and so we call the protocols in this model γ₂ protocols. We show that log-inverse-discrepancy is a lower-bound for this model. We then show how to compute equality (deterministically) using O(1) bits of γ₂-communication, which implies that KW games are easy in the model.
When relaxing the second of the two formulations we obtain what we call quantum lab protocols. This model happens to have a functional description, wherein Alice and Bob communicate solely via the outcomes of binary measurements of a shared quantum state (whose initial state is independent of the inputs). They are required to give the correct output with zero error probability. We use our QPHP to prove a lower-bound of n against two-round quantum lab protocols for equality. However we also show that any Boolean function f can be computed in three rounds and four measurements.
Cite as
Pavel Dvořák, Bruno Loff, and Suhail Sherif. A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dvorak_et_al:LIPIcs.STACS.2026.35,
author = {Dvo\v{r}\'{a}k, Pavel and Loff, Bruno and Sherif, Suhail},
title = {{A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {35:1--35:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.35},
URN = {urn:nbn:de:0030-drops-255243},
doi = {10.4230/LIPIcs.STACS.2026.35},
annote = {Keywords: Proofs, Semidefinite Programs, Quantum Pigeonhole Principle, Communication Complexity}
}
Document
Time-Optimal Construction of String Synchronizing Sets
Abstract
A powerful design principle behind many modern string algorithms is local consistency: breaking the symmetry between string positions based on their small contexts so that matching fragments are handled consistently. Among the most influential instantiations of this principle are string synchronizing sets [Kempa & Kociumaka; STOC 2019]. A τ-synchronizing set of a string of length n is a set of O(n/τ) string positions, chosen using their length-2τ contexts, such that (outside of highly periodic regions) every block of τ consecutive positions contains at least one element of the set. Synchronizing sets have found dozens of applications in diverse settings, from quantum and dynamic algorithms to fully compressed computation. In the classic word RAM model, particularly for strings over small alphabets, they enabled faster solutions to core problems in data compression, text indexing, and string similarity.
In this work, we show that any string T ∈ [0 .. σ)ⁿ can be preprocessed in O(n log σ / log n) time so that, for any given integer τ ∈ [1 .. n], a τ-synchronizing set of T can be constructed in O((n log τ)/(τ log n)) time. Both bounds are optimal in the word RAM model with machine word size w = Θ(log n), matching the information-theoretic minimum for the input and output sizes, respectively. Previously, constructing a τ-synchronizing set required O(n/τ) time after an O(n)-time preprocessing [Kociumaka, Radoszewski, Rytter, and Waleń; SICOMP 2024], or, in the restricted regime of τ < 0.2 log_σ n, without any preprocessing needed [Kempa & Kociumaka; STOC 2019].
A simple instantiation of our method outputs the synchronizing set as a sorted list in O(n/τ) time, or as a bitmask in O(n/log n) time. Our optimal construction produces a compact fully indexable dictionary, supporting select queries in O(1) time and rank queries in O(log ((log τ)/(log log n))) time. The latter complexity matches known unconditional cell-probe lower bounds for τ ≤ n^{1-Ω(1)}.
To achieve this, we introduce a general framework for efficiently processing sparse integer sequences via a custom variable-length encoding. We also augment the optimal variant of van Emde Boas trees [Pătraşcu & Thorup; STOC 2006] with a deterministic linear-time construction. When the set is represented as a bitmask under our sparse encoding, the same guarantees for select and rank queries hold after preprocessing in time proportional to the size of our encoding (in words).
Cite as
Jonas Ellert and Tomasz Kociumaka. Time-Optimal Construction of String Synchronizing Sets. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 36:1-36:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ellert_et_al:LIPIcs.STACS.2026.36,
author = {Ellert, Jonas and Kociumaka, Tomasz},
title = {{Time-Optimal Construction of String Synchronizing Sets}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {36:1--36:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.36},
URN = {urn:nbn:de:0030-drops-255258},
doi = {10.4230/LIPIcs.STACS.2026.36},
annote = {Keywords: synchronizing sets, local consistency, packed strings}
}
Document
Random Models and Guarded Logic
Abstract
Building on ideas of Gurevich and Shelah for the Gödel Class, we present a new probabilistic proof of the finite model property for the Guarded Fragment of First-Order Logic. Our proof is conceptually simple and yields the optimal doubly-exponential upper bound on the size of minimal models. We precisely analyse the obtained bound, up to constant factors in the exponents, and construct sentences that enforce models of tightly matching size. The probabilistic approach adapts naturally to the Triguarded Fragment, an extension of the Guarded Fragment that also subsumes the Two-Variable Fragment. Finally, we derandomise the probabilistic proof by providing an explicit model construction which replaces randomness with deterministic hash functions.
Cite as
Oskar Fiuk. Random Models and Guarded Logic. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 37:1-37:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fiuk:LIPIcs.STACS.2026.37,
author = {Fiuk, Oskar},
title = {{Random Models and Guarded Logic}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {37:1--37:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.37},
URN = {urn:nbn:de:0030-drops-255269},
doi = {10.4230/LIPIcs.STACS.2026.37},
annote = {Keywords: guarded fragment, finite model property, probabilistic method}
}
Document
Fully Dynamic Spectral Sparsification for Directed Hypergraphs
Abstract
There has been a surge of interest in spectral hypergraph sparsification, a natural generalization of spectral sparsification for graphs. In this paper, we present a simple fully dynamic algorithm for maintaining spectral hypergraph sparsifiers of directed hypergraphs. Our algorithm achieves a near-optimal size of O(n² / ε ² log ⁷ m) and amortized update time of O(r² log ³ m), where n is the number of vertices, and m and r respectively upper bound the number of hyperedges and the rank of the hypergraph at any time.
We also extend our approach to the parallel batch-dynamic setting, where a batch of any k hyperedge insertions or deletions can be processed with O(kr² log ³ m) amortized work and O(log ² m) depth. This constitutes the first spectral-based sparsification algorithm in this setting.
Cite as
Sebastian Forster, Gramoz Goranci, and Ali Momeni. Fully Dynamic Spectral Sparsification for Directed Hypergraphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{forster_et_al:LIPIcs.STACS.2026.38,
author = {Forster, Sebastian and Goranci, Gramoz and Momeni, Ali},
title = {{Fully Dynamic Spectral Sparsification for Directed Hypergraphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {38:1--38:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.38},
URN = {urn:nbn:de:0030-drops-255272},
doi = {10.4230/LIPIcs.STACS.2026.38},
annote = {Keywords: Spectral sparsification, Dynamic algorithms, (Directed) hypergraphs, Data structures}
}
Document
Planting and MCMC Sampling from the Potts Model
Abstract
We consider the problem of sampling from the ferromagnetic q-state Potts model on the random d-regular graph with parameter β > 0. A key difficulty that arises in sampling from the model is the existence of a "metastability" window β ∈ (β_u,β_u'), where roughly the distribution has two competing modes, the so-called disordered and ordered phases. This causes classical Markov-chain algorithms to be slow mixing from worst-case initialisations. Nevertheless, Helmuth, Jenssen and Perkins (SODA '19) designed a sampling algorithm that works for all β, when d ≥ 5 and q = d^{Ω(d)}, using polymers and cluster expansion methods; more recently, their analysis technique has been adapted to show that a Markov chain (random-cluster dynamics) mixes fast when initialised appropriately, in the same regime of q,d,β.
Despite these positive algorithmic results, a well-known bottleneck behind cluster-expansion arguments is that they inherently only work for large q, whereas it is widely conjectured that sampling on the random d-regular graph is possible for all q,d ≥ 3. The only result so far that applies to general q,d ≥ 3 is by Blanca and Gheissari who showed that the random-cluster dynamics mixes fast in the "uniqueness" regime β < β_u where roughly only the disordered mode exists. For β ≥ β_u however, a second subdominant mode emerges creating bottlenecks and giving rise to correlations which have been hard to handle, especially for small values of q and d.
Our main contribution is to perform a delicate analysis of the Potts distribution and the random-cluster dynamics that goes beyond the threshold β_u. We use planting as the main tool, a technique used in the analysis of random CSPs to capture how the space of solutions is correlated with the structure of the random instance. While planting arguments provide only weak sampling guarantees generically, here we instead combine planting with the analysis of random-cluster dynamics to obtain significantly stronger guarantees. We are thus able to show that the random-cluster dynamics initialised from all-out mixes fast for all integers q,d ≥ 3 beyond the uniqueness threshold β_u, all the way to the optimal threshold β_c ∈ (β_u,β_u') where the dominant mode switches from disordered to ordered. A more involved analysis also applies to the ordered regime β > β_c where we obtain an algorithm for all d ≥ 3 and q ≥ (5d)⁵, improving significantly upon the previous range of q,d by Carlson, Davies, Fraiman, Kolla, Potukuchi, and Yap (FOCS'22).
Cite as
Andreas Galanis, Leslie Ann Goldberg, and Paulina Smolarova. Planting and MCMC Sampling from the Potts Model. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{galanis_et_al:LIPIcs.STACS.2026.39,
author = {Galanis, Andreas and Goldberg, Leslie Ann and Smolarova, Paulina},
title = {{Planting and MCMC Sampling from the Potts Model}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {39:1--39:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.39},
URN = {urn:nbn:de:0030-drops-255280},
doi = {10.4230/LIPIcs.STACS.2026.39},
annote = {Keywords: approximate sampling, Glauber dynamics, Potts model, random cluster model}
}
Document
Maker-Maker Games of Rank 4 Are PSPACE-Complete
Abstract
The Maker-Maker convention of positional games is played on a hypergraph whose edges are interpreted as winning sets. Two players take turns picking a previously unpicked vertex, aiming at being first to pick all the vertices of some edge. Optimal play can only lead to a first player win or a draw, and deciding between the two is known to be PSPACE-complete even for 6-uniform hypergraphs. We establish PSPACE-completeness for hypergraphs of rank 4. As an intermediary, we use the recently introduced achievement positional games, a more general convention in which each player has their own winning sets (blue and red). We show that deciding whether the blue player has a winning strategy as the first player is PSPACE-complete even with blue edges of size 2 or 3 and pairwise disjoint red edges of size 2. The result for hypergraphs of rank 4 in the Maker-Maker convention follows as a simple corollary.
Cite as
Florian Galliot and Jonas Sénizergues. Maker-Maker Games of Rank 4 Are PSPACE-Complete. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{galliot_et_al:LIPIcs.STACS.2026.40,
author = {Galliot, Florian and S\'{e}nizergues, Jonas},
title = {{Maker-Maker Games of Rank 4 Are PSPACE-Complete}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {40:1--40:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.40},
URN = {urn:nbn:de:0030-drops-255298},
doi = {10.4230/LIPIcs.STACS.2026.40},
annote = {Keywords: Game theory, Positional games, Combinatorial games, Complexity, Hypergraphs}
}
Document
On the Complexity of Computing Strahler Numbers
Abstract
It is shown that the problem of computing the Strahler number of a binary tree given as a term is complete for the circuit complexity class uniform NC¹. For several variants, where the binary tree is given by a pointer structure or in a succinct form by a directed acyclic graph or a tree straight-line program, the complexity of computing the Strahler number is determined as well. The problem, whether a given context-free grammar in Chomsky normal form produces a derivation tree (resp., an acyclic derivation tree), whose Strahler number is at least a given number k is shown to be P-complete (resp., PSPACE-complete).
Cite as
Moses Ganardi and Markus Lohrey. On the Complexity of Computing Strahler Numbers. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ganardi_et_al:LIPIcs.STACS.2026.41,
author = {Ganardi, Moses and Lohrey, Markus},
title = {{On the Complexity of Computing Strahler Numbers}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {41:1--41:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.41},
URN = {urn:nbn:de:0030-drops-255301},
doi = {10.4230/LIPIcs.STACS.2026.41},
annote = {Keywords: Strahler number, circuit complexity classes, context-free grammars}
}
Document
Computing Twin-Width via Treedepth and Vertex Integrity
Abstract
Twin-width is a graph parameter that has become central to explaining the fixed-parameter tractability of first-order model checking across many graph classes. Despite its algorithmic importance, computing twin-width remains poorly understood: even recognizing graphs of twin-width at most four is NP-hard, and no fixed-parameter approximations parameterized by twin-width itself are known. A recent approach towards breaking this barrier focuses on first developing fixed-parameter algorithms for computing or approximating twin-width under parameterizations distinct from twin-width.
Our first result establishes that approximating twin-width is fixed-parameter tractable when parameterized by treedepth, thereby breaking the long-standing barrier that all previous tractable parameterizations were based on deletion distance. The proof proceeds via oriented twin-width, yielding the first constructive evidence that this variant may be easier to handle algorithmically. As our second main result, we show that computing twin-width exactly is fixed-parameter tractable with respect to vertex integrity. This constitutes the first non-trivial parameterized algorithm for computing optimal contraction sequences.
Cite as
Robert Ganian and Mathis Rocton. Computing Twin-Width via Treedepth and Vertex Integrity. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ganian_et_al:LIPIcs.STACS.2026.42,
author = {Ganian, Robert and Rocton, Mathis},
title = {{Computing Twin-Width via Treedepth and Vertex Integrity}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {42:1--42:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.42},
URN = {urn:nbn:de:0030-drops-255318},
doi = {10.4230/LIPIcs.STACS.2026.42},
annote = {Keywords: twin-width, fixed-parameter algorithms, treedepth, vertex integrity}
}
Document
Stealing from the Dragon’s Hoard: Online Unbounded Knapsack With Removal
Abstract
We introduce the Online Unbounded Knapsack Problem with Removal, a variation of the well-known Online Knapsack Problem. Items, each with a weight and value, arrive online and an algorithm must decide on whether or not to pack them into a knapsack with a fixed weight limit. An item may be packed an arbitrary number of times and items may be removed from the knapsack at any time without cost. The goal is to maximize the total value of items packed, while respecting a weight limit. We show that this is one of the very few natural online knapsack variants that allow for competitive deterministic algorithms in the general setting, by providing an algorithm with competitivity 1.6911. We complement this with a lower bound of 1.5877.
We also analyze the proportional setting, where the weight and value of any single item agree, and show that deterministic algorithms can be exactly 3/2-competitive. Lastly, we give lower and upper bounds of 6/5 and 4/3 on the competitivity of randomized algorithms in this setting.
Cite as
Matthias Gehnen and Moritz Stocker. Stealing from the Dragon’s Hoard: Online Unbounded Knapsack With Removal. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 43:1-43:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gehnen_et_al:LIPIcs.STACS.2026.43,
author = {Gehnen, Matthias and Stocker, Moritz},
title = {{Stealing from the Dragon’s Hoard: Online Unbounded Knapsack With Removal}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {43:1--43:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.43},
URN = {urn:nbn:de:0030-drops-255327},
doi = {10.4230/LIPIcs.STACS.2026.43},
annote = {Keywords: online problems, online knapsack, unbounded knapsack, removal}
}
Document
Optimal Average Disk-Inspection via Fermat’s Principle
Abstract
This work resolves the optimal average-case cost of the Disk-Inspection problem, a variant of Bellman’s 1955 lost-in-a-forest problem. In Disk-Inspection, a mobile agent starts at the center of a unit disk and follows a trajectory that inspects perimeter points whenever the disk does not obstruct visibility. The worst-case cost was solved optimally in 1957 by Isbell [Isbell, 1957], but the average-case version remained open, with heuristic upper bounds proposed by Gluss [Gluss, 1961] in 1961 and improved only recently in [Conley and Georgiou, 2025].
Our approach applies Fermat’s Principle of Least Time from optics to the discretization framework of [Conley and Georgiou, 2025], showing that optimal solutions are captured by a one-parameter family of recurrences independent of the discretization size. In the continuum limit these recurrences give rise to a single-parameter optimal control problem, whose trajectories coincide with limiting solutions of the original Disk-Inspection problem. A crucial step is proving that the optimal initial condition generates a trajectory that avoids the unit disk, thereby validating the optics formulation and reducing the many-variable optimization to a rigorous one-parameter problem. In particular, this disproves Gluss’s conjecture [Gluss, 1961] that optimal trajectories must touch the disk.
Our analysis determines the exact optimal average-case inspection cost, equal to 3.549259… and certified to at least six digits of accuracy.
Cite as
Konstantinos Georgiou. Optimal Average Disk-Inspection via Fermat’s Principle. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{georgiou:LIPIcs.STACS.2026.44,
author = {Georgiou, Konstantinos},
title = {{Optimal Average Disk-Inspection via Fermat’s Principle}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {44:1--44:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.44},
URN = {urn:nbn:de:0030-drops-255331},
doi = {10.4230/LIPIcs.STACS.2026.44},
annote = {Keywords: Inspection, Disk, Average-Case Performance}
}
Document
The Complexity of Homomorphism Reconstruction Revisited
Abstract
We revisit the algorithmic problem of reconstructing a graph from homomorphism counts that has first been studied in (Böker et al., STACS 2024): given graphs F₁,…,F_k and counts m₁,…,m_k, decide if there is a graph G such that the number of homomorphisms from F_i to G is m_i, for all i. We prove that the problem is NEXP-hard if the counts m_i are specified in binary and Σ₂^p-complete if they are in unary.
Furthermore, as a positive result, we show that the unary version can be solved in polynomial time if the constraint graphs are stars of bounded size.
Cite as
Timo Gervens, Martin Grohe, Louis Härtel, and Philipp da Silva Fonseca. The Complexity of Homomorphism Reconstruction Revisited. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gervens_et_al:LIPIcs.STACS.2026.45,
author = {Gervens, Timo and Grohe, Martin and H\"{a}rtel, Louis and da Silva Fonseca, Philipp},
title = {{The Complexity of Homomorphism Reconstruction Revisited}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {45:1--45:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.45},
URN = {urn:nbn:de:0030-drops-255342},
doi = {10.4230/LIPIcs.STACS.2026.45},
annote = {Keywords: graph homomorphism, nexp-complete, counting complexity}
}
Document
Smaller Circuits for Bit Addition
Abstract
Bit addition arises virtually everywhere in digital circuits: arithmetic operations, increment/decrement operators, computing addresses and table indices, and so on. Since bit addition is such a basic task in Boolean circuit synthesis, a lot of research has been done on constructing efficient circuits for various special cases of it. A vast majority of these results are devoted to optimizing the circuit depth (also known as delay).
In this paper, we investigate the circuit size (also known as area) over the full binary basis of bit addition. Most of the known circuits are built from Half Adders and Full Adders as suggested by Dadda in 1965 for designing multiplier circuits. Applying these ideas to the bit addition function, one gets a 5n-3m upper bound on its circuit size, where n is the number of input bits and m is the number of output bits. We prove an upper bound 4.5n-2m. In the regimes where m is small compared to n (for example, for computing the sum of n bits or multiplying two n-bit integers), this leads to 10% improvement. We also show that it is provably impossible to improve the two upper bounds above to 5n-3.01m or 4.5n-2.51m. We achieve this by establishing that the circuit size of the increment function (a special case of the bit addition function with m = n+1) is equal to 2n.
We complement our theoretical result by an open-source implementation of generators producing circuits for bit addition and multiplication. The generators allow one to produce the corresponding circuits in two lines of code and to compare them to existing designs.
Cite as
Mikhail Goncharov, Alexander S. Kulikov, and Georgie Levtsov. Smaller Circuits for Bit Addition. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{goncharov_et_al:LIPIcs.STACS.2026.46,
author = {Goncharov, Mikhail and Kulikov, Alexander S. and Levtsov, Georgie},
title = {{Smaller Circuits for Bit Addition}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {46:1--46:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.46},
URN = {urn:nbn:de:0030-drops-255356},
doi = {10.4230/LIPIcs.STACS.2026.46},
annote = {Keywords: bit addition, summation, multiplier, multiplication, Boolean, circuit, synthesis, combinational, digital}
}
Document
On the Hardness of the One-Sided Code Sparsifier Problem
Abstract
The notion of code sparsification was introduced by Khanna, Putterman and Sudan (SODA 2024) as an analogue to the more established notion of cut sparsification in graphs and hypergraphs. In particular, for α ∈ (0,1), an (unweighted) one-sided α-sparsifier for a linear code 𝒞 ⊆ 𝐅₂ⁿ is a subset S ⊆ [n] such that the weight of each codeword projected onto the coordinates in S is preserved up to an α fraction. Recently, Gharan and Sahami (arXiv:2502.02799) show the existence of one-sided 1/2-sparsifiers of size n/2+O(√{kn}) for any linear code, where k is the dimension of 𝒞. In this paper, we consider the computational problem of finding a one-sided 1/2-sparsifier of minimal size, and show that it is NP-hard, via a reduction from the classical nearest codeword problem. We also show hardness of approximation results.
Cite as
Elena Grigorescu and Alice Moayyedi. On the Hardness of the One-Sided Code Sparsifier Problem. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 47:1-47:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{grigorescu_et_al:LIPIcs.STACS.2026.47,
author = {Grigorescu, Elena and Moayyedi, Alice},
title = {{On the Hardness of the One-Sided Code Sparsifier Problem}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {47:1--47:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.47},
URN = {urn:nbn:de:0030-drops-255365},
doi = {10.4230/LIPIcs.STACS.2026.47},
annote = {Keywords: Code sparsifiers, NP-hardness, Approximation hardness}
}
Document
Polynomial Complementation of Nondeterministic Two-Way Finite Automata by 1-Limited Automata
Abstract
We prove that, paying a polynomial increase in size only, every unrestricted two-way nondeterministic finite automaton (2NFA) can be complemented by a 1-limited automaton (1-LA), a nondeterministic extension of 2NFAs still characterizing regular languages. The resulting machine is actually a restricted form of 1-LAs - known as 2NFAs with common guess - and is self-verifying. A corollary of our construction is that a single exponential is necessary and sufficient for complementing 1-LAs.
Cite as
Bruno Guillon, Luca Prigioniero, and Javad Taheri. Polynomial Complementation of Nondeterministic Two-Way Finite Automata by 1-Limited Automata. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{guillon_et_al:LIPIcs.STACS.2026.48,
author = {Guillon, Bruno and Prigioniero, Luca and Taheri, Javad},
title = {{Polynomial Complementation of Nondeterministic Two-Way Finite Automata by 1-Limited Automata}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {48:1--48:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.48},
URN = {urn:nbn:de:0030-drops-255378},
doi = {10.4230/LIPIcs.STACS.2026.48},
annote = {Keywords: descriptional complexity, inductive counting, common-guess}
}
Document
2D Minimal Graph Rigidity is in NC for One-Crossing-Minor-Free Graphs
Abstract
Minimally rigid graphs can be decided and embedded in the plane efficiently, i.e. in polynomial time. There is also an efficient randomized parallel algorithm, i.e. in RNC. We present an NC-algorithm to decide whether one-crossing-minor-free graphs are minimally rigid. In the special case of K_{3,3}-free graphs, we also compute an infinitesimally rigid embedding in NC.
Cite as
Rohit Gurjar, Kilian Rothmund, and Thomas Thierauf. 2D Minimal Graph Rigidity is in NC for One-Crossing-Minor-Free Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gurjar_et_al:LIPIcs.STACS.2026.49,
author = {Gurjar, Rohit and Rothmund, Kilian and Thierauf, Thomas},
title = {{2D Minimal Graph Rigidity is in NC for One-Crossing-Minor-Free Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {49:1--49:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.49},
URN = {urn:nbn:de:0030-drops-255385},
doi = {10.4230/LIPIcs.STACS.2026.49},
annote = {Keywords: Graph Rigidity, Parallel Algorithms, Polynomial Identity Testing, Derandomization}
}
Document
A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs
Abstract
Given a planar graph, a subset of its vertices called terminals, and k ∈ ℕ, the Face Cover Number problem asks whether the terminals lie on the boundaries of at most k faces of some embedding of the input graph. When a plane graph is given in the input, the problem is known to have a polynomial kernel [Valentin Garnero et al., 2017]. In this paper, we present the first polynomial kernel for Face Cover Number when the input is a planar graph (without a fixed embedding). Our approach overcomes the challenge of not having a predefined set of face boundaries by building a kernel bottom-up on an SPR-tree while preserving the essential properties of the face cover along the way.
Cite as
Thekla Hamm, Sukanya Pandey, and Krisztina Szilágyi. A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hamm_et_al:LIPIcs.STACS.2026.50,
author = {Hamm, Thekla and Pandey, Sukanya and Szil\'{a}gyi, Krisztina},
title = {{A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {50:1--50:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.50},
URN = {urn:nbn:de:0030-drops-255392},
doi = {10.4230/LIPIcs.STACS.2026.50},
annote = {Keywords: Kernelization, Planar Graphs, SPQR-tree}
}
Document
Structural Parameterization of Steiner Tree Packing
Abstract
Steiner Tree Packing (STP) is a notoriously hard problem in classical complexity theory, which is of practical relevance to VLSI circuit design. Previous research has approached this problem by providing heuristic or approximate algorithms. In this paper, we show the first FPT algorithms for STP parameterized by structural parameters of the input graph. In particular, we show that STP is fixed-parameter tractable by the tree-cut width as well as the fracture number of the input graph.
To achieve our results, we generalize techniques from Edge-Disjoint Paths (EDP) to Generalized Steiner Tree Packing (GSTP), which generalizes both STP and EDP. First, we derive the notion of the augmented graph for GSTP analogous to EDP. We then show that GSTP is FPT by
- the tree-cut width of the augmented graph,
- the fracture number of the augmented graph,
- the slim tree-cut width of the input graph.
The latter two results were previously known for EDP; our results generalize these to GSTP and improve the running time for the parameter fracture number. On the other hand, it was open whether EDP is FPT parameterized by the tree-cut width of the augmented graph, despite extensive research on the structural complexity of the problem. We settle this question affirmatively.
Cite as
Niko Hastrich and Kirill Simonov. Structural Parameterization of Steiner Tree Packing. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hastrich_et_al:LIPIcs.STACS.2026.51,
author = {Hastrich, Niko and Simonov, Kirill},
title = {{Structural Parameterization of Steiner Tree Packing}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {51:1--51:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.51},
URN = {urn:nbn:de:0030-drops-255405},
doi = {10.4230/LIPIcs.STACS.2026.51},
annote = {Keywords: Steiner tree packing, structural parameters, fixed-parameter tractability}
}
Document
Upper and Lower Bounds for the Linear Ordering Principle
Abstract
Korten and Pitassi (FOCS, 2024) defined a new complexity class L₂^P as the polynomial-time Turing closure of the Linear Ordering Principle (a total function extending finding the minimum of an order [M. Chiari and J. Krajíček, 1998] to the case where the order is not linear). They put it between MA (Merlin-Arthur protocols) and S₂^P (the second symmetric level of the polynomial hierarchy).
In this paper we sandwich L₂^P between P^prMA and P^prSBP. (The oracles here are promise problems, and SBP is the only known class between MA and AM.) The containment in P^prSBP is proved via an iterative process that uses a prSBP oracle to estimate the average order rank of a subset and find the minimum of a linear order.
Another containment result of this paper is P^prO₂^P ⊆ O₂^P (where O₂^P is the input-oblivious version of S₂^P). These containment results altogether have several byproducts:
- We give an affirmative answer to an open question posed by Chakaravarthy and Roy (Computational Complexity, 2011) whether P^prMA ⊆ S₂^P, thereby settling the relative standing of the existing (non-oblivious) Karp–Lipton–style collapse results of [V. T. Chakaravarthy and S. Roy, 2011] and [J.-Y. Cai, 2007],
- We give an affirmative answer to an open question of Korten and Pitassi whether a Karp-Lipton-style collapse can be proven for L₂^P,
- We show that the Karp-Lipton-style collapse to P^prOMA is actually better than both known collapses to P^prMA due to Chakaravarthy and Roy (Computational Complexity, 2011) and to O₂^P also due to Chakaravarthy and Roy (STACS, 2006). Thus we resolve the controversy between previously incomparable Karp-Lipton collapses stemming from these two lines of research.
Cite as
Edward A. Hirsch and Ilya Volkovich. Upper and Lower Bounds for the Linear Ordering Principle. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 52:1-52:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hirsch_et_al:LIPIcs.STACS.2026.52,
author = {Hirsch, Edward A. and Volkovich, Ilya},
title = {{Upper and Lower Bounds for the Linear Ordering Principle}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {52:1--52:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.52},
URN = {urn:nbn:de:0030-drops-255410},
doi = {10.4230/LIPIcs.STACS.2026.52},
annote = {Keywords: Complexity Classes, Structural Complexity Theory, Linear Ordering Principle, Symmetric Alternation, Merlin-Arthur Protocols, Karp-Lipton Collapse}
}
Document
Maximum Reachability Orientation of Mixed Graphs
Abstract
We aim to find orientations of mixed graphs optimizing the total reachability, a problem that has applications in causality and biology. For given a digraph D, we use P(D) for the set of ordered pairs of distinct vertices in V(D) and we define κ_D:P(D) → {0,1} by κ_D(u,v) = 1 if v is reachable from u in D, and κ_D(u,v) = 0, otherwise. We use R(D) = ∑_{(u,v) ∈ P(D)}κ_D(u,v).
Now, given a mixed graph G, we aim to find an orientation x⃑{G} of G that maximizes R(x⃑{G}). Hakimi, Schmeichel, and Young proved that the problem can be solved in polynomial time when restricted to undirected inputs. They inquired about the complexity in mixed graphs.
We answer this question by showing that this problem is NP-hard, and, moreover, APX-hard.
We then develop a finer understanding of how quickly the problem becomes difficult when going from undirected to mixed graphs. To this end, we consider the parameterized complexity of the problem with respect to the number k of preoriented arcs of G, a poorly studied form of parameterization.
We show that the problem can be solved in time n^{O(k)} and that a (1-ε)-approximation can be computed in time f(k,ε)n^{O(1)} for any ε > 0.
Cite as
Florian Hörsch. Maximum Reachability Orientation of Mixed Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 53:1-53:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{horsch:LIPIcs.STACS.2026.53,
author = {H\"{o}rsch, Florian},
title = {{Maximum Reachability Orientation of Mixed Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {53:1--53:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.53},
URN = {urn:nbn:de:0030-drops-255421},
doi = {10.4230/LIPIcs.STACS.2026.53},
annote = {Keywords: orientations, mixed graphs, reachability, parameterized complexity, approximation}
}
Document
A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm
Abstract
In the Strip Packing problem, we are given a vertical strip of fixed width and unbounded height, along with a set of axis‑parallel rectangles. The task is to place all rectangles within the strip, without overlaps, while minimizing the height of the packing. This problem is known to be NP-hard. The Bottom-Left Algorithm is a simple and widely used heuristic for Strip Packing. Given a fixed order of the rectangles, it places them one by one, always choosing the lowest feasible position in the strip and, in case of ties, the leftmost one. Baker, Coffman, and Rivest proved in 1980 that the Bottom-Left Algorithm has approximation ratio 3 if the rectangles are sorted by decreasing width [Brenda S. Baker et al., 1980]. For the past 45 years, no alternative ordering has been found that improves this bound. We introduce a new rectangle ordering and show that with this ordering the Bottom-Left Algorithm achieves a 13/6 approximation for the Strip Packing problem.
Cite as
Stefan Hougardy and Bart Zondervan. A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hougardy_et_al:LIPIcs.STACS.2026.54,
author = {Hougardy, Stefan and Zondervan, Bart},
title = {{A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {54:1--54:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.54},
URN = {urn:nbn:de:0030-drops-255432},
doi = {10.4230/LIPIcs.STACS.2026.54},
annote = {Keywords: Approximation Algorithm, Strip Packing, Bottom-Left Algorithm, Rectangle Packing}
}
Document
Testing H-Freeness on Sparse Graphs, the Case of Bounded Expansion
Abstract
In property testing, a tester makes queries to (an oracle for) a graph and, on a graph having or being far from having a property P, it decides with high probability whether the graph satisfies P or not. Often, testers are restricted to a constant number of queries. While the graph properties for which there exists such a tester are somewhat well characterized in the dense graph model, it is not the case for sparse graphs. In this area, Czumaj and Sohler (FOCS’19) proved that H-freeness (i.e. the property of excluding the graph H as a subgraph) can be tested with constant queries on planar graphs as well as on graph classes excluding a minor.
Using results from the sparsity toolkit, we propose a simpler alternative to the proof of Czumaj and Sohler, for a statement generalized to the broader notion of bounded expansion. That is, we prove that for any class 𝒞 with bounded expansion and any graph H, testing H-freeness can be done with constant query complexity on any graph G in 𝒞, where the constant depends on H and 𝒞, but is independent of G.
While classes excluding a minor are prime examples of classes with bounded expansion, so are, for example, cubic graphs, graph classes with bounded maximum degree, or graphs of bounded book thickness. Additionally, random graphs with bounded average degree almost surely have bounded expansion.
Cite as
Samuel Humeau, Mamadou Moustapha Kanté, Daniel Mock, Timothé Picavet, and Alexandre Vigny. Testing H-Freeness on Sparse Graphs, the Case of Bounded Expansion. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 55:1-55:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{humeau_et_al:LIPIcs.STACS.2026.55,
author = {Humeau, Samuel and Kant\'{e}, Mamadou Moustapha and Mock, Daniel and Picavet, Timoth\'{e} and Vigny, Alexandre},
title = {{Testing H-Freeness on Sparse Graphs, the Case of Bounded Expansion}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {55:1--55:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.55},
URN = {urn:nbn:de:0030-drops-255441},
doi = {10.4230/LIPIcs.STACS.2026.55},
annote = {Keywords: Property testing, Sparsity, Bounded expansion, Treedepth}
}
Document
A Practical 73/50 Approximation for Contiguous Monotone Moldable Job Scheduling
Abstract
In moldable job scheduling, we are provided m identical machines and n jobs that can be executed on a variable number of machines. The execution time of each job depends on the number of machines assigned to execute that job. For the specific problem of monotone moldable job scheduling, jobs are assumed to have a processing time that is non-increasing in the number of machines.
The previous best-known algorithms are: (1) a Polynomial Time Approximation Scheme (PTAS) with time complexity Ω(n^{g(1/ε)}), where g(⋅) is a super-exponential function [Jansen and Thöle '08; Jansen and Land '18], (2) a Fully Polynomial Time Approximation Scheme (FPTAS) for the case of m ≥ 8n/(ε) [Jansen and Land '18], and (3) a 3/2 approximation with time complexity O(nmlog(mn)) [Wu, Zhang, and Chen '23].
We present a new practically efficient algorithm with an approximation ratio of ≈ (1.4593 + ε) and a time complexity of O(nm log 1/(ε)). Our result also applies to the contiguous variant of the problem. In addition to our theoretical results, we implement the presented algorithm and show that the practical performance is significantly better than the theoretical worst-case approximation ratio.
Cite as
Klaus Jansen and Felix Ohnesorge. A Practical 73/50 Approximation for Contiguous Monotone Moldable Job Scheduling. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jansen_et_al:LIPIcs.STACS.2026.56,
author = {Jansen, Klaus and Ohnesorge, Felix},
title = {{A Practical 73/50 Approximation for Contiguous Monotone Moldable Job Scheduling}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {56:1--56:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.56},
URN = {urn:nbn:de:0030-drops-255453},
doi = {10.4230/LIPIcs.STACS.2026.56},
annote = {Keywords: computing, machine scheduling, moldable, polynomial approximation}
}
Document
Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach
Abstract
Lagarias and Odlyzko (J.ACM 1985) proposed a polynomial-time algorithm for solving "almost all" instances of the Subset Sum problem with n integers of size Ω(Γ_LO), where log₂(Γ_LO) > n² log₂(γ) and γ is a parameter of the lattice basis reduction (γ > √{4/3} for LLL). The algorithm of Lagarias and Odlyzko is a cornerstone of cryptography. However, the theoretical guarantee on the density of feasible instances has remained unimproved for almost 40 years.
In this paper, we propose an algorithm that solves "almost all" instances of Subset Sum with integers of size Ω(√{Γ_LO}) after a single call to lattice reduction. Additionally, our approach allows solving the Subset Sum problem for multiple targets, whereas the previous method could handle only one target per call to lattice basis reduction. We introduce a modular arithmetic approach to the Subset Sum problem, leveraging lattice reduction to solve a linear system modulo a suitably large prime. By analyzing the lengths of the LLL-reduced basis vectors of both the primal and dual lattices simultaneously, we show that density guarantees can be improved.
Cite as
Antoine Joux and Karol Węgrzycki. Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 57:1-57:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{joux_et_al:LIPIcs.STACS.2026.57,
author = {Joux, Antoine and W\k{e}grzycki, Karol},
title = {{Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {57:1--57:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.57},
URN = {urn:nbn:de:0030-drops-255462},
doi = {10.4230/LIPIcs.STACS.2026.57},
annote = {Keywords: Average-Case Analysis, Subset Sum, Lattice Reduction, LLL}
}
Document
A Polylogarithmic Competitive Algorithm for Stochastic Online Sorting and TSP
Abstract
In Online Sorting, an array of n initially empty cells is given. At each time step t, an element x_t ∈ [0,1] arrives and must be irrevocably placed in an empty cell without knowledge of future arrivals. We aim to minimize the sum of absolute differences between pairs of elements placed in consecutive array cells, seeking an online placement strategy that results in a final array close to a sorted one. An interesting multidimensional generalization, referred to as the Online Traveling Salesperson Problem, arises when the request sequence consists of points in the d-dimensional unit cube and the objective is to minimize the sum of Euclidean distances between points in consecutive cells. Motivated by the recent work of (Abrahamsen, Bercea, Beretta, Klausen and Kozma; ESA 2024), we consider the stochastic version of Online Sorting (resp. Online TSP), where each element (resp. point) x_t is an i.i.d. sample from the uniform distribution on [0, 1] (resp. [0,1]^d). By carefully decomposing the request sequence into a hierarchy of balls-into-bins instances, where the balls to bins ratio is large enough so that bin occupancy is sharply concentrated around its mean and small enough so that we can efficiently deal with the elements placed in the same bin, we obtain an online algorithm that approximates the optimal cost within a factor of O(log² n) with high probability. Our result comprises an exponential improvement over the previously best known competitive ratio of Õ(n^{1/4}) for Stochastic Online Sorting due to (Abrahamsen et al.; ESA 2024) and O(√n) for (adversarial) Online TSP due to (Bertram, ESA 2025).
Cite as
Andreas Kalavas, Charalampos Platanos, and Thanos Tolias. A Polylogarithmic Competitive Algorithm for Stochastic Online Sorting and TSP. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 58:1-58:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kalavas_et_al:LIPIcs.STACS.2026.58,
author = {Kalavas, Andreas and Platanos, Charalampos and Tolias, Thanos},
title = {{A Polylogarithmic Competitive Algorithm for Stochastic Online Sorting and TSP}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {58:1--58:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.58},
URN = {urn:nbn:de:0030-drops-255473},
doi = {10.4230/LIPIcs.STACS.2026.58},
annote = {Keywords: sorting, online algorithm, balls-into-bins, TSP}
}
Document
When Is Local Search Both Effective and Efficient?
Abstract
Combinatorial optimization problems implicitly define fitness landscapes that combine the numeric structure of the "fitness" function to be maximized with the combinatorial structure of which assignments are "adjacent". Local search starts at an assignment in this landscape and successively moves assignments until no further improvement is possible among the adjacent assignments. Classic analyses of local search algorithms have focused more on the question of effectiveness ("did we find a good solution?") and often implicitly assumed that there are no doubts about their efficiency ("did we find it quickly?"). But there are many reasons to doubt the efficiency of local search. Even if we focus on fitness landscapes on the hypercube that are single peaked on every subcube (known as semismooth fitness landscapes, completely unimodal pseudo-Boolean functions, or acyclic unique sink orientations) where effectiveness is obvious, many local search algorithms are known to be inefficient. Since fitness landscapes are unwieldy exponentially large objects, we focus on their polynomial-sized representations by instances of valued constraint satisfaction problems (VCSP). We define a "direction" for valued constraints such that directed VCSPs generate semismooth fitness landscapes. We call directed VCSPs oriented if they do not have any pair of variables with arcs in both directions. Since recognizing if a VCSP-instance is directed or oriented is coNP-complete, we generalized oriented VCSPs as conditionally-smooth fitness landscapes where the structural property of "conditionally-smooth" is recognizable in polynomial time for a VCSP-instance. We prove that many popular local search algorithms like random ascent, simulated annealing, history-based rules, jumping rules, and the Kernighan-Lin heuristic are very efficient on conditionally-smooth landscapes. But conditionally-smooth landscapes are still expressive enough so that other well-regarded local search algorithms like steepest ascent and random facet require a super-polynomial number of steps to find the fitness peak.
Cite as
Artem Kaznatcheev and Sofia Vazquez Alferez. When Is Local Search Both Effective and Efficient?. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 59:1-59:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kaznatcheev_et_al:LIPIcs.STACS.2026.59,
author = {Kaznatcheev, Artem and Vazquez Alferez, Sofia},
title = {{When Is Local Search Both Effective and Efficient?}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {59:1--59:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.59},
URN = {urn:nbn:de:0030-drops-255480},
doi = {10.4230/LIPIcs.STACS.2026.59},
annote = {Keywords: valued constraint satisfaction problem, local search, algorithm analysis, constraint graphs, pseudo-Boolean functions, parameterized complexity}
}
Document
The Asymptotic Size of Finite Irreducible Semigroups of Rational Matrices
Abstract
We study finite semigroups of n × n matrices with rational entries. Such semigroups provide a rich generalization of transition monoids of unambiguous (and, in particular, deterministic) finite automata. In this paper we determine the maximum size of finite semigroups of rational n × n matrices, with the goal of shedding more light on the structure of such matrix semigroups.
While in general such semigroups can be arbitrarily large in terms of n, a classical result of Schützenberger from 1962 implies an upper bound of 2^{𝒪(n² log n)} for irreducible semigroups, i.e., the only subspaces of ℚⁿ that are invariant for all matrices in the semigroup are ℚⁿ and the subspace consisting only of the zero vector. Irreducible matrix semigroups can be viewed as the building blocks of general matrix semigroups, and as such play an important role in mathematics and computer science. From the point of view of automata theory, they generalize strongly connected automata.
Using a very different technique from that of Schützenberger, we improve the upper bound on the cardinality to 3^{n²}. This is the main result of the paper. The bound is in some sense tight, as we show that there exists, for every n, a finite irreducible semigroup with 3^{⌊ n²/4 ⌋} rational matrices. Our main result also leads to an improvement of a bound, due to Almeida and Steinberg, on the mortality threshold. The mortality threshold is a number 𝓁 such that if the zero matrix is in the semigroup, then the zero matrix can be written as a product of at most 𝓁 matrices from any subset that generates the semigroup.
Cite as
Stefan Kiefer and Andrew Ryzhikov. The Asymptotic Size of Finite Irreducible Semigroups of Rational Matrices. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 60:1-60:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kiefer_et_al:LIPIcs.STACS.2026.60,
author = {Kiefer, Stefan and Ryzhikov, Andrew},
title = {{The Asymptotic Size of Finite Irreducible Semigroups of Rational Matrices}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {60:1--60:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.60},
URN = {urn:nbn:de:0030-drops-255496},
doi = {10.4230/LIPIcs.STACS.2026.60},
annote = {Keywords: finite matrix semigroups, irreducible matrix semigroups, matrix mortality, aperiodic semigroups, unambiguous automata, transition monoids}
}
Document
On Effective Banach-Mazur Games and an Application to the Poincaré Recurrence Theorem for Category
Abstract
The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we provide a game-theoretic proof of an effective theorem in dynamical systems, namely the category version of Poincaré Recurrence. The Poincaré Recurrence Theorem for category states that for a homeomorphism without open wandering sets, the set of non recurrent points forms a first category (meager) set. As an application of the effectivization of the Banach-Mazur game, we show that such a result holds true in effective settings as well.
Cite as
Prajval Koul and Satyadev Nandakumar. On Effective Banach-Mazur Games and an Application to the Poincaré Recurrence Theorem for Category. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{koul_et_al:LIPIcs.STACS.2026.61,
author = {Koul, Prajval and Nandakumar, Satyadev},
title = {{On Effective Banach-Mazur Games and an Application to the Poincar\'{e} Recurrence Theorem for Category}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {61:1--61:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.61},
URN = {urn:nbn:de:0030-drops-255509},
doi = {10.4230/LIPIcs.STACS.2026.61},
annote = {Keywords: Recurrence, Topology, Category, Computable Analysis, Computable Toplogy, Dynamical Systems}
}
Document
Relative Compressed Reverse Suffix Array
Abstract
Suffix trees and suffix arrays are two fundamental data structures in the field of string algorithms. For a string (a.k.a. text or sequence) of length n over an alphabet of size σ, these structures typically require O(nlog n) bits of space. The FM-index provides a compressed representation of the suffix array in ≈ nlog σ bits, allowing for efficient queries on both the suffix array and its inverse array in near logarithmic time. In certain applications, such as approximate pattern matching (i.e., with wildcards, mismatches, edits), there is a need to access the suffix array of a text, as well as the suffix array of text’s reverse. Motivated by this, we explore the possibility of encoding the suffix array of the reversed text in a compact form, assuming the availability of the FM-index for the original text. Our first solution is an O(n)-bit (relative) encoding of the suffix array of the reversed text, with the time for decoding an entry being only O(log^*n) times that of decoding an entry in the text’s suffix array using FM-index. We then demonstrate how to reduce the space to O(n/κ) bits for a parameter κ, while multiplicative factor in time becomes approximately O(κlog^*n+κ³). We can also support inverse suffix array and longest common extension queries on the reversed text. These results are achieved through some careful and non-trivial application of various succinct data structure techniques.
Cite as
Muhammed Oguzhan Kulekci, Mano Prakash Parthasarathi, Rahul Shah, and Sharma V. Thankachan. Relative Compressed Reverse Suffix Array. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 62:1-62:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kulekci_et_al:LIPIcs.STACS.2026.62,
author = {Kulekci, Muhammed Oguzhan and Parthasarathi, Mano Prakash and Shah, Rahul and Thankachan, Sharma V.},
title = {{Relative Compressed Reverse Suffix Array}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {62:1--62:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.62},
URN = {urn:nbn:de:0030-drops-255512},
doi = {10.4230/LIPIcs.STACS.2026.62},
annote = {Keywords: String Matching, Text Indexing, Data Structures, Suffix Trees}
}
Document
Generalised Quantifiers Based on Rabin-Mostowski Index
Abstract
In this work we introduce new generalised quantifiers which allow us to express the Rabin-Mostowski index of automata. Our main results study expressive power and decidability of the monadic second-order (MSO) logic extended with these quantifiers. We study these problems in the realm of both ω-words and infinite trees. As it turns out, the pictures in these two cases are very different. In the case of ω-words the new quantifiers can be effectively expressed in pure MSO logic. In contrast, in the case of infinite trees, addition of these quantifiers leads to an undecidable formalism.
To realise index-quantifier elimination, we consider the extension of MSO by game quantifiers. As a tool, we provide a specific quantifier-elimination procedure for them. Moreover, we introduce a novel construction of transducers realising strategies in ω-regular games with monadic parameters.
Cite as
Denis Kuperberg, Damian Niwiński, Paweł Parys, and Michał Skrzypczak. Generalised Quantifiers Based on Rabin-Mostowski Index. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 63:1-63:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kuperberg_et_al:LIPIcs.STACS.2026.63,
author = {Kuperberg, Denis and Niwi\'{n}ski, Damian and Parys, Pawe{\l} and Skrzypczak, Micha{\l}},
title = {{Generalised Quantifiers Based on Rabin-Mostowski Index}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {63:1--63:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.63},
URN = {urn:nbn:de:0030-drops-255526},
doi = {10.4230/LIPIcs.STACS.2026.63},
annote = {Keywords: monadic quantifiers, decidability, quantifier elimination, parity automata, game quantifier, Rabin-Mostowski index}
}
Document
One-Clock Synthesis Problems
Abstract
We study a generalisation of Büchi-Landweber games to the timed setting. The winning condition is specified by a non-deterministic timed automaton, and one of the players can elapse time. We perform a systematic study of synthesis problems in all variants of timed games, depending on which player’s winning condition is specified, and which player’s strategy (or controller, a finite-memory strategy) is sought.
As our main result we prove ubiquitous undecidability in all the variants, both for strategy and controller synthesis, already for winning conditions specified by one-clock automata. This strengthens and generalises previously known undecidability results. We also fully characterise those cases where finite memory is sufficient to win, namely existence of a strategy implies existence of a controller.
All our results are stated in the timed setting, while analogous results hold in the data setting where one-clock automata are replaced by one-register ones.
Cite as
Sławomir Lasota, Mathieu Lehaut, Julie Parreaux, and Radosław Piórkowski. One-Clock Synthesis Problems. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 64:1-64:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lasota_et_al:LIPIcs.STACS.2026.64,
author = {Lasota, S{\l}awomir and Lehaut, Mathieu and Parreaux, Julie and Pi\'{o}rkowski, Rados{\l}aw},
title = {{One-Clock Synthesis Problems}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {64:1--64:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.64},
URN = {urn:nbn:de:0030-drops-255533},
doi = {10.4230/LIPIcs.STACS.2026.64},
annote = {Keywords: timed automata, register automata, B\"{u}chi-Landweber games, Church synthesis problem, reactive synthesis problem}
}
Document
Demystifying Codensity Monads via Duality
Abstract
Codensity monads provide a universal method to generate complex monads from simple functors. Recently, a wide range of important monads in logic, denotational semantics, and probabilistic computation, such as several incarnations of the ultrafilter monad, the Vietoris monad, and the Giry monad, have been presented as codensity monads, using complex arguments. We propose a unifying categorical approach to codensity presentations of monads, based on the idea of relating the presenting functor to a dense functor via a suitable duality between categories. We prove a general presentation result applying to every such situation and demonstrate that most codensity presentations known in the literature emerge from this strikingly simple duality-based setup, drastically alleviating the complexity of their proofs and in many cases completely reducing them to standard duality results. Additionally, we derive a number of novel codensity presentations using our framework, including the first non-trivial codensity presentations for the filter monads on sets and topological spaces, the lower Vietoris monad on topological spaces, and the expectation monad on sets.
Cite as
Fabian Lenke, Nico Wittrock, Stefan Milius, and Henning Urbat. Demystifying Codensity Monads via Duality. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lenke_et_al:LIPIcs.STACS.2026.65,
author = {Lenke, Fabian and Wittrock, Nico and Milius, Stefan and Urbat, Henning},
title = {{Demystifying Codensity Monads via Duality}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {65:1--65:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.65},
URN = {urn:nbn:de:0030-drops-255549},
doi = {10.4230/LIPIcs.STACS.2026.65},
annote = {Keywords: Codensity, Monad, Duality}
}
Document
Computational Hardness of Estimating Quantum Entropies via Binary Entropy Bounds
Abstract
We investigate the computational hardness of estimating the quantum α-Rényi entropy S^𝚁_α(ρ) = (ln Tr(ρ^α))/(1-α) and the quantum q-Tsallis entropy S^𝚃_q(ρ) = (1-Tr(ρ^q))/(q-1), both converging to the von Neumann entropy as the order approaches 1. The promise problems Quantum α-Rényi Entropy Approximation (RényiQEA_α) and Quantum q-Tsallis Entropy Approximation (TsallisQEA_q) ask whether S^𝚁_α(ρ) or S^𝚃_q(ρ), respectively, is at least τ_Y or at most τ_N, where τ_Y - τ_N is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order 1) and some cases of the quantum q-Tsallis entropy, while existing approaches do not readily extend to other orders.
We establish that for all positive real orders, the rank-2 variants Rank2RényiQEA_α and Rank2TsallisQEA_q are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), our results imply:
- For all real order α > 0 and 0 < q ≤ 1, LowRankRényiQEA_α and LowRankTsallisQEA_q are BQP-complete, where both are restricted versions of RényiQEA_α and TsallisQEA_q with ρ of polynomial rank.
- For all real order q > 1, TsallisQEA_q is BQP-complete.
Our hardness results stem from reductions based on new inequalities relating the α-Rényi or q-Tsallis binary entropies of different orders, where the reductions differ substantially from previous approaches, and the inequalities are also of independent interest.
Cite as
Yupan Liu. Computational Hardness of Estimating Quantum Entropies via Binary Entropy Bounds. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 66:1-66:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{liu:LIPIcs.STACS.2026.66,
author = {Liu, Yupan},
title = {{Computational Hardness of Estimating Quantum Entropies via Binary Entropy Bounds}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {66:1--66:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.66},
URN = {urn:nbn:de:0030-drops-255550},
doi = {10.4230/LIPIcs.STACS.2026.66},
annote = {Keywords: computational hardness, quantum state testing, quantum R\'{e}nyi entropy, quantum Tsallis entropy, von Neumann entropy}
}
Document
Pumping-Like Results for Copyless Cost Register Automata and Polynomially Ambiguous Weighted Automata
Abstract
In this work we consider two rich subclasses of weighted automata over fields: polynomially ambiguous weighted automata and copyless cost register automata. Primarily we are interested in understanding their expressiveness power. Over the field of rationals and 1-letter alphabets, it is known that the two classes coincide; they are equivalent to linear recurrence sequences (LRS) whose exponential bases are roots of rationals. We develop a tool we call Pumping Sequence Families, which, by exploiting the simple single-letter behaviour of the models, yields two pumping-like results over arbitrary fields with unrestricted alphabets, one for each class. As a corollary of these results, we present examples proving that the two classes become incomparable over the field of rationals with unrestricted alphabets.
We complement the results by analysing the zeroness and equivalence problems. For weighted automata (even unrestricted) these problems are well understood: there are polynomial time, and even NC² algorithms. For copyless cost register automata we show that the two problems are PSpace-complete, where the difficulty is to show the lower bound.
Cite as
Filip Mazowiecki, Antoni Puch, and Daniel Smertnig. Pumping-Like Results for Copyless Cost Register Automata and Polynomially Ambiguous Weighted Automata. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 67:1-67:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mazowiecki_et_al:LIPIcs.STACS.2026.67,
author = {Mazowiecki, Filip and Puch, Antoni and Smertnig, Daniel},
title = {{Pumping-Like Results for Copyless Cost Register Automata and Polynomially Ambiguous Weighted Automata}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {67:1--67:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.67},
URN = {urn:nbn:de:0030-drops-255568},
doi = {10.4230/LIPIcs.STACS.2026.67},
annote = {Keywords: weighted automata, cost register automata, ambiguity, linear recurrence sequences, equivalence problem}
}
Document
Dynamic Pattern Matching with Wildcards
Abstract
We study the fully dynamic pattern matching problem where the pattern may contain up to k wildcard symbols, each matching any symbol of the alphabet. Both the text and the pattern are subject to updates (insert, delete, change). We design an algorithm with 𝒪(n log² n) preprocessing and update/query time 𝒪̃(kn^{k/{k+1}} + k² log n). The bound is truly sublinear for a constant k, and sublinear when k = o(log n). We further complement our results with a conditional lower bound: assuming subquadratic preprocessing time, achieving truly sublinear update time for the case k = Ω(log n) would contradict the Strong Exponential Time Hypothesis (SETH). Finally, we develop sublinear algorithms for two special cases:
- If the pattern contains w non-wildcard symbols, we give an algorithm with preprocessing time 𝒪(nw) and update time 𝒪(w + log n), which is truly sublinear whenever w is truly sublinear.
- Using FFT technique combined with block decomposition, we design a deterministic truly sublinear algorithm with preprocessing time 𝒪(n^{1.8}) and update time 𝒪(n^{0.8} log n) for the case that there are at most two non-wildcards.
Cite as
Arshia Ataee Naeini, Amir-Parsa Mobed, Masoud Seddighin, and Saeed Seddighin. Dynamic Pattern Matching with Wildcards. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 68:1-68:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{naeini_et_al:LIPIcs.STACS.2026.68,
author = {Naeini, Arshia Ataee and Mobed, Amir-Parsa and Seddighin, Masoud and Seddighin, Saeed},
title = {{Dynamic Pattern Matching with Wildcards}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {68:1--68:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.68},
URN = {urn:nbn:de:0030-drops-255579},
doi = {10.4230/LIPIcs.STACS.2026.68},
annote = {Keywords: pattern matching, wildcards, dynamic algorithms, string algorithms, data structures}
}
Document
Approximating q → p Norms of Non-Negative Matrices in Nearly-Linear Time
Abstract
We provide the first nearly-linear time algorithm for approximating 𝓁_{q → p}-norms of non-negative matrices, for q ≥ p ≥ 1. Our algorithm returns a (1-ε)-approximation to the matrix norm in time Õ(1/(q ε) ⋅ nnz(A)), where A is the input matrix, and improves upon the previous state of the art, which either proved convergence only in the limit [Boyd '74], or had very high polynomial running times [Bhaskara-Vijayraghavan, SODA '11]. Our algorithm is extremely simple, and is largely inspired from the coordinate-scaling approach used for positive linear program solvers. Our algorithm can readily be used in the [Englert-Räcke, FOCS '09] to improve the running time of constructing O(log n)-competitive 𝓁_p-oblivious routings.
Cite as
Etienne Objois and Adrian Vladu. Approximating q → p Norms of Non-Negative Matrices in Nearly-Linear Time. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 69:1-69:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{objois_et_al:LIPIcs.STACS.2026.69,
author = {Objois, Etienne and Vladu, Adrian},
title = {{Approximating q → p Norms of Non-Negative Matrices in Nearly-Linear Time}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {69:1--69:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.69},
URN = {urn:nbn:de:0030-drops-255585},
doi = {10.4230/LIPIcs.STACS.2026.69},
annote = {Keywords: matrix norm, Perron-Frobenius theory, oblivious routings, input-sparsity time, lp norm}
}
Document
A Permanental Analog of the Rank-Nullity Theorem for Symmetric Matrices
Abstract
The rank of an n × n matrix A is equal to the maximum order of a square submatrix with a nonzero determinant; it can be computed in O(n^{2.37}) time. Analogously, the maximum order of a square submatrix with nonzero permanent is defined as the permanental rank ρ_{per}(A). Computing the permanent or the coefficients of the permanental polynomial per(xI-A) is #P-complete. The permanental nullity η_{per}(A) is defined as the multiplicity of zero as a root of the permanental polynomial. We establish a permanental analog of the rank–nullity theorem, ρ_{per}(A) + η_{per}(A) = n for symmetric nonnegative matrices, positive semidefinite matrices, and adjacency matrices of balanced signed graphs. Using this theorem, we can compute the permanental nullity for these classes in polynomial time. For {0,± 1}-matrices, we also provide a complete characterization of when the permanental rank-nullity identity holds.
Cite as
Priyanshu Pant, Surabhi Chakrabartty, and Ranveer Singh. A Permanental Analog of the Rank-Nullity Theorem for Symmetric Matrices. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 70:1-70:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{pant_et_al:LIPIcs.STACS.2026.70,
author = {Pant, Priyanshu and Chakrabartty, Surabhi and Singh, Ranveer},
title = {{A Permanental Analog of the Rank-Nullity Theorem for Symmetric Matrices}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {70:1--70:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.70},
URN = {urn:nbn:de:0030-drops-255590},
doi = {10.4230/LIPIcs.STACS.2026.70},
annote = {Keywords: permanent, matrix rank, #P-completeness, graph algorithms, permanental polynomial, spectral graph theory}
}
Document
Broadcast in Almost Mixing Time
Abstract
We study the problem of broadcasting multiple messages in the CONGEST model. In this problem, a dedicated source node s possesses a set M of messages with every message of size O(log n) where n is the total number of nodes. The objective is to ensure that every node in the network learns all messages in M. The execution of an algorithm progresses in rounds, and we focus on optimizing the round complexity of broadcasting multiple messages.
Our primary contribution is a randomized algorithm for networks with expander topology. The algorithm succeeds with high probability and achieves a round complexity that is optimal up to a factor of the network’s mixing time and polylogarithmic terms. It leverages a multi-COBRA primitive, which uses multiple branching random walks running in parallel. A crucial aspect of our method is the use of these branching random walks to construct an optimal (up to a polylogarithmic factor) tree packing of a random graph, which is then used for efficient broadcasting.
We also prove the problem to be NP-hard in a centralized setting and provide insights into why lower bounds that can be matched in expanders, namely graph diameter and |M|/minCut, cannot be tight in general graphs.
Cite as
Anton Paramonov and Roger Wattenhofer. Broadcast in Almost Mixing Time. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 71:1-71:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{paramonov_et_al:LIPIcs.STACS.2026.71,
author = {Paramonov, Anton and Wattenhofer, Roger},
title = {{Broadcast in Almost Mixing Time}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {71:1--71:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.71},
URN = {urn:nbn:de:0030-drops-255603},
doi = {10.4230/LIPIcs.STACS.2026.71},
annote = {Keywords: Distributed algorithms, Expander Graphs, Random graphs, Broadcast, Branching random walks, Tree packing, CONGEST model}
}
Document
Circle Graphs Can Be Recognized in Linear Time
Abstract
To date, the best circle graph recognition algorithm, due to Gioan et al. [E. Gioan et al., 2014] runs in almost linear time as it relies on a split decomposition algorithm [E. Gioan et al., 2014] that uses the union-find data-structure [B.A. Galler and M.J. Fischer, 1964; R. Tarjan, 1975]. We show that in the case of circle graphs, the PC-tree data-structure [W. K. Shih and W. L. Hsu, 1999] allows one to avoid the union-find data-structure to compute the split decomposition in linear time. As a consequence, we obtain the first linear-time recognition algorithm for circle graphs.
Cite as
Christophe Paul and Ignaz Rutter. Circle Graphs Can Be Recognized in Linear Time. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 72:1-72:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{paul_et_al:LIPIcs.STACS.2026.72,
author = {Paul, Christophe and Rutter, Ignaz},
title = {{Circle Graphs Can Be Recognized in Linear Time}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {72:1--72:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.72},
URN = {urn:nbn:de:0030-drops-255614},
doi = {10.4230/LIPIcs.STACS.2026.72},
annote = {Keywords: graph classes, circle graphs, graph algorithms}
}
Document
Colouring Probe H-Free Graphs
Abstract
The NP-complete problems Colouring and k-Colouring (k ≥ 3) are well studied on H-free graphs, i.e., graphs that do not contain some fixed graph H as an induced subgraph. We research to what extent the known polynomial-time algorithms for H-free graphs can be generalized if we only know some of the edges of the input graph. We do this by considering the classical probe graph model introduced in the early nineties. For a graph H, a partitioned probe H-free graph (G,P,N) consists of a graph G = (V,E), together with a set P ⊆ V of probes and an independent set N = V ⧵ P of non-probes, such that G+F is H-free for some edge set F ⊆ binom(N,2). We show the following:
- We fully classify Colouring on partitioned probe H-free graphs and show that the obtained complexity dichotomy differs from the known dichotomy of Colouring for H-free graphs.
- We fully classify 3-Colouring on partitioned probe P_t-free graphs: we prove polynomial-time solvability for t ≤ 5 and NP-completeness for t ≥ 6. In contrast, 3-Colouring on P_t-free graphs is known to be polynomial-time solvable for t ≤ 7 and quasi-polynomial-time solvable for t ≥ 8. Our main result is our polynomial-time algorithm for 3-Colouring on partitioned P₅-free graphs. For this result, and also for all our other polynomial-time results, we do not need to know the edge set F; we only need to know its existence. Moreover, the class of probe P₅-free graphs includes not only paths of arbitrary length but even all bipartite graphs and is much richer than the class of P₅-free graphs. The latter is also evidenced by the fact that there exist graph problems, such as Matching Cut, that are known to be polynomial-time solvable for P₅-free graphs but NP-complete for partitioned probe P₅-free graphs. In particular, unlike the class of 3-colourable P₅-free graphs, the class of 3-colourable probe P₅-free graphs has unbounded mim-width. Hence, our polynomial-time result for 3-Colouring for probe P₅-free graphs suggests that there may be another, deeper overarching reason why 3-Colouring is polynomial-time solvable for P₅-free graphs.
Cite as
Daniël Paulusma, Johannes Rauch, and Erik Jan van Leeuwen. Colouring Probe H-Free Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{paulusma_et_al:LIPIcs.STACS.2026.73,
author = {Paulusma, Dani\"{e}l and Rauch, Johannes and van Leeuwen, Erik Jan},
title = {{Colouring Probe H-Free Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {73:1--73:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.73},
URN = {urn:nbn:de:0030-drops-255621},
doi = {10.4230/LIPIcs.STACS.2026.73},
annote = {Keywords: colouring, probe graph, forbidden induced subgraph, complexity dichotomy}
}
Document
List Coloring Ordered Graphs with Forbidden Induced Subgraphs
Abstract
In the List k-Coloring problem we are given a graph whose every vertex is equipped with a list, which is a subset of {1,…,k}. We need to decide if G admits a proper coloring, where every vertex receives a color from its list.
The complexity of the problem in classes defined by forbidding induced subgraphs is a widely studied topic in algorithmic graph theory. Recently, Hajebi, Li, and Spirkl [SIAM J. Discr. Math. 38 (2024)] initiated the study of List 3-Coloring in ordered graphs, i.e., graphs with fixed linear ordering of vertices. Forbidding ordered induced subgraphs allows us to investigate the boundary of tractability more closely.
We continue this direction of research, focusing mostly on the case of List 4-Coloring. We present several algorithmic and hardness results, which altogether provide an almost complete dichotomy for classes defined by forbidding one fixed ordered graph: our investigations leave one minimal open case.
Cite as
Marta Piecyk and Paweł Rzążewski. List Coloring Ordered Graphs with Forbidden Induced Subgraphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 74:1-74:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{piecyk_et_al:LIPIcs.STACS.2026.74,
author = {Piecyk, Marta and Rz\k{a}\.{z}ewski, Pawe{\l}},
title = {{List Coloring Ordered Graphs with Forbidden Induced Subgraphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {74:1--74:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.74},
URN = {urn:nbn:de:0030-drops-255634},
doi = {10.4230/LIPIcs.STACS.2026.74},
annote = {Keywords: coloring, ordered graphs, forbidden induced subgraphs}
}
Document
Effective Versions of Strong Measure Zero
Abstract
Effective versions of strong measure zero sets are developed for various levels of complexity and computability. It is shown that the sets can be equivalently defined using a generalization of supermartingales called odds supermartingales, success rates on supermartingales, predictors, and coverings. We show Borel’s conjecture that a set has strong measure zero if and only if it is countable holds in the time and space bounded setting. At the level of computability this does not hold. We show the computable level contains sequences at arbitrary levels of the hyperarithmetical hierarchy. This is done by proving a correspondence principle yielding a condition for the sets of computable strong measure zero to agree with the classical sets of strong measure zero.
An algorithmic version of strong measure zero using lower semicomputability is defined. We show that this notion is equivalent to the set of NCR reals studied by Reimann and Slaman, thereby giving new characterizations of this set.
Effective strong packing dimension zero is investigated by requiring success with respect to the limit inferior instead of the limit superior. It is proven that every sequence in the corresponding algorithmic class is decidable. At the level of computability, the sets coincide with a notion of weak countability that we define.
Cite as
Matthew Rayman. Effective Versions of Strong Measure Zero. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rayman:LIPIcs.STACS.2026.75,
author = {Rayman, Matthew},
title = {{Effective Versions of Strong Measure Zero}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {75:1--75:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.75},
URN = {urn:nbn:de:0030-drops-255648},
doi = {10.4230/LIPIcs.STACS.2026.75},
annote = {Keywords: Strong measure zero, NCR, Effective fractal dimensions, Borel’s Conjecture, Hausdorff dimension, Packing dimension}
}
Document
Modularity of Preferential Attachment Graphs
Abstract
We study a preferential attachment model G_n^h. The graph G_n^h is generated from a finite initial graph by adding new vertices one at a time. Each new vertex connects to h ≥ 1 already existing vertices, and these are chosen with probability proportional to their current degrees. We are particularly interested in the community structure of G_n^h, which is expressed in terms of the so-called modularity. We prove that the modularity of G_n^h is, with high probability, upper bounded by a function that tends to 0 as h tends to infinity. This resolves a conjecture of Prokhorenkova, Prałat, and Raigorodskii from 2016.
As a byproduct, we obtain novel concentration results (which are interesting in their own right) for the volume and edge density parameters of vertex subsets of G_n^h. The key ingredient here is the definition of a function μ, which serves as a natural measure for vertex subsets, and is proportional to the average size of their volumes. This extends previous results on the topic by Frieze, Pérez-Giménez, Prałat, and Reiniger from 2019.
Cite as
Katarzyna Rybarczyk and Małgorzata Sulkowska. Modularity of Preferential Attachment Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 76:1-76:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rybarczyk_et_al:LIPIcs.STACS.2026.76,
author = {Rybarczyk, Katarzyna and Sulkowska, Ma{\l}gorzata},
title = {{Modularity of Preferential Attachment Graphs}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {76:1--76:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.76},
URN = {urn:nbn:de:0030-drops-255658},
doi = {10.4230/LIPIcs.STACS.2026.76},
annote = {Keywords: Modularity, preferential attachment model, edge expansion}
}
Document
An Improved Version of Hmelevskii’s Theorem on Three-Variable Word Equations
Abstract
Hmelevskii proved in 1971 that every constant-free three-variable word equation has a parametric solution. We prove an improved version of this result by showing that every such equation has a parametric solution using only three numerical parameters and with only two levels of nesting. This means that the structure of the solution sets of these equations is considerably simpler than has been known before.
Cite as
Aleksi Saarela. An Improved Version of Hmelevskii’s Theorem on Three-Variable Word Equations. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{saarela:LIPIcs.STACS.2026.77,
author = {Saarela, Aleksi},
title = {{An Improved Version of Hmelevskii’s Theorem on Three-Variable Word Equations}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {77:1--77:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.77},
URN = {urn:nbn:de:0030-drops-255664},
doi = {10.4230/LIPIcs.STACS.2026.77},
annote = {Keywords: Combinatorics on words, word equation, parametric word}
}
Document
Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors
Abstract
Let ℱ be a finite family of graphs. In the ℱ-Deletion problem, one is given a graph G and an integer k, and the goal is to find k vertices whose deletion results in a graph with no minor from the family ℱ. This may be regarded as a far-reaching generalization of Vertex Cover and Feedback vertex Set. In their seminal work, Fomin, Lokshtanov, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family ℱ contains a planar graph. As the size of their kernel is g(ℱ) ⋅ k^{f(ℱ)}, a natural follow-up question was whether the dependence on ℱ in the exponent of k can be avoided. The answer turned out to be negative: Giannopoulou, Jansen, Lokshtanov & Saurabh [TALG 2017] proved that this is already inevitable for the special case of the Treewidth-η-Deletion problem.
In this work, we show that this non-uniformity can be avoided at the expense of a small loss. First, we present a simple 2-approximate kernelization algorithm for Treewidth-η-Deletion with a kernel size g(η) ⋅ k⁶. Next, we show that the approximation factor can be made arbitrarily close to 1, if we settle for a kernelization protocol with 𝒪(1) calls to an oracle that solves instances of size bounded by a uniform polynomial in k. We extend the above results to general ℱ-Deletion, whenever ℱ contains a planar graph, as long as an oracle for Treewidth-η-Deletion is available for small instances. Notably, all our constants are computable functions of ℱ and our techniques work also when some graphs in ℱ may be disconnected.
Our results rely on two novel techniques. First, we transform so-called "near-protrusion decompositions" into true protrusion decompositions by sacrificing a small accuracy loss. Secondly, we show how to optimally compress such a decomposition with respect to general ℱ-Deletion. Using our second technique, we also obtain linear kernels on sparse graph classes when ℱ contains a planar graph, whereas the previously known theorems required all graphs in ℱ to be connected. Specifically, we generalize the kernelization algorithm by Kim, Langer, Paul, Reidl, Rossmanith, Sau & Sikdar [TALG 2015] on graph classes that exclude a topological minor.
Cite as
Roohani Sharma and Michał Włodarczyk. Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 78:1-78:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{sharma_et_al:LIPIcs.STACS.2026.78,
author = {Sharma, Roohani and W{\l}odarczyk, Micha{\l}},
title = {{Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {78:1--78:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.78},
URN = {urn:nbn:de:0030-drops-255674},
doi = {10.4230/LIPIcs.STACS.2026.78},
annote = {Keywords: kernelization, graph minors, treewidth, uniform kernels, minor hitting}
}
Document
Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More
Abstract
Despite much research, hard weighted problems still resist super-polynomial improvements over their textbook solution. On the other hand, the unweighted versions of these problems have recently witnessed the sought-after speedups. Currently, the only way to repurpose the algorithm of the unweighted version for the weighted version is to employ a polynomial embedding of the input weights. This, however, introduces a pseudo-polynomial factor into the running time, which becomes impractical for arbitrarily weighted instances.
In this paper, we introduce a new way to repurpose the algorithm of the unweighted problem. Specifically, we show that the time complexity of several well-known NP-hard problems operating over the (min, +) and (max, +) semirings, such as TSP, Weighted Max-Cut, and Edge-Weighted k-Clique, is proportional to that of their unweighted versions when the set of input weights has small doubling. We achieve this by a meta-algorithm that converts the input weights into polynomially bounded integers using the recent constructive Freiman’s theorem by Randolph and Węgrzycki [ESA 2024] before applying the polynomial embedding.
Cite as
Mihail Stoian. Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 79:1-79:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{stoian:LIPIcs.STACS.2026.79,
author = {Stoian, Mihail},
title = {{Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {79:1--79:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.79},
URN = {urn:nbn:de:0030-drops-255680},
doi = {10.4230/LIPIcs.STACS.2026.79},
annote = {Keywords: doubling constant parametrization, weighted problems, traveling salesman, weighted max-cut, edge-weighted k-clique}
}
Document
Efficient Compression in Semigroups
Abstract
Straight-line programs are a central tool in several areas of computer science, including data compression, algebraic complexity theory, and the algorithmic solution of algebraic equations. In the algebraic setting, where straight-line programs can be interpreted as circuits over algebraic structures such as semigroups or groups, they have led to deep insights in computational complexity.
A key result by Babai and Szemerédi (1984) showed that finite groups afford efficient compression via straight-line programs, enabling the design of a black-box computation model for groups. Building on their result, Fleischer (2019) placed the Cayley table membership problem for certain classes (pseudovarieties) of finite semigroups in NPOLYLOGTIME, and in some cases even in FOLL. He also provided a complete classification of pseudovarieties of finite monoids affording efficient compression.
In this work, we complete this classification program initiated by Fleischer, characterizing precisely those pseudovarieties of finite semigroups that afford efficient compression via straight-line programs. Along the way, we also improve several known bounds on the length and width of straight-line programs over semigroups, monoids, and groups. These results lead to new upper bounds for the membership problem in the Cayley table model: for all pseudovarieties that afford efficient compression and do not contain any nonsolvable group, we obtain FOLL algorithms. In particular, we resolve a conjecture of Barrington, Kadau, Lange, and McKenzie (2001), showing that the membership problem for all solvable groups is in FOLL.
Cite as
Alexander Thumm and Armin Weiß. Efficient Compression in Semigroups. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 80:1-80:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{thumm_et_al:LIPIcs.STACS.2026.80,
author = {Thumm, Alexander and Wei{\ss}, Armin},
title = {{Efficient Compression in Semigroups}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {80:1--80:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.80},
URN = {urn:nbn:de:0030-drops-255694},
doi = {10.4230/LIPIcs.STACS.2026.80},
annote = {Keywords: Semigroups, straight-line programs, compression, membership problem}
}
Document
Counting Unit Circular Arc Intersections
Abstract
Given a set of n circular arcs of the same radius in the plane, we consider the problem of computing the number of intersections among the arcs. The problem was studied before and the previously best algorithm solves the problem in O(n^{4/3+ε}) time [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993], for any constant ε > 0. No progress has been made on the problem for more than 30 years. We present a new algorithm of O(n^{4/3}log^{16/3} n) time and improve it to O(n^{1+ε}+K^{1/3}n^{2/3}((n²)/(n+K))^{ε}log^{16/3}n) time for small K, where K is the number of intersections of all arcs.
Cite as
Haitao Wang. Counting Unit Circular Arc Intersections. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 81:1-81:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wang:LIPIcs.STACS.2026.81,
author = {Wang, Haitao},
title = {{Counting Unit Circular Arc Intersections}},
booktitle = {43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
pages = {81:1--81:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-412-3},
ISSN = {1868-8969},
year = {2026},
volume = {364},
editor = {Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.81},
URN = {urn:nbn:de:0030-drops-255707},
doi = {10.4230/LIPIcs.STACS.2026.81},
annote = {Keywords: circular arc intersections, unit circles, arrangements, cuttings, segment intersections}
}