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LIPIcs, Volume 388
34th Annual European Symposium on Algorithms (ESA 2026)
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Leibniz International Proceedings in Informatics (LIPIcs)
Conference: European Symposium on Algorithms (ESA)
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Publication Details
- published at: 2026-08-25
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-445-1
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Complete Volume
LIPIcs, Volume 388, ESA 2026, Complete Volume
Abstract
LIPIcs, Volume 388, ESA 2026, Complete Volume
Cite as
34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 1-3048, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{bille_et_al:LIPIcs.ESA.2026,
title = {{LIPIcs, Volume 388, ESA 2026, Complete Volume}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {1--3048},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026},
URN = {urn:nbn:de:0030-drops-277119},
doi = {10.4230/LIPIcs.ESA.2026},
annote = {Keywords: LIPIcs, Volume 388, ESA 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
34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 0:i-0:xxii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bille_et_al:LIPIcs.ESA.2026.0,
author = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {0:i--0:xxii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.0},
URN = {urn:nbn:de:0030-drops-271363},
doi = {10.4230/LIPIcs.ESA.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
History Independence as an Algorithmic Tool (Invited Talk)
Abstract
In this talk, I’ll argue that history independence - a widely-studied privacy property in the data-structures literature - is actually a powerful algorithmic tool, not just for privacy, but for building faster and better data structures. Along the way, I’ll discuss a recent sequence of results on the list-labeling problem, where history independence played a pivotal role in overturning a 30-year-old conjecture.
Cite as
William Kuszmaul. History Independence as an Algorithmic Tool (Invited Talk). In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kuszmaul:LIPIcs.ESA.2026.1,
author = {Kuszmaul, William},
title = {{History Independence as an Algorithmic Tool}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {1:1--1:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.1},
URN = {urn:nbn:de:0030-drops-271377},
doi = {10.4230/LIPIcs.ESA.2026.1},
annote = {Keywords: Data Structures, History Independence, List-Labeling, Randomized Algorithms}
}
Document
Invited Talk
Proofs of Useful Work from Arbitrary Matrix Multiplication (Invited Talk)
Abstract
We revisit the longstanding open problem of implementing Nakamoto’s proof-of-work (PoW) consensus based on a real-world computational task T(x) (as opposed to artificial random hashing), in a truly permissionless setting where the miner itself chooses the input x. The challenge in designing such a Proof-of-Useful-Work (PoUW) protocol is to use the native computation of T(x) to produce a PoW certificate with prescribed hardness and with negligible computational overhead over the worst-case complexity of T(⋅). This ensures malicious miners cannot "game the system" by fooling the verifier to accept with higher probability than honest miners while using similar resources. Indeed, obtaining a PoUW with O(1)-factor overhead is trivial for any task T, but also useless.
Our main result is a PoUW for the task of matrix multiplication MatMul(A,B) of arbitrary matrices, with 1+o(1) multiplicative overhead compared to naïve MatMul. We conjecture that our protocol has optimal security, in the sense that a malicious prover cannot obtain any significant advantage over an honest prover. This conjecture reduces the hardness of our protocol to the task of solving a batch of correlated low-rank random linear systems, which is of independent interest.
Since matrix multiplications are the bottleneck of AI compute as well as countless industry-scale applications, this primitive suggests a concrete design of a new L1 base-layer protocol which nearly eliminates the energy waste of Bitcoin mining, allowing GPU consumers to reduce their AI training and inference costs by "re-using" them for blockchain consensus, in exchange for block rewards (2-for-1).
Cite as
Ilan Komargodski and Omri Weinstein. Proofs of Useful Work from Arbitrary Matrix Multiplication (Invited Talk). In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{komargodski_et_al:LIPIcs.ESA.2026.2,
author = {Komargodski, Ilan and Weinstein, Omri},
title = {{Proofs of Useful Work from Arbitrary Matrix Multiplication}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {2:1--2:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.2},
URN = {urn:nbn:de:0030-drops-271385},
doi = {10.4230/LIPIcs.ESA.2026.2},
annote = {Keywords: Proof of work, blockchain, matrix multiplication, fine-grained complexity, random self-reducibility}
}
Document
Advances in Exact and Approximate Group Closeness Centrality Maximization
Abstract
In the NP-hard Group Closeness Centrality Maximization problem, the input is a graph G = (V,E) and a positive integer k, and the task is to find a set S ⊆ V of size k that minimizes group farness f(S) = ∑_{v ∈ V} min_{s ∈ S} dist(v,s). The state-of-the-art exact algorithm iteratively solves ILPs of increasing size until the final ILP can provably represent an optimal solution. We introduce a new data reduction technique that eliminates variables from the ILP by proving that certain vertices have their distance to any optimal solution structurally determined by a neighbor. Additionally, we bootstrap the exact solver with an approximate solution to produce near-sufficient ILPs from the first iteration, reducing the number of needed iterations. Our improvements yield a speedup by a factor of 4.5 over the next best exact algorithm and can achieve speedups by up to a factor of 34.1. Furthermore, we add reduction techniques to a 1/5-approximation algorithm, and show that these adaptations do not compromise its approximation guarantee. The improved algorithm achieves mean speedups of up to 1.6 and a maximum speedup of 9.6 times. Finally, we settle an open question by proving that a widely used greedy algorithm admits arbitrarily poor approximation ratios.
Cite as
Christian Schulz, Jakob Ternes, and Henning Woydt. Advances in Exact and Approximate Group Closeness Centrality Maximization. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 3:1-3:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{schulz_et_al:LIPIcs.ESA.2026.3,
author = {Schulz, Christian and Ternes, Jakob and Woydt, Henning},
title = {{Advances in Exact and Approximate Group Closeness Centrality Maximization}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {3:1--3:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.3},
URN = {urn:nbn:de:0030-drops-271394},
doi = {10.4230/LIPIcs.ESA.2026.3},
annote = {Keywords: Group Closeness Centrality, Exact Algorithms, Approximation Algorithms}
}
Document
Data-Dependent Evaluations for Budgeted Submodular Maximization
Abstract
Submodular maximization is an important building block for developing algorithms in many areas such as machine learning and data mining. Due to the NP-hardness of the problem, analysis of submodular maximization algorithms typically provides pessimistic worst-case approximation factors only. It is not easy to evaluate how close a produced solution is to an optimal one for a given problem instance. In this paper, we develop new data-dependent upper bounds for submodular maximization with a knapsack constraint. We theoretically prove that they dominate the optimal solution and empirically demonstrate their advantages in certifying how close to optimal a solution is through experiments with real-world datasets.
Cite as
Lejian Zhang, Xueyan Tang, and Jing Tang. Data-Dependent Evaluations for Budgeted Submodular Maximization. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 4:1-4:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{zhang_et_al:LIPIcs.ESA.2026.4,
author = {Zhang, Lejian and Tang, Xueyan and Tang, Jing},
title = {{Data-Dependent Evaluations for Budgeted Submodular Maximization}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {4:1--4:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.4},
URN = {urn:nbn:de:0030-drops-271407},
doi = {10.4230/LIPIcs.ESA.2026.4},
annote = {Keywords: submodular maximization, knapsack constraint, approximation guarantee}
}
Document
Persistent Homology on GPU for 1d and 2d Cubical Filtrations
Abstract
This paper describes the algorithmic choices made in a pure GPU implementation to compute persistent homology for cubical complexes in dimension 1 or 2. The main ingredients are the parallel detection of apparent pairs, the simple reduction of the remaining complex when we only consider H₀, and the possibility to iterate those two steps. The result can be 200 times faster than a sequential CPU implementation.
Cite as
Marc Glisse. Persistent Homology on GPU for 1d and 2d Cubical Filtrations. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{glisse:LIPIcs.ESA.2026.5,
author = {Glisse, Marc},
title = {{Persistent Homology on GPU for 1d and 2d Cubical Filtrations}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {5:1--5:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.5},
URN = {urn:nbn:de:0030-drops-271416},
doi = {10.4230/LIPIcs.ESA.2026.5},
annote = {Keywords: Persistent homology, cubical complex, parallelism, GPU}
}
Document
Dynamic Grammar-Compressed Self-Index in δ-Optimal Space
Abstract
A compressed self-index stores a string in compressed form while supporting locate queries without decompression. For highly repetitive strings - arising in web crawls, versioned documents, and genomic collections - static self-indexes can match the δ-optimal lower bound of Ω(δ log(n log σ / (δ log n)) log n) bits up to constant factors, where n is the string length, σ is the alphabet size, and δ is the substring complexity. Their dynamic counterparts, however, remain scarce: every existing dynamic self-index either fails to attain δ-optimal space, pays Ω(log n) time per reported occurrence during locate, or has an update time that grows with the maximum value in the longest common prefix (LCP) array of the text. We present the dynamic RR-index, a dynamic grammar-compressed self-index built on the restricted recompression run-length straight-line program (RLSLP). To our knowledge, it is the first dynamic self-index to attain δ-optimal space. The index occupies expected 𝒪(δ log(n log σ / (δ log n)) log n) bits, answers locate queries in expected 𝒪(m + log m log² n + occ (log n / log log n)) time - where m is the pattern length and occ is the number of occurrences - and supports insertion of a length-m' string and deletion of a length-m' substring in expected amortized 𝒪(m' log² n + log³ n) time, with no dependence on the maximum LCP value. On eleven highly repetitive corpora, including a 37 GB Wikipedia dump and a 59 GB human-chromosome collection, the dynamic RR-index is up to 77× faster than the dynamic r-index on updates and up to 11× faster than other dynamic indexes on locate.
Cite as
Takaaki Nishimoto and Yasuo Tabei. Dynamic Grammar-Compressed Self-Index in δ-Optimal Space. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nishimoto_et_al:LIPIcs.ESA.2026.6,
author = {Nishimoto, Takaaki and Tabei, Yasuo},
title = {{Dynamic Grammar-Compressed Self-Index in \delta-Optimal Space}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {6:1--6:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.6},
URN = {urn:nbn:de:0030-drops-271422},
doi = {10.4230/LIPIcs.ESA.2026.6},
annote = {Keywords: Compressed string indexes, grammar-compression, dynamic data structures}
}
Document
Symmetry-Preserving Graph Compression
Abstract
Exploiting symmetry is a well-established technique to eliminate redundant work in combinatorial solvers, yet it often incurs computational overhead that limits its practical impact. In particular, practical instances arising from applications are frequently large and give rise to graphs whose size becomes a major bottleneck for algorithms dealing with symmetry.
We propose a method for symmetry-preserving graph compression that reduces graph size while preserving the symmetries of the original graph in a controlled way. Our approach identifies and merges equivalent vertex colors under conditions that guarantee the recoverability of all symmetries. We provide both a theoretical foundation and efficient practical criteria for such merges, show that computing optimal and approximately optimal compression is intractable, and introduce a linear-time, practical heuristic. Extensive experiments on a vast library of graphs demonstrate that our new technique achieves significant compression ratios. Implemented in the state-of-the-art symmetry detection tool dejavu, we achieve an overall speedup of 1.39, with large modern SAT and MIP benchmarks benefiting the most.
Cite as
Markus Anders, Manuel Penschuck, and Pascal Schweitzer. Symmetry-Preserving Graph Compression. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{anders_et_al:LIPIcs.ESA.2026.7,
author = {Anders, Markus and Penschuck, Manuel and Schweitzer, Pascal},
title = {{Symmetry-Preserving Graph Compression}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {7:1--7:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.7},
URN = {urn:nbn:de:0030-drops-271430},
doi = {10.4230/LIPIcs.ESA.2026.7},
annote = {Keywords: symmetry detection, graph automorphisms, preprocessing, color refinement, approximation hardness, mathematical programming}
}
Document
Practical Bit Vectors Supporting Constant Time Rank and Select in Optimal Space
Abstract
Bit vectors with support for fast rank and select are a fundamental building block for compressed data structures. We close a gap between theory and practice by mapping a design space of promising data structures, analyzing it, and experimentally evaluating a promising region. The result are implementations of rank and select data structures for bit vectors with worst-case constant query time, leading practical performance, and a space-overhead reaching below 1 %. For difficult inputs, we are ≈ 8 times faster than the best previous implementations.
Cite as
Florian Kurpicz, Niccolò Rigi-Luperti, and Peter Sanders. Practical Bit Vectors Supporting Constant Time Rank and Select in Optimal Space. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kurpicz_et_al:LIPIcs.ESA.2026.8,
author = {Kurpicz, Florian and Rigi-Luperti, Niccol\`{o} and Sanders, Peter},
title = {{Practical Bit Vectors Supporting Constant Time Rank and Select in Optimal Space}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {8:1--8:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.8},
URN = {urn:nbn:de:0030-drops-271443},
doi = {10.4230/LIPIcs.ESA.2026.8},
annote = {Keywords: succinct data structures, bit vectors, rank and select}
}
Document
T-REX: Fast and Dynamic Journey Planning for Continental-Scale Public Transit Networks
Abstract
We present T-REX (Transfer-Ranked EXploration), a new algorithm for journey planning in public transit networks on the country and continental scale. Our algorithm applies the principles of multi-level overlays to Trip-Based Public Transit Routing (TB). Using a multi-level partition of the network, T-REX identifies transfers between trips that are relevant for long-distance travel in a short precomputation phase. This information is then used to prune irrelevant local transfers during a query. Like other state-of-the-art algorithms, T-REX Pareto-optimizes arrival time and the number of used trips. T-REX dramatically outperforms previous overlay-based algorithms for three key reasons: (1) a better partition, (2) reducing the search space by focusing on transfers rather than trips, and (3) a redesigned query algorithm with improved memory efficiency and throughput. As a result, T-REX answers queries in less than 10ms on a network of Europe, including local and long-distance transit. This constitutes a speedup of 20 compared to TB and 80 compared to algorithms without preprocessing. The memory footprint is moderate and the precomputation takes only two minutes, while real-time schedule updates can be incorporated in a few seconds. These properties make T-REX the first public transit journey planning algorithm that fulfills the requirements of interactive real-time applications on the continental scale.
Cite as
Jonas Sauer, Patrick Steil, and Sascha Witt. T-REX: Fast and Dynamic Journey Planning for Continental-Scale Public Transit Networks. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 9:1-9:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{sauer_et_al:LIPIcs.ESA.2026.9,
author = {Sauer, Jonas and Steil, Patrick and Witt, Sascha},
title = {{T-REX: Fast and Dynamic Journey Planning for Continental-Scale Public Transit Networks}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {9:1--9:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.9},
URN = {urn:nbn:de:0030-drops-271458},
doi = {10.4230/LIPIcs.ESA.2026.9},
annote = {Keywords: Public transit routing, graph algorithms, algorithm engineering}
}
Document
Scheduling Tasks Towards Energy Autarky: Benefits and Computational Costs of Flexibility
Abstract
We study the autarky problem: given an energy forecast, a battery, and a set of energy-consuming jobs with time windows, decide whether all jobs can be scheduled without requiring external energy. We analyze the problem through the lens of job flexibility, defined as the number of time steps at which a job may be scheduled. We show that the problem is NP-hard already for flexibility two, even in restricted settings. On the positive side, we identify settings in which the problem is polynomial-time solvable, even for large flexibilities. Moreover, we obtain fixed-parameter tractability for combined parameters involving flexibility, such as the number of jobs. In contrast, we establish W-hardness when parameterized by maximum flexibility alone, even in a restricted setting. To complement our theoretical results, we formulate an integer linear program (ILP) that computes the minimum required external energy and evaluate it experimentally on instances derived from real-world energy-consumption and radiation data. The experiments indicate that increased job flexibility substantially reduces the need for external energy at moderate computational cost.
Cite as
Robert Bredereck, Till Fluschnik, and Klaus Heeger. Scheduling Tasks Towards Energy Autarky: Benefits and Computational Costs of Flexibility. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bredereck_et_al:LIPIcs.ESA.2026.10,
author = {Bredereck, Robert and Fluschnik, Till and Heeger, Klaus},
title = {{Scheduling Tasks Towards Energy Autarky: Benefits and Computational Costs of Flexibility}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {10:1--10:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.10},
URN = {urn:nbn:de:0030-drops-271469},
doi = {10.4230/LIPIcs.ESA.2026.10},
annote = {Keywords: computational sustainability, NP-hardness, parameterized complexity and algorithms, integer linear programming}
}
Document
Bicriteria Polygon Aggregation with Arbitrary Shapes
Abstract
We study the problem of aggregating a set of polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Equivalently, this can be seen as a fence enclosure problem, where the goal is to surround the polygons with a set of closed curves. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions, which naturally extends other fencing problems and has applications in geographical information systems. Previously, this objective was only studied in a restricted variant, in which the boundary curves of the regions must be selected from a fixed subdivision of the plane. It is natural to ask whether the problem is still tractable if this restriction is removed, allowing output regions to be bounded by arbitrary curves. We provide a positive answer in the form of an algorithm with runtime Õ(n⁴), where n is the number of input vertices. To achieve this, we fully characterize the optimal solutions by showing that their boundaries are composed of input edges and circular arcs of constant radius. Additionally, we consider the parametric problem, where for every weighting factor we seek a solution that is optimal for that trade-off of area and perimeter. We show that 𝒪 combinatorial solutions suffice to describe all optimal solutions across all weighting factors, and provide both an exact algorithm and an approximation scheme. To make the algorithms scalable in practice, we develop engineering techniques that exploit structural properties of the solutions. Our experimental evaluation on real-world data shows linear runtime in practice, even for the parametric variant.
Cite as
Lotte Blank, David Eppstein, Jan-Henrik Haunert, Herman Haverkort, Benedikt Kolbe, Philip Mayer, Petra Mutzel, Alexander Naumann, and Jonas Sauer. Bicriteria Polygon Aggregation with Arbitrary Shapes. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blank_et_al:LIPIcs.ESA.2026.11,
author = {Blank, Lotte and Eppstein, David and Haunert, Jan-Henrik and Haverkort, Herman and Kolbe, Benedikt and Mayer, Philip and Mutzel, Petra and Naumann, Alexander and Sauer, Jonas},
title = {{Bicriteria Polygon Aggregation with Arbitrary Shapes}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {11:1--11:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.11},
URN = {urn:nbn:de:0030-drops-271472},
doi = {10.4230/LIPIcs.ESA.2026.11},
annote = {Keywords: polygon aggregation, fencing, minimum-perimeter clustering, map generalization, urban analytics}
}
Document
High-Quality Multi-Constraint Hypergraph Partitioning via Greedy Rebalancing
Abstract
Multi-constraint hypergraph partitioning is a generalization of balanced partitioning, where the vertex set of a hypergraph is partitioned such that the inter-block connectivity of hyperedges is minimized while balancing the vertices with regard to d distinct constraints. A prominent class of applications is data distribution tasks, where this allows to achieve good load balance for d different kinds of resources and simultaneously minimize the communication volume.
Although the best approaches for single-constraint partitioning are usually complex (multilevel) algorithms with many components, we show that replacing only one component already leads to high-quality multi-constraint partitions: the rebalancing step, which restores balance for a partition that has (hopefully) small connectivity but violates the constraints. We design a multi-constraint rebalancing algorithm based on greedy local search, proving that balance is always restored for d = 2 and bounded maximum weight. The key is to ensure monotonically decreasing global imbalance by choosing an imbalance metric where there is always a balance-improving move available. Integrating our algorithm into the state-of-the-art partitioner Mt-KaHyPar, we demonstrate an 11.5 % geometric mean connectivity reduction compared to the next best competitor (Metis) and better reliability regarding partition balance, even though the majority of inputs is outside of the theoretical guarantee.
Cite as
Nikolai Maas. High-Quality Multi-Constraint Hypergraph Partitioning via Greedy Rebalancing. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{maas:LIPIcs.ESA.2026.12,
author = {Maas, Nikolai},
title = {{High-Quality Multi-Constraint Hypergraph Partitioning via Greedy Rebalancing}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {12:1--12:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.12},
URN = {urn:nbn:de:0030-drops-271480},
doi = {10.4230/LIPIcs.ESA.2026.12},
annote = {Keywords: Hypergraph Partitioning, Multi-Constraint Partitioning, Graph Algorithms, Multilevel Algorithms, Local Search, Vector Scheduling, Multidimensional Load Balancing}
}
Document
Strong ILP Formulations for the p-Regions Problem
Abstract
Regionalization is a fundamental task in spatial analysis that seeks to partition a larger area - such as a country - into smaller regions that are homogeneous with respect to a given attribute. A popular model for regionalization is the p-regions problem, in which regions are formed by grouping the areas of an input planar subdivision. Given the subdivision’s adjacency graph G and pairwise dissimilarities between vertices, the goal is to partition G into a fixed number p of connected subgraphs, such as to minimize the sum of dissimilarities over all vertex pairs in the same subgraph. The problem is NP-hard and even small instances are difficult to solve to provable optimality.
In this paper, we present the new ILP model ER-S for the p-regions problem, exploiting a connection between the p-regions objective and the k-partitioning problem. Furthermore, we strengthen the known ILP model Tree with a new type of subtour elimination inequality specific to the p-regions problem. Combining ER-S and the strengthened version of Tree yields the model ER-S-Tree, which dominates the state-of-the-art models in polyhedral strength. This theoretical advantage is reflected in its superior performance in our experimental evaluation. In particular, the new models ER-S and ER-S-Tree enable the solution of problem instances for major European countries that were previously intractable.
Cite as
Daniel Faber, Jan-Henrik Haunert, and Petra Mutzel. Strong ILP Formulations for the p-Regions Problem. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{faber_et_al:LIPIcs.ESA.2026.13,
author = {Faber, Daniel and Haunert, Jan-Henrik and Mutzel, Petra},
title = {{Strong ILP Formulations for the p-Regions Problem}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {13:1--13:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.13},
URN = {urn:nbn:de:0030-drops-271498},
doi = {10.4230/LIPIcs.ESA.2026.13},
annote = {Keywords: p-regions problem, connected graph partitioning, area aggregation, integer linear programming, branch-and-cut}
}
Document
Virtual-Memory Powersort
Abstract
We give a more space-efficient implementation of adaptive mergesort: Virtual-Memory Powersort. Using internal buffering techniques, we significantly reduce the memory consumption of the algorithm; specifically, for sorting n objects the required buffer area is reduced from space for n/2 objects to O(√{n log n}) objects. While this space-efficiency can be achieved (indeed reduced to O(1)) conceptually very easily with known inplace merging algorithms, using these as a drop-in replacement for the standard merge algorithm incurs a substantial slow-down. Virtual-Memory Powersort, by contrast, uses the same number of moves and comparisons as previous Powersort implementations up to an additive O(n) term. We report on an empirical running-time study comparing our implementation against other Powersort variants and state-of-the-art stable sorting methods, demonstrating that almost in-place stable sorting can be achieved with negligible overhead in many scenarios.
Cite as
Finn Moltmann, Tamio-Vesa Nakajima, and Sebastian Wild. Virtual-Memory Powersort. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{moltmann_et_al:LIPIcs.ESA.2026.14,
author = {Moltmann, Finn and Nakajima, Tamio-Vesa and Wild, Sebastian},
title = {{Virtual-Memory Powersort}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {14:1--14:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.14},
URN = {urn:nbn:de:0030-drops-271501},
doi = {10.4230/LIPIcs.ESA.2026.14},
annote = {Keywords: adaptive sorting, inplace sorting, inplace merging, library sort, virtual memory, internal buffering, Powersort, Timsort}
}
Document
Deconstructed "Learned" Indexes and Their Smoothed Analysis
Abstract
Data structures that maintain a sorted sequence are crucial for many applications. There is a zoo of variants with recent particular interest in "learned" indexes that accelerate operations by learning the distribution of the data. This paper helps to bring some order to this complex situation. We identify important building blocks and model the input using smoothed analysis where an adversary can control the dynamically changing input except for a small amount of noise. Within a resulting design space of data structures, we prove that already a simple 2-level data structure with minimal learning can achieve constant operation times in many situations: PARROT partitions the input into equal size parts, within which keys are approximately uniformly distributed. In many of our experiments, PARROT performs very well compared to state-of-the-art learned indexes, being 2× faster than the well known ALEX and LIPP indexes on large datasets, and 10× faster than a well engineered standard B-Tree.
Cite as
Stefan Hermann, Mattia Odorisio, Peter Sanders, and Stefan Walzer. Deconstructed "Learned" Indexes and Their Smoothed Analysis. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 15:1-15:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hermann_et_al:LIPIcs.ESA.2026.15,
author = {Hermann, Stefan and Odorisio, Mattia and Sanders, Peter and Walzer, Stefan},
title = {{Deconstructed "Learned" Indexes and Their Smoothed Analysis}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {15:1--15:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.15},
URN = {urn:nbn:de:0030-drops-271518},
doi = {10.4230/LIPIcs.ESA.2026.15},
annote = {Keywords: Learned data structure, sorted sequence, index data structure, smoothed analysis}
}
Document
SimdQuickHeap: The QuickHeap Reconsidered
Abstract
Motivation. Priority queues are data structures that maintain a dynamic collection of elements and allow inserting new elements and removing the smallest element. The most widely known and used priority queue is likely the implicit binary heap, even though it has frequent cache misses and is hard to optimize using e.g. SIMD instructions.
Contributions. We introduce the SimdQuickHeap, a variant of the QuickHeap that was introduced by Navarro and Paredes in 2010. As suggested by the name, the data structure bears some similarity to QuickSort. We modify the data layout of the original QuickHeap to have all pivots adjacent in memory, with elements between consecutive pivots stored in dedicated buckets. This allows efficient SIMD implementations for both partitioning of buckets and scanning the list of pivots to find the bucket to append newly inserted elements to.
The SimdQuickHeap has amortized expected complexity O(log n) per operation, which improves to O((1/W) log n) in non-degenerate cases, where W is the number of words in a SIMD register. In this case, the I/O-complexity is amortized O(1/B) per push and O((1/B) log₂(n/M)) per pop.
Results. In synthetic benchmarks, the SimdQuickHeap is 1.2× to 1.7× as fast as the monotone radix heap, the next-best competitor, and 1.4× to 2.8× as fast as the superscalar sample queue, the fastest comparison-based priority queue. The SimdQuickHeap needs around 1.5log₂ n comparisons and log₂ n nanoseconds per pair of push and pop operations. On graph benchmarks with Dijkstra’s shortest path algorithm and Jarník-Prim’s minimum spanning tree algorithm, the SimdQuickHeap is consistently the fastest.
Cite as
Johannes Breitling, Ragnar Groot Koerkamp, and Marvin Williams. SimdQuickHeap: The QuickHeap Reconsidered. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{breitling_et_al:LIPIcs.ESA.2026.16,
author = {Breitling, Johannes and Groot Koerkamp, Ragnar and Williams, Marvin},
title = {{SimdQuickHeap: The QuickHeap Reconsidered}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {16:1--16:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.16},
URN = {urn:nbn:de:0030-drops-271527},
doi = {10.4230/LIPIcs.ESA.2026.16},
annote = {Keywords: Heap, SIMD, Priority Queue}
}
Document
Minimizing Total Travel Time for Collaborative Package Delivery with Heterogeneous Drones
Abstract
Given a fleet of drones with different speeds and a set of package delivery requests, the collaborative delivery problem asks for a schedule for the drones to collaboratively carry out all package deliveries, with the objective of minimizing the total travel time of all drones. We show that the best non-preemptive schedule (where a package that is picked up at its source is immediately delivered to its destination by one drone) is within a factor of three of the best preemptive schedule (where several drones can participate in the delivery of a single package). Then, we present a constant-factor approximation algorithm for the problem of computing the best non-preemptive schedule. The algorithm reduces the problem to a tree combination problem and uses a primal-dual approach to solve the latter. We have implemented a version of the algorithm optimized for practical efficiency and report the results of experiments on large-scale instances with synthetic and real-world data, demonstrating that our algorithm is scalable and delivers schedules of excellent quality.
Cite as
Thomas Erlebach, Kelin Luo, and Wen Zhang. Minimizing Total Travel Time for Collaborative Package Delivery with Heterogeneous Drones. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{erlebach_et_al:LIPIcs.ESA.2026.17,
author = {Erlebach, Thomas and Luo, Kelin and Zhang, Wen},
title = {{Minimizing Total Travel Time for Collaborative Package Delivery with Heterogeneous Drones}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {17:1--17:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.17},
URN = {urn:nbn:de:0030-drops-271531},
doi = {10.4230/LIPIcs.ESA.2026.17},
annote = {Keywords: Approximation algorithms, Primal-dual method, Heterogeneous pickup and delivery problem, Algorithm engineering}
}
Document
Efficient Uniform Negative Edge Weights
Abstract
We consider a maximum entropy edge weight model that allows for negative weights. Given a graph G and possible weights W typically consisting of positive and negative values, the model selects edge weights w ∈ W^m uniformly at random from all weights that do not introduce a negative cycle. We propose an MCMC process and show that it converges to the required distribution. We then engineer an implementation of the process using a dynamic version of Johnson’s algorithm in connection with a bidirectional Dijkstra search as well as an innovative resampling method. We empirically study the performance characteristics of these novel sampling algorithms as well as the output produced by the model.
Cite as
Lukas Geis, Daniel Allendorf, Thomas Bläsius, Alexander Leonhardt, Ulrich Meyer, Manuel Penschuck, and Hung Tran. Efficient Uniform Negative Edge Weights. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 18:1-18:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{geis_et_al:LIPIcs.ESA.2026.18,
author = {Geis, Lukas and Allendorf, Daniel and Bl\"{a}sius, Thomas and Leonhardt, Alexander and Meyer, Ulrich and Penschuck, Manuel and Tran, Hung},
title = {{Efficient Uniform Negative Edge Weights}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {18:1--18:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.18},
URN = {urn:nbn:de:0030-drops-271542},
doi = {10.4230/LIPIcs.ESA.2026.18},
annote = {Keywords: Random Graphs, Shortest Path, Random Edge Weights, Negative Cycles}
}
Document
No Time to Interact: Simulating Population Protocols at Scale
Abstract
We consider the simulation of population protocols with a random scheduler. A population protocol consists of n agents each associated a state q_i ∈ Q where |Q| = 𝒪(log n) for many protocols. An execution consists of a large number I ≫ n of interactions, each time selecting two agents uniformly at random and updating their state solely based on the current states of the pair.
We propose - to the best of our knowledge - a novel and faithful simulation technique: we show that a protocol’s (undirected) interaction multi-graph follows the configuration model with a Poisson degree distribution (up to some rejection). We then introduce a sampling algorithm for the component structure of the configuration model which we believe to be of independent interest. In the subcritical regime, its runtime scales in the number of component types rather than the graph size. Finally, we show how to accelerate the simulation of a protocol by using the component structure as an execution plan template. For any fixed integer parameter τ ≥ 2 and I ≫ n, the time taken per n interactions is in 𝒪(|Q|^{2-2/τ} n^{1/τ}) (where the hidden constant depends heavily on τ).
In an experimental evaluation, we give evidence to the practicality of our approach. For large systems, we observe a speedup of several orders of magnitude over state-of-the-art simulators.
Cite as
Lukas Hintze and Manuel Penschuck. No Time to Interact: Simulating Population Protocols at Scale. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hintze_et_al:LIPIcs.ESA.2026.19,
author = {Hintze, Lukas and Penschuck, Manuel},
title = {{No Time to Interact: Simulating Population Protocols at Scale}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {19:1--19:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.19},
URN = {urn:nbn:de:0030-drops-271556},
doi = {10.4230/LIPIcs.ESA.2026.19},
annote = {Keywords: Stochastic Simulation, Population protocols, Configuration model, Component structure, Sampling}
}
Document
Non-Minimal k-Perfect Hashing: Tight Lower Bounds and an Application to Fast Static Hash Tables
Abstract
A minimal perfect hash function (minimal PHF) is a data structure mapping a static set of n keys to n bins without collisions. Two natural generalizations are minimal k-PHFs where n keys are mapped to n/k bins of capacity k each, and (non-minimal) PHFs with load factor α < 1 where the number of bins is increased by a factor of 1/α, resulting in spare capacity.
While there has been a recent surge of interest in perfect hashing generally, non-minimal k-PHFs have not been systematically studied despite a natural use case of speeding up static hash tables: The idea is that a small cache-resident k-PHF maps each key x to a cache-line-sized bin of capacity k where x resides. Ideally, this yields a branchless lookup operation with a single cache miss working at high load factors for positive and negative queries alike.
Our main theoretical contribution is to determine tight space lower bounds for k-PHFs for all pairs of α ∈ (0,1] and k ≥ 1. It turns out that combining α < 1 and k ≥ 2 drastically reduces the space of k-PHFs, e.g. for (k,α) = (16,0.8) the space lower bound is 0.027 bits per key while for (k,α) = (16,1.0) and (k,α) = (1,0.8) the lower bounds are higher by factors of ≈ 8 and ≈ 32, respectively. On the practical side, we develop a k-PHF based on PtrHash and tune it for use in static hash tables. Empirically, our implementation produces k-PHFs of size roughly 50% above the lower bound. A static hash set based on this k-PHF is consistently at least as fast as other hash sets for negative and mixed queries. On two of the three tested architectures it achieves up to 1.5× speedup for large n ≥ 30M where a 1-PHF does not fit in cache.
Cite as
Ragnar Groot Koerkamp, Stefan Hermann, Peter Sanders, and Stefan Walzer. Non-Minimal k-Perfect Hashing: Tight Lower Bounds and an Application to Fast Static Hash Tables. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{grootkoerkamp_et_al:LIPIcs.ESA.2026.20,
author = {Groot Koerkamp, Ragnar and Hermann, Stefan and Sanders, Peter and Walzer, Stefan},
title = {{Non-Minimal k-Perfect Hashing: Tight Lower Bounds and an Application to Fast Static Hash Tables}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {20:1--20:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.20},
URN = {urn:nbn:de:0030-drops-271565},
doi = {10.4230/LIPIcs.ESA.2026.20},
annote = {Keywords: Compressed Data Structures, k-Perfect Hashing, Hash Table, Space Lower Bound}
}
Document
Optimality-Preserving Data Reduction for Maximum k-Cut
Abstract
Preprocessing has become an increasingly important part of solving Maximum Cut to optimality, enabling exact solvers to tackle significantly larger instances. This suggests that exact solvers for the more general Maximum k-Cut problem could also benefit from sophisticated preprocessing. However, to the best of our knowledge, no preprocessing techniques that are effective for k > 2 have been published.
In this paper, we introduce structured cut sets, a novel data reduction technique for Maximum k-Cut. We provide criteria under which deleting cut sets is optimality-preserving, yielding a decomposition into connected components that can be solved independently and whose solutions can be combined into an optimal solution for the original graph. Furthermore, we extend several preprocessing techniques from Maximum Cut to Maximum k-Cut. To show that our rules are optimality-preserving, we develop a new proof framework based on the addition of weighted graphs.
We complement our theoretical results by engineering a preprocessing framework for Maximum k-Cut and show its effectiveness in a computational study. The preprocessed instances are typically significantly smaller. Integrating our preprocessing into an exact solver yields significant speed-ups and enables solving more instances to optimality.
Cite as
Michael Kaibel and Petra Mutzel. Optimality-Preserving Data Reduction for Maximum k-Cut. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kaibel_et_al:LIPIcs.ESA.2026.21,
author = {Kaibel, Michael and Mutzel, Petra},
title = {{Optimality-Preserving Data Reduction for Maximum k-Cut}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {21:1--21:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.21},
URN = {urn:nbn:de:0030-drops-271578},
doi = {10.4230/LIPIcs.ESA.2026.21},
annote = {Keywords: Data Reduction, Preprocessing, Maximum k-Cut}
}
Document
Benchmarking and Engineering Data Structures for Spherical Range Queries
Abstract
Spherical range queries are a fundamental primitive for working with spatial data. Many spatial data structures have been developed to answer these queries, but choosing the optimal one for a specific application is a difficult task. This is because theoretical worst-case bounds are often overly pessimistic, and existing average-case analyses are rather restricted and hard to compare.
We address this problem with two main contributions. First, we present a comprehensive evaluation of state-of-the-art spatial indices across a diverse set of benchmarks. This includes a new benchmark based on graph embeddings alongside multiple real-world datasets from the literature. Our benchmark covers instances scaling up to 10M points and ranging between 2 and 960 dimensions.
Second, we introduce the Sorted-Projection Radius KD-tree (SPRK-tree), a high-performance KD-tree variant. The SPRK-tree combines aggressive subtree pruning via radius reduction, sorted projection-based leaf nodes, and careful implementation optimizations. It consistently achieves the fastest query times in almost all benchmarks, and ranks second in the few remaining cases.
Cite as
Thomas Bläsius, Jean-Pierre von der Heydt, Tobias Kempf, Dennis Kobert, and Nikolai Maas. Benchmarking and Engineering Data Structures for Spherical Range Queries. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blasius_et_al:LIPIcs.ESA.2026.22,
author = {Bl\"{a}sius, Thomas and von der Heydt, Jean-Pierre and Kempf, Tobias and Kobert, Dennis and Maas, Nikolai},
title = {{Benchmarking and Engineering Data Structures for Spherical Range Queries}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {22:1--22:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.22},
URN = {urn:nbn:de:0030-drops-271585},
doi = {10.4230/LIPIcs.ESA.2026.22},
annote = {Keywords: Spherical Range Queries, Fixed-Radius Nearest Neighbor Search, Spatial Indexing, KD-tree, Benchmarking, Graph Embedding, SPRK-Tree}
}
Document
Exact Ratio Preservation via Outliers for Fair k-Center Clustering
Abstract
We study the k-center clustering problem under demographic fairness constraints, where the point set is partitioned into groups, and the aim is to compute clusters that exhibit a given group proportion. Previous work in this direction assumes that the entire point set already respects the desired proportions or uses relaxed notions of fairness.
In this work, we propose a model that facilitates the creation of clusters that exactly match given target ratios, even when the input point set does not. We combine the well-known fair clustering model initiated by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [Flavio Chierichetti et al., 2017] with the notion of outliers to obtain a practical combinatorial framework that provides constant-factor approximate solutions for all proportion settings from 1:1 for two groups to t₁:t₂:…:t_m for m ≥ 2 groups, where t₁,…,t_m are integers.
We implement and evaluate our algorithms, compare different variants, and provide evidence of the practicability of this approach.
Cite as
Anna Arutyunova, Irina Fast, Annika Hennes, Carsten Krollmann, Daniel R. Schmidt, and Melanie Schmidt. Exact Ratio Preservation via Outliers for Fair k-Center Clustering. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 23:1-23:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{arutyunova_et_al:LIPIcs.ESA.2026.23,
author = {Arutyunova, Anna and Fast, Irina and Hennes, Annika and Krollmann, Carsten and Schmidt, Daniel R. and Schmidt, Melanie},
title = {{Exact Ratio Preservation via Outliers for Fair k-Center Clustering}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {23:1--23:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.23},
URN = {urn:nbn:de:0030-drops-271592},
doi = {10.4230/LIPIcs.ESA.2026.23},
annote = {Keywords: Fairness, k-center, approximation algorithms}
}
Document
On the See-Through Watchman Route Problem and the Quota-TSP Problem on Infinite Lines
Abstract
The classic Watchman Route Problem (WRP) seeks to compute a shortest tour in a polygonal domain that sees every point of the domain. We introduce and study a novel generalization of the WRP, the See-Through Watchman Route Problem (STWRP), in which, in addition to vision-blocking "walls" of an input domain, there are obstacles to motion that are not opaque to vision: the watchman can see through certain obstacles or portions of the boundary of a polygonal domain P. This setting is motivated by real-world situations that may include transparent barriers (e.g., glass walls), obstacles that obstruct movement but not vision (e.g., lakes, flowerbeds, or potholes), and robotic sensors with penetration capabilities (e.g., microwave imaging). To the best of our knowledge, this version of the problem is new to the algorithms community.
Our main result is an FPTAS for the STWRP in the case that P is an opaque-walled simple polygon having within it a set of transparent obstacles. A closely related problem that arises in this setting is that of the Traveling Salesperson problem with neighborhoods (TSPN) on a set of lines in the plane, with obstacles. We give the first FPTAS for the Quota-TSPN on infinite lines with polygonal obstacles. Additionally, we show tightness of our FPTAS, in that the Quota-TSPN on infinite lines with obstacles is weakly NP-hard. In the case of the STWRP within a simple polygon P with portions of the boundary, ∂ P, being transparent, we prove that the problem is NP-hard to approximate within a factor better than O(log n).
Cite as
Joseph S. B. Mitchell and Linh Nguyen. On the See-Through Watchman Route Problem and the Quota-TSP Problem on Infinite Lines. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 24:1-24:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mitchell_et_al:LIPIcs.ESA.2026.24,
author = {Mitchell, Joseph S. B. and Nguyen, Linh},
title = {{On the See-Through Watchman Route Problem and the Quota-TSP Problem on Infinite Lines}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {24:1--24:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.24},
URN = {urn:nbn:de:0030-drops-271605},
doi = {10.4230/LIPIcs.ESA.2026.24},
annote = {Keywords: Watchman route problem, TSP with neighborhoods}
}
Document
Algorithms for Standard-Form ILP Problems via Komlós' Discrepancy Setting
Abstract
We study the standard-form ILP problem c^⊤ x → max Ax = b, x ∈ ℤ_{≥ 0}ⁿ, where A ∈ ℤ^{k× n} has full row rank. We obtain refined FPT algorithms parameterized by k and Δ, the maximum absolute value of a k× k minor of A. Our approach combines discrepancy-based dynamic programming with matrix discrepancy bounds in Komlós' setting. Let κ_k denote the maximum discrepancy over all matrices with k columns whose columns have Euclidean norm at most 1. Up to polynomial factors in the input size, the optimization problem can be solved in time O(κ_k)^{2k} Δ², and the corresponding feasibility problem in time O(κ_k)^kΔ. Using the best currently known bound κ_k = Õ(log^{1/4}k), this yields running times O(log k)^{k/2(1+o(1))} Δ² and O(log k)^{k/4(1+o(1))} Δ, respectively. Under the Komlós conjecture, the dependence on k in both running times reduces to 2^O(k).
Cite as
Dmitry Gribanov, Tagir Khayaleyev, Mikhail Cherniavskii, Maxim Klimenko, Dmitry Malyshev, and Stanislav Moiseev. Algorithms for Standard-Form ILP Problems via Komlós' Discrepancy Setting. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gribanov_et_al:LIPIcs.ESA.2026.25,
author = {Gribanov, Dmitry and Khayaleyev, Tagir and Cherniavskii, Mikhail and Klimenko, Maxim and Malyshev, Dmitry and Moiseev, Stanislav},
title = {{Algorithms for Standard-Form ILP Problems via Koml\'{o}s' Discrepancy Setting}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {25:1--25:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.25},
URN = {urn:nbn:de:0030-drops-271610},
doi = {10.4230/LIPIcs.ESA.2026.25},
annote = {Keywords: Parameterized complexity, FPT algorithms, Integer linear programming, Koml\'{o}s' conjecture, Discrepancy}
}
Document
Sliding Cubes in Parallel
Abstract
In the classic sliding cube model for programmable matter in three dimensions, the task is to find a reconfiguration sequence between two connected configurations of n indistinguishable unit cube modules by sliding modules along their neighbors' faces. Depending on the objective, this sequence should minimize either the total energy expended (the number of moves) or the total elapsed time (the makespan). We give a number of results for the three-dimensional setting, including (i) the first algorithm that achieves worst-case optimal makespan under parallel motion in three dimensions, (ii) a proof of log-APX-hardness to decide either the optimal makespan or the optimal number of moves, which is the strongest known inapproximability bound in any related model, and (iii) a proof of NP-hardness to decide the optimal makespan under parallel motion, even if the two configurations differ only by one module and the optimal makespan is at most two. Our results strengthen the inapproximability claim from [Hugo A. Akitaya et al., 2022] and answer a question of [Akitaya et al., 2025] in the negative.
Cite as
Hugo A. Akitaya, Joseph Dorfer, Peter Kramer, Christian Rieck, Gabriel Shahrouzi, and Frederick Stock. Sliding Cubes in Parallel. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{a.akitaya_et_al:LIPIcs.ESA.2026.26,
author = {A. Akitaya, Hugo and Dorfer, Joseph and Kramer, Peter and Rieck, Christian and Shahrouzi, Gabriel and Stock, Frederick},
title = {{Sliding Cubes in Parallel}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {26:1--26:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.26},
URN = {urn:nbn:de:0030-drops-271621},
doi = {10.4230/LIPIcs.ESA.2026.26},
annote = {Keywords: Sliding squares, parallel motion, reconfigurability, three dimensions, constant makespan, log-APX-hardness, NP-hardness, worst-case optimality}
}
Document
On the Stability of Minimum-Weight Perfect Matching on the Line
Abstract
Computing a minimum-weight perfect matching for a point set P in Euclidean space is a classic geometric optimization problem. We consider the problem in a dynamic setting, where pairs of points may be added to or removed from the set P. Our focus is on maintaining an approximately optimal solution without making too many changes to the solution. More precisely, we are interested in k-stable algorithms, which change at most k edges in the matching after each update to the set P. In other words, we consider an online setting (with insertions and deletions) with bounded recourse.
We study trade-offs between the stability of the algorithm and the approximation ratio of the maintained solution for point sets in ℝ¹. First, we present an O(√n)-stable algorithm that maintains a 2-approximation, which we show to be optimal among all algorithms with sublinear stability. Second, we prove that any o(log n)-stable algorithm has unbounded approximation ratio. Our lower bounds hold even in the insertion-only case, while our algorithm works in the fully dynamic case. Moreover, our lower bounds also hold for the bipartite variant of the problem.
Cite as
Mark de Berg, Ulrike Schmidt-Kraepelin, and Andree-Ovidiu Ștef. On the Stability of Minimum-Weight Perfect Matching on the Line. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{deberg_et_al:LIPIcs.ESA.2026.27,
author = {de Berg, Mark and Schmidt-Kraepelin, Ulrike and Ștef, Andree-Ovidiu},
title = {{On the Stability of Minimum-Weight Perfect Matching on the Line}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {27:1--27:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.27},
URN = {urn:nbn:de:0030-drops-271632},
doi = {10.4230/LIPIcs.ESA.2026.27},
annote = {Keywords: Euclidean matching, stable approximation algorithms, dynamic algorithms, online algorithms, bounded recourse}
}
Document
Rerouting Curves on Surfaces
Abstract
We study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In this process, we reroute one edge at a time while maintaining crossing-free intermediate embeddings. This problem was introduced by Ito et al. [TALG 2025], who showed that even if the graph is a matching of two edges, reconfiguration is not always possible in the plane, but is always possible on the torus. For matchings of two or more edges, they gave a necessary and sufficient condition for reconfigurable embeddings in the plane, but not on the torus. Our main result is that for matchings, trees and forests, reconfiguration is always possible on the torus, and consequently, on any orientable surface of genus at least one. In addition, we provide sufficient conditions for reconfiguration on orientable surfaces of genus at least one and in the projective plane. For more general graphs, we show that reconfiguration is not always possible.
Cite as
Timo Brand, Stefan Felsner, Henry Förster, Stephen Kobourov, Anna Lubiw, Yoshio Okamoto, János Pach, Csaba D. Tóth, Géza Tóth, Torsten Ueckerdt, and Pavel Valtr. Rerouting Curves on Surfaces. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brand_et_al:LIPIcs.ESA.2026.28,
author = {Brand, Timo and Felsner, Stefan and F\"{o}rster, Henry and Kobourov, Stephen and Lubiw, Anna and Okamoto, Yoshio and Pach, J\'{a}nos and T\'{o}th, Csaba D. and T\'{o}th, G\'{e}za and Ueckerdt, Torsten and Valtr, Pavel},
title = {{Rerouting Curves on Surfaces}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {28:1--28:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.28},
URN = {urn:nbn:de:0030-drops-271641},
doi = {10.4230/LIPIcs.ESA.2026.28},
annote = {Keywords: Combinatorial reconfiguration, orientable surface, non-orientable surface, rerouting}
}
Document
Multiple-Choice Knapsack with Small Items
Abstract
The Multiple-Choice Knapsack problem is a generalization of the Knapsack problem in which items are partitioned into disjoint classes and exactly one item must be selected from each class. Besides the standard dynamic program running in time 𝒪(nc), where n is the total number of items and c is the knapsack capacity, no other pseudopolynomial-time algorithm is currently known. We present an 𝒪(n+w_max^4)-time algorithm, where w_max = max_i (w_{i n_i}-w_{i 1}) is the maximum weight range within a class.
Cite as
Jakub Pawlewicz. Multiple-Choice Knapsack with Small Items. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 29:1-29:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{pawlewicz:LIPIcs.ESA.2026.29,
author = {Pawlewicz, Jakub},
title = {{Multiple-Choice Knapsack with Small Items}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {29:1--29:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.29},
URN = {urn:nbn:de:0030-drops-271653},
doi = {10.4230/LIPIcs.ESA.2026.29},
annote = {Keywords: multiple-choice knapsack, knapsack problem, small items, pseudopolynomial algorithms, proximity bounds, subset sums}
}
Document
Maximum Weight Independent Set in Hereditary Classes of Ordered Graphs
Abstract
The complexity of classical computational problems in graph classes defined by forbidding induced subgraphs is one of the central topics of algorithmic graph theory. Recently, there has been a growing interest in the complexity of such problems in ordered graphs, i.e., graphs with a fixed linear ordering of vertices. Such an approach allows us to investigate the boundary of tractability more closely. However, most results so far concern coloring problems.
In this paper, we focus on the complexity of the Maximum Weight Independent Set (MWIS) problem in classes of ordered graphs. For every ordered graph H, we classify the complexity of MWIS in ordered graphs that exclude H as an induced subgraph into one of the following cases: (1) solvable in polynomial time, (2) solvable in quasipolynomial time, (3) solvable in subexponential time, and (4) NP-hard.
Notably, case (3) contains only one well-structured family of H obtained from two nested edges by adding isolated vertices in a specific way. Thus, our results yield an almost complete complexity dichotomy for MWIS in classes of ordered graphs defined by a single forbidden induced subgraph into cases solvable in quasipolynomial time and those that are NP-hard.
Cite as
Paweł Rafał Bieliński, Marta Piecyk, and Paweł Rzążewski. Maximum Weight Independent Set in Hereditary Classes of Ordered Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bielinski_et_al:LIPIcs.ESA.2026.30,
author = {Bieli\'{n}ski, Pawe{\l} Rafa{\l} and Piecyk, Marta and Rz\k{a}\.{z}ewski, Pawe{\l}},
title = {{Maximum Weight Independent Set in Hereditary Classes of Ordered Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {30:1--30:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.30},
URN = {urn:nbn:de:0030-drops-271662},
doi = {10.4230/LIPIcs.ESA.2026.30},
annote = {Keywords: Max Independent Set, ordered graphs, hereditary classes}
}
Document
Unconditional Lower Bounds for Degree Fault Tolerant Spanners
Abstract
We study multiplicative graph spanners in the f-degree fault tolerant (f-DFT) model, in which the spanner must approximately preserve distances even after any subset of edges of maximum degree f temporarily "fails" and is removed from the graph. We prove that there are n-node lower bound graphs for which any f-DFT (2k-1)-stretch spanner H must have size |E(H)| ≥ Ω(f^{1-1/k} n^{1+1/k}) . This matches a lower bound that was previously only known to hold conditionally, under the 1963 girth conjecture of Erdős. It also matches the current upper bounds, up to a factor of exp(k). Our proof is an analysis of the so-called Wenger graphs (J. Comb. Theory 1991), via their recent reinterpretation by Szabó and by Conlon (Am. Math. Monthly 2021).
Cite as
Greg Bodwin and Aleksey Lopez. Unconditional Lower Bounds for Degree Fault Tolerant Spanners. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 31:1-31:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bodwin_et_al:LIPIcs.ESA.2026.31,
author = {Bodwin, Greg and Lopez, Aleksey},
title = {{Unconditional Lower Bounds for Degree Fault Tolerant Spanners}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {31:1--31:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.31},
URN = {urn:nbn:de:0030-drops-271673},
doi = {10.4230/LIPIcs.ESA.2026.31},
annote = {Keywords: Spanners, Fault Tolerance, Girth Conjecture, Wenger Graphs}
}
Document
Polynomial-Size Encoding of All Cuts of Small Value in Integer-Valued Symmetric Submodular Functions
Abstract
We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining 0 on the empty set. For a connectivity function f on an n-element set V and an integer k ≥ 0, we show that the family of all sets X ⊆ V with f(X) = k admits a polynomial-size representation: it can be described by a list of at most O(n^{4k}) items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets X with f(X) = k. We also give an algorithm that constructs this representation in time O(n^{2k+7}γ+n^{2k+8}+n^{4k+2}), where γ is the oracle time to evaluate f. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, LICS 2026] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed k, we obtain a polynomial-time algorithm for finding a set A with f(A) = k and a prescribed cardinality constraint on A.
Cite as
Sang-il Oum and Marek Sokołowski. Polynomial-Size Encoding of All Cuts of Small Value in Integer-Valued Symmetric Submodular Functions. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 32:1-32:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{oum_et_al:LIPIcs.ESA.2026.32,
author = {Oum, Sang-il and Soko{\l}owski, Marek},
title = {{Polynomial-Size Encoding of All Cuts of Small Value in Integer-Valued Symmetric Submodular Functions}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {32:1--32:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.32},
URN = {urn:nbn:de:0030-drops-271680},
doi = {10.4230/LIPIcs.ESA.2026.32},
annote = {Keywords: Connectivity function, Symmetric Submodular Function, Submodular Minimum Bisection}
}
Document
Conflict-Free Coloring Planar Graphs with 4 Colors
Abstract
We efficiently conflict-free color every planar graph with 4 colors. An (open-neighborhood) conflict-free coloring assigns colors to vertices in a way that every vertex v has a neighbor w such that the color of w is distinct from the colors of the other neighbors of v (i.e., the color of w is unique in the open neighborhood of v). A previous best upper bound on the conflict-free chromatic number of planar graphs was 5, and it is known that 4 colors are sometimes necessary. Deciding whether, e.g., a planar graph admits a conflict-free coloring with 3 colors is NP-complete. Our approach uses a refined variant of the classical Gallai-Edmonds decomposition and the Four Color Theorem. In fact, our result is equivalent to the Four Color Theorem.
Cite as
Petr Hliněný and Lukáš Málik. Conflict-Free Coloring Planar Graphs with 4 Colors. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hlineny_et_al:LIPIcs.ESA.2026.33,
author = {Hlin\v{e}n\'{y}, Petr and M\'{a}lik, Luk\'{a}\v{s}},
title = {{Conflict-Free Coloring Planar Graphs with 4 Colors}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {33:1--33:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.33},
URN = {urn:nbn:de:0030-drops-271696},
doi = {10.4230/LIPIcs.ESA.2026.33},
annote = {Keywords: conflict-free coloring, planar graph, matching, Gallai-Edmonds decomposition}
}
Document
Triangle Nearest-Neighbor Searching in 3-Space
Abstract
We study various nearest-neighbor searching problems involving points, lines, segments and triangles in ℝ³. Among many results, we present a linear-size data structure for answering nearest-neighbor queries with lines amid n points in ℝ³, in O^*(n^{1/2}) time per query (where the O^*(⋅) notation hides subpolynomial factors). Our solution is based on parametric search, where the problem is reduced to range emptiness queries amid points in ℝ³ with cylindrical queries. For the latter problem we show that reporting all k points lying inside a cylinder query costs an additional term of O(k). We also study setups where both data and query objects are lines, segments, or triangles, and obtain improved solutions for the two extreme regimes of (near-)linear storage and of fast query time. These results also yield tradeoff bounds, where the cost of a query depends on the storage allocated to the structure. This work is a continuation of a recent work by the authors [Pankaj K. Agarwal et al., 2024].
Cite as
Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Triangle Nearest-Neighbor Searching in 3-Space. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 34:1-34:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{agarwal_et_al:LIPIcs.ESA.2026.34,
author = {Agarwal, Pankaj K. and Ezra, Esther and Sharir, Micha},
title = {{Triangle Nearest-Neighbor Searching in 3-Space}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {34:1--34:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.34},
URN = {urn:nbn:de:0030-drops-271709},
doi = {10.4230/LIPIcs.ESA.2026.34},
annote = {Keywords: line-point nearest neighbors, range searching, vertical decomposition in 3-space, test sets}
}
Document
Dynamic Detours
Abstract
Fix a parameter k ∈ ℕ. We give dynamic data structures that for a fully dynamic undirected graph G, updated over time by edge insertions and edge deletions, can answer the following queries:
- Long (u,v)-path: Given u,v ∈ V(G), is there a path from u to v of length at least k?
- Long (u,v)-detour: Given u,v ∈ V(G), is there a path from u to v of length at least dist_G(u,v)+k?
- Even/odd (u,v)-path: Given u,v ∈ V(G), is there a path from u to v of even/odd length? The amortized time of executing an update or answering a query is 2^𝒪(k³) log n + 𝒪(log² n log² log n) in the first two cases, and 𝒪(log² n log² log n) in the last, where n is the number of vertices of G.
The first result is in sharp contrast with known conditional lower bounds for reporting paths of length at most k. Specifically, there is no data structure supporting queries about (u,v)-paths of length at most two in time n^o(1) unless the Triangle Conjecture fails.
Our main technical contribution is a mechanism of "delayed edge insertion" that works locally on the level of biconnected components.
Cite as
Daniel Dadush, Michał Pilipczuk, Amadeus Reinald, Marek Sokołowski, and Michał Włodarczyk. Dynamic Detours. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dadush_et_al:LIPIcs.ESA.2026.35,
author = {Dadush, Daniel and Pilipczuk, Micha{\l} and Reinald, Amadeus and Soko{\l}owski, Marek and W{\l}odarczyk, Micha{\l}},
title = {{Dynamic Detours}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {35:1--35:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.35},
URN = {urn:nbn:de:0030-drops-271712},
doi = {10.4230/LIPIcs.ESA.2026.35},
annote = {Keywords: Dynamic algorithms, Fixed-parameter tractability}
}
Document
A Quadratic Lower Bound for Stable Roommates Solvability
Abstract
In their seminal work on the Stable Marriage Problem (SM), Gale and Shapley introduced a generalization of SM referred to as the Stable Roommates Problem (SR). An instance of SR consists of a set of 2n agents, and each agent has preferences in the form of a ranked list of all other agents. The goal is to find a one-to-one matching between the agents that is stable in the sense that no pair of agents have a mutual incentive to deviate from the matching. Unlike the (bipartite) stable marriage problem, in SR, stable matchings need not exist. Irving devised an algorithm that finds a stable matching or reports that none exists in O(n²) time. In their influential 1989 text, Gusfield and Irving posed the question of whether Ω(n²) time is required for SR solvability - the task of deciding if an SR instance admits a stable matching.
In this paper, we show that any (randomized) algorithm that decides SR solvability requires Ω(n²) adaptive Boolean queries to the agents' preferences (in expectation). Our argument follows from a reduction from the communication complexity of the set disjointness function. The query lower bound implies quadratic time lower bounds for Turing machines, and memory access lower bounds for random access machines. Thus, we establish that Irving’s algorithm is optimal (up to a logarithmic factor) in a very strong sense.
Cite as
Will Rosenbaum. A Quadratic Lower Bound for Stable Roommates Solvability. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rosenbaum:LIPIcs.ESA.2026.36,
author = {Rosenbaum, Will},
title = {{A Quadratic Lower Bound for Stable Roommates Solvability}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {36:1--36:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.36},
URN = {urn:nbn:de:0030-drops-271726},
doi = {10.4230/LIPIcs.ESA.2026.36},
annote = {Keywords: Stable Roommates, Stable Matching, Lower Bound, Communication Complexity}
}
Document
Quantum Time-Space Tradeoffs for Exponential Dynamic Programming
Abstract
We investigate the quantum algorithms for dynamic programming by Ambainis et al. (SODA'19). While giving provable complexity speedups and applicable to a variety of NP-hard problems, these algorithms have a notable drawback: they require a large amount of Quantum Random Access Memory (QRAM), which potentially could be very challenging to implement in a physical quantum computer. In this work, we study how the space complexity can be improved by trading it for time, while still retaining a speedup over the classical algorithms. We show novel quantum time-space tradeoffs by combining different classical approaches with quantum techniques. For instance, we show that the Travelling Salesman Problem can be solved quantumly in Õ(1.859ⁿ) time and Õ(1.315ⁿ) QRAM space.
Cite as
Susanna Caroppo, Jevgēnijs Vihrovs, Dārta Zajakina, and Aleksejs Zajakins. Quantum Time-Space Tradeoffs for Exponential Dynamic Programming. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{caroppo_et_al:LIPIcs.ESA.2026.37,
author = {Caroppo, Susanna and Vihrovs, Jevg\={e}nijs and Zajakina, D\={a}rta and Zajakins, Aleksejs},
title = {{Quantum Time-Space Tradeoffs for Exponential Dynamic Programming}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {37:1--37:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.37},
URN = {urn:nbn:de:0030-drops-271733},
doi = {10.4230/LIPIcs.ESA.2026.37},
annote = {Keywords: Quantum Algorithms, Time-Space Tradeoffs, NP-Hard Problems, Dynamic Programming, Divide \& Conquer}
}
Document
Shifting Is Optimal Under Gap-ETH: A Lower Bound Framework for Geometric Approximation Schemes
Abstract
The shifting technique of Hochbaum and Maass [J.ACM'85] produces PTASes with the fastest known running times n^O(1/ε^{d-1}) for several d dimensional geometric problems. However, it is only known, due to Marx [FOCS'07], that these algorithms are indeed optimal for dimension d = 2. We show that these running times are optimal under Gap-ETH for every constant dimension. More precisely, we develop a framework that enables us to prove the conditional optimality of the shifting algorithms for several problems on unit ball graphs, such as maximum independent set, maximum induced forest, and others, as well as for the problem of piercing unit balls. Our framework is built using the cube wiring theorem of De Berg et al. [SICOMP'20] and the reduction steps of Marx and Sidiropoulos [SoCG'14] to create a convenient maximization version of geometric CSP that can be used as a basis for reductions.
Cite as
Manuel Cáceres, Sándor Kisfaludi-Bak, and Saeed Odak. Shifting Is Optimal Under Gap-ETH: A Lower Bound Framework for Geometric Approximation Schemes. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{caceres_et_al:LIPIcs.ESA.2026.38,
author = {C\'{a}ceres, Manuel and Kisfaludi-Bak, S\'{a}ndor and Odak, Saeed},
title = {{Shifting Is Optimal Under Gap-ETH: A Lower Bound Framework for Geometric Approximation Schemes}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {38:1--38:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.38},
URN = {urn:nbn:de:0030-drops-271749},
doi = {10.4230/LIPIcs.ESA.2026.38},
annote = {Keywords: Gap-ETH Lower Bounds, Geometric PTASes, Geometric CSP, Shifting Technique, Maximum Induced Forest, Constant Dimension}
}
Document
Deterministic Online Embedding of Metric Spaces into Low Dimensional Spaces
Abstract
We study online embeddings of metric spaces into Euclidean spaces of a constant dimension d > 1, against an adaptive adversary. While the case of d = 1 is well understood, for higher dimensions little is known. In particular, even for d = 2 it remains unknown whether the worst-case distortion grows exponentially with the number of exposed points, as it does in the case for the line, or whether it is polynomial, as in the case for unbounded d.
Our first result is about fixed solid graphs, i.e., K₅, whose edges are solid intervals, equipped with the shortest-path metric. We show that if the input points arrive from such a metric space, they can indeed be online-embedded into ℝ² with a polynomial distortion. This refutes the previously believed conjecture that the topological non-embeddability of K₅ into the plane could be exploited for establishing exponential lower bounds.
The second results is about online embeddings of tree metrics of a certain type, including, e.g., ultrametrics and HST’s. Somewhat surprisingly, we show that for metrics from this class the worst-case online embedding into ℝ^d is not much worse that the offline embedding, both being n^Θ(1/d), and this holds even when d = Θ(log n). This is in a stark contrast to the more common situation where the online-offline gap is typically huge, and even exponential. This result allows us to transfer results about probabilistic embeddings of metrics into HST’s to low-dimensional Euclidean spaces, in an almost optimal possible manner.
Cite as
Noam Licht, Ilan Newman, and Yuri Rabinovich. Deterministic Online Embedding of Metric Spaces into Low Dimensional Spaces. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 39:1-39:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{licht_et_al:LIPIcs.ESA.2026.39,
author = {Licht, Noam and Newman, Ilan and Rabinovich, Yuri},
title = {{Deterministic Online Embedding of Metric Spaces into Low Dimensional Spaces}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {39:1--39:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.39},
URN = {urn:nbn:de:0030-drops-271750},
doi = {10.4230/LIPIcs.ESA.2026.39},
annote = {Keywords: online embedding, metric embedding, online algorithms, design of algorithms}
}
Document
Adaptive Sparsification for Linear Programming
Abstract
We provide a generic toolkit for sparsifying the constraint set of linear programs (LPs). To this end, we reduce solving a linear program with n constraints and d variables (n≫ d), to solving a sequence of LPs defined over only a small subset of the constraints, obtained by adaptively sub-sampling the original set. We provide results for both the low and high precision regimes. To achieve the former result, we streamline and generalize the techniques from [Assadi '25] for approximately computing maximum matchings in the semi-streaming setting to the case of general LPs. For the latter, we robustify the methods of [Clarkson '95], which were originally designed for exact LP solvers. As a consequence we obtain fast approximate LP solvers which reduce the dependence on width and error from quadratic to linear, compared to vanilla multiplicative-weights based approaches.
Additionally, we leverage our findings to obtain fast LP solvers in the quantum query access model, where the running time scales with √n. This completely decouples the component responsible for quantum speed-ups, solely represented by a generalization of Grover’s search, from its classical algorithmic counterpart.
Cite as
Etienne Objois and Adrian Vladu. Adaptive Sparsification for Linear Programming. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{objois_et_al:LIPIcs.ESA.2026.40,
author = {Objois, Etienne and Vladu, Adrian},
title = {{Adaptive Sparsification for Linear Programming}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {40:1--40:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.40},
URN = {urn:nbn:de:0030-drops-271768},
doi = {10.4230/LIPIcs.ESA.2026.40},
annote = {Keywords: linear programming, sparsification, sampling, quantum algorithms}
}
Document
The Prophet and the Voronoi Diagram
Abstract
Consider a stream of n random points (say, from the unit square) arriving one by one, where a player must make an immediate and irreversible decision upon each point’s arrival, whether to pick it. The player must pick exactly one such point, and the payoff is the area of the cell of the picked point, in the final Voronoi diagram of all the points. We show that there is a simple strategy so that with probability ≥ 1 - Õ(1/√n), the player’s payoff is only a constant factor smaller than the optimal choice (i.e., the one made by the prophet). This competitiveness is somewhat surprising, as both the optimal payoff and this strategy’s payoff are larger by a factor of Θ(log n) than the average payoff.
Cite as
Sariel Har-Peled. The Prophet and the Voronoi Diagram. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 41:1-41:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{harpeled:LIPIcs.ESA.2026.41,
author = {Har-Peled, Sariel},
title = {{The Prophet and the Voronoi Diagram}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {41:1--41:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.41},
URN = {urn:nbn:de:0030-drops-271779},
doi = {10.4230/LIPIcs.ESA.2026.41},
annote = {Keywords: Voronoi diagram, prophet inequality, secretary problem}
}
Document
Nearly Optimal Bounds for Computing Decision Tree Splits in Data Streams
Abstract
We establish nearly optimal upper and lower bounds for approximating decision tree splits in data streams. For regression with labels in the range {0,1,…,M}, we give a one-pass algorithm using 𝒪̃(M²/ε) space that outputs a split within additive ε error of the optimal split, improving upon the two-pass algorithm of Pham et al. (ISIT 2025). Furthermore, we provide a matching one-pass lower bound showing that Ω(M²/ε) space is indeed necessary.
For classification, we also obtain a one-pass algorithm using 𝒪̃(1/ε) space for approximating the optimal Gini split, improving upon the previous 𝒪̃(1/ε²)-space algorithm. We complement these results with matching space lower bounds: Ω(1/ε) for Gini impurity and Ω(1/ε) for misclassification (which matches the upper bound obtained by sampling).
Our algorithms exploit the Lipschitz property of the loss functions and use reservoir sampling along with Count-Min sketches with range queries. Our lower bounds follow from careful reductions from the Index problem.
Cite as
Hoang Ta and Hoa T. Vu. Nearly Optimal Bounds for Computing Decision Tree Splits in Data Streams. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ta_et_al:LIPIcs.ESA.2026.42,
author = {Ta, Hoang and Vu, Hoa T.},
title = {{Nearly Optimal Bounds for Computing Decision Tree Splits in Data Streams}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {42:1--42:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.42},
URN = {urn:nbn:de:0030-drops-271785},
doi = {10.4230/LIPIcs.ESA.2026.42},
annote = {Keywords: Decision trees, Streaming algorithms, Lower bounds}
}
Document
Algorithm Exercises Skyline and Young Tableau: Divide-and-Conquer Revisited
Abstract
We revisit the two classic algorithm exercises skyline and Young tableau, often given to students in an introduction to algorithms course. For both problems we present alternative, still very simple, divide-and-conquer solutions achieving running-times better than what is traditionally asked to achieve in these exercises, in the sense that the running times we achieve are output and input sensitively, respectively. Computing the skyline of a list of n buildings is a classic algorithm problem, solvable with various sweep line and divide-and-conquer approaches in O(nlg n) time. The classic divide-and-conquer solution resembles mergesort, merging skylines of subsets of the buildings. In this paper we describe an alternative simple divide-and-conquer approach (using Kirkpatrick and Seidel’s marriage-before-conquest technique), achieving an optimal output sensitive running time of O(n lg k), where k is the number of buildings contributing to the skyline. For the Young tableau problem, we consider searching rectangular m × n matrices which are both row and column monotone, and where the original problem asks to find an algorithm with running time O(m+n). We present a simple divide-and-conquer algorithm for searching matrices in worst-case optimal running time O(m (1 + lg n/m)), where m ≤ n. We also present an input sensitive search algorithm with optimal running time O(k (1 + lg n/k)), where k is the complexity of the boundary in the matrix between values smaller and larger than the query value (k is the number of vertical line segments on the boundary, where k ≤ m). Here optimality refers to the best possible running time when expressing the running time in terms of m and n, or m, n and k, respectively.
Cite as
Gerth Stølting Brodal. Algorithm Exercises Skyline and Young Tableau: Divide-and-Conquer Revisited. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 43:1-43:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brodal:LIPIcs.ESA.2026.43,
author = {Brodal, Gerth St{\o}lting},
title = {{Algorithm Exercises Skyline and Young Tableau: Divide-and-Conquer Revisited}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {43:1--43:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.43},
URN = {urn:nbn:de:0030-drops-271790},
doi = {10.4230/LIPIcs.ESA.2026.43},
annote = {Keywords: Divide and conquer, marriage before conquest, output-sensitive running time}
}
Document
Smallest Convex Hulls of Polygons
Abstract
We study the problem of minimizing the area of the convex hull of k polygons with a total of n vertices in the plane, under translations and rigid motions for any fixed k ≥ 3. For any ε ∈ (0, 1), we give (1 + ε)-approximation algorithms running in O(ε^{-1/2} log n + ε^{1/2 - k}) time for translations, and in O(ε^{-1/2} log n + ε^{3/2 - 2k}) time for rigid motions. We also consider minimizing the perimeter of the convex hull under translations and obtain a (1 + ε)-approximation algorithm running in O(ε^{-1/2} log n + ε^{1/2 - k}log^{k-1}(1/ε)) time. To the best of our knowledge, these are the first results of this kind for k ≥ 3 polygons. Furthermore, for the special case of two polygons with n₀ and n₁ vertices (n₀ ≥ n₁), respectively, we give an O(n₀+n₁log²(n₀+n₁))-time algorithm for the minimum-perimeter problem. This significantly improves upon the best-known O((n₀+n₁)log²(n₀+n₁)) bound by eliminating the logarithmic overhead associated with the larger input size n₀.
Cite as
Mook Kwon Jung and Hee-Kap Ahn. Smallest Convex Hulls of Polygons. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jung_et_al:LIPIcs.ESA.2026.44,
author = {Jung, Mook Kwon and Ahn, Hee-Kap},
title = {{Smallest Convex Hulls of Polygons}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {44:1--44:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.44},
URN = {urn:nbn:de:0030-drops-271803},
doi = {10.4230/LIPIcs.ESA.2026.44},
annote = {Keywords: Convex hull, packing, bundling, approximation algorithms, cuttings}
}
Document
Near-Optimal Working-Set Heaps and Dijkstra on Pointer Machines
Abstract
A heap is a dynamic data structure that stores a set of labeled values under the following operations: pop returns the minimum value of the heap, Push(x_i) pushes a new value x_i onto the heap, and DecreaseKey(i, v) decreases the value x_i to v. A working-set heap is a heap that supports the x_i ← pop() operation in O(log Γ(x_i)) time where Γ(x_i) is the size of the working set: the number of elements that were pushed onto the heap while x_i was in the heap. The goal of working set heap design is to maintain the working set property while minimizing the overhead of the Push and DecreaseKey operations. On a word RAM, there exist working set heaps that support Push and DecreaseKey in amortized constant time. In this paper, we show via a simple construction that pointer machines, one of the most general and least-assuming computational models, support working set heaps that support Push in amortized constant time and DecreaseKey in inverse-Ackermann time. A by-product of this analysis is that Dijkstra’s shortest path algorithm can be near-universally optimal on a pointer machine - incurring only an additive O(m α(m)) overhead compared to the optimal running time for distance ordering, where m denotes the number of edges in the graph.
Cite as
Ivor van der Hoog, John Iacono, Eva Rotenberg, and Daniel Rutschmann. Near-Optimal Working-Set Heaps and Dijkstra on Pointer Machines. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vanderhoog_et_al:LIPIcs.ESA.2026.45,
author = {van der Hoog, Ivor and Iacono, John and Rotenberg, Eva and Rutschmann, Daniel},
title = {{Near-Optimal Working-Set Heaps and Dijkstra on Pointer Machines}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {45:1--45:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.45},
URN = {urn:nbn:de:0030-drops-271814},
doi = {10.4230/LIPIcs.ESA.2026.45},
annote = {Keywords: Data structures, graph algorithms, amortized analysis}
}
Document
Greedy Completion for Weighted (α,β)-Spanners
Abstract
We study (α,β)-spanners for weighted graphs. We propose a simple greedy completion procedure which starts from a sparse initial graph, and repeatedly fixes pairs of vertices with a bad stretch, generalizing Knudsen’s additive completion [SWAT 2014]. As an application, we construct (k,k-1)-spanners for weighted graphs of size Õ(n^{1+1/k}), which were previously unknown.
Cite as
Elad Tzalik. Greedy Completion for Weighted (α,β)-Spanners. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{tzalik:LIPIcs.ESA.2026.46,
author = {Tzalik, Elad},
title = {{Greedy Completion for Weighted (\alpha,\beta)-Spanners}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {46:1--46:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.46},
URN = {urn:nbn:de:0030-drops-271821},
doi = {10.4230/LIPIcs.ESA.2026.46},
annote = {Keywords: Graph spanners, Greedy algorithms, Weighted graphs}
}
Document
Dichotomies for #CSP on Graphs That Forbid a Clique as a Minor
Abstract
We establish complexity dichotomies for #CSP and #R_D-CSP problems (not necessarily symmetric) with Boolean domain and complex range over several key minor-closed graph classes. These dichotomies give a complete characterization of the complexity of #CSP and #R_D-CSP over graph classes that forbid a complete graph as a minor. Notably, our algorithms apply not only to #CSP and #R_D-CSP problems but also directly to the Holant setting.
Cite as
Boning Meng and Yicheng Pan. Dichotomies for #CSP on Graphs That Forbid a Clique as a Minor. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{meng_et_al:LIPIcs.ESA.2026.47,
author = {Meng, Boning and Pan, Yicheng},
title = {{Dichotomies for #CSP on Graphs That Forbid a Clique as a Minor}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {47:1--47:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.47},
URN = {urn:nbn:de:0030-drops-271834},
doi = {10.4230/LIPIcs.ESA.2026.47},
annote = {Keywords: Graph Minor, Tree Decomposition, Computational Complexity, Counting, Constraint Satisfaction Problem, Holant, Matchgate}
}
Document
Matching in Geometric Uniform Hypergraphs
Abstract
Let P be a set of n points in ℝ^d, d ≥ 2, and let t ≥ 2 be an integer. Let H_t(P) denote the t-uniform hypergraph on P, whose hyperedges consist of all t-tuples T ⊂ P for which ‖p-q‖ ≤ 1, for any two points p,q ∈ T. A matching in H_t(P) is a collection of vertex-disjoint hyperedges. We present a PTAS for finding a maximum matching in H_t(P). In particular, we present the first PTAS for the well-studied problem known as maximum (vertex-disjoint) triangle packing in unit disk graphs. Our approach consists of a sparsification stage, which replaces P by a subset Q with favorable properties, followed by an implementation of a PTAS for a maximum matching in H_t(Q). The two stages follow the high-level machinery in [Édouard Bonnet et al., 2023] and [Rom Aschner et al., 2013], respectively, but are considerably more involved.
Cite as
Matthew J. Katz, Yuval Nidam, Rachel Saban, and Micha Sharir. Matching in Geometric Uniform Hypergraphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{katz_et_al:LIPIcs.ESA.2026.48,
author = {Katz, Matthew J. and Nidam, Yuval and Saban, Rachel and Sharir, Micha},
title = {{Matching in Geometric Uniform Hypergraphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {48:1--48:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.48},
URN = {urn:nbn:de:0030-drops-271848},
doi = {10.4230/LIPIcs.ESA.2026.48},
annote = {Keywords: Geometric hypergraphs, maximum matching, PTAS, sparsification, local search}
}
Document
Computational Boundaries for Escaping Rectangles
Abstract
Ma and Wong [IEEE TCAD '12] introduced and studied the Rectangle Escape problem, motivated by bus escape routing in printed circuit board design. In this problem, we are given an axis-parallel rectangle R, a set 𝒮 of axis-parallel rectangles fully contained in R, and an integer d. The goal is to determine whether each rectangle in 𝒮 can be extended in one of the four axis-parallel directions (up, down, left, or right) to the boundary of R such that no point is covered by more than d extended rectangles. We revisit Rectangle Escape and resolve several open complexity questions.
Ahmadinejad et al. [TCS '17] studied Rectangle Escape and its variants where rectangles are only allowed to be extended in a subset of directions - most notably, in two directions, a variant they termed Bidirectional REP. They showed that the problem is NP-complete when extensions are limited to two adjacent directions and d = 3, but left open the complexity of the case when d = 2. Additionally, the case for two opposite directions remained unresolved for any d ≥ 2. We resolve the first question by showing that Bidirectional REP is NP-complete even when extensions are restricted to two adjacent directions and d = 2. We also settle the complexity of Rectangle Escape with two opposite directions by proving that the problem is NP-complete when d is part of the input but solvable in 𝒪(n log n) time for any constant d. Finally, we consider the special case where all extended rectangles must be disjoint, that is, d = 1. We show an unconditional lower bound of Ω(n log n) with a matching upper bound of 𝒪(n log n) for all variants. This improves upon a sequence of algorithms for the setting with all four directions allowed and d = 1, starting with an 𝒪(n⁶)-time algorithm, later improved to 𝒪(n⁴), and then to O(n³).
Cite as
Akanksha Agrawal, Akanksha Agrawal, Pradeesha Ashok, Pradeesha Ashok, Matthias Bentert, Matthias Bentert, Satyabrata Jana, Satyabrata Jana, Abhishek Sahu, Abhishek Sahu, Saket Saurabh, Saket Saurabh, Kushal Singanporia, and Kushal Singanporia. Computational Boundaries for Escaping Rectangles. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{agrawal_et_al:LIPIcs.ESA.2026.49,
author = {Agrawal, Akanksha and Ashok, Pradeesha and Bentert, Matthias and Jana, Satyabrata and Sahu, Abhishek and Saurabh, Saket and Singanporia, Kushal},
title = {{Computational Boundaries for Escaping Rectangles}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {49:1--49:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.49},
URN = {urn:nbn:de:0030-drops-271858},
doi = {10.4230/LIPIcs.ESA.2026.49},
annote = {Keywords: NP-hardness, Sweep-line algorithm, Fixed-parameter tractability, Tight lower and upper bounds}
}
Document
Moderately Beyond Clique-Width: Reduced Component Max-Leaf and Related Parameters
Abstract
Reduced parameters [BKW, JCTB '26; BKRT, SODA '22] are defined via contraction sequences. Based on this framework, we introduce the reduced component max-leaf, denoted by cml^↓, where component max-leaf is the maximum number of leaves in any spanning tree of any connected component. Reduced component max-leaf is strictly sandwiched between clique-width and reduced bandwidth, it is bounded in unit interval graphs, and unbounded in planar graphs. We design polynomial-time algorithms for problems such as Maximum Independent Set, Maximum Clique, Maximum Induced d-Regular Subgraph, and Induced Disjoint Paths in graphs given with a contraction sequence witnessing low cml^↓, unifying and extending tractability results for classes of bounded clique-width and unit interval graphs.
We get the following collapses in sparse classes of bounded cml^↓: bounded maximum degree implies bounded treewidth, whereas K_{t,t}-subgraph-freeness implies strongly sublinear treewidth; we show the latter, more generally, for classes of bounded reduced cutwidth. We establish the former result by showing that graphs with bounded cml^↓ admit balanced separators dominated by a bounded number of vertices. In contrast, there are graphs G of arbitrarily large girth and treewidth Θ(|V(G)|^{1/2}) such that cml^↓(G) ⩽ 3.
We then showcase an application of the reduced parameters to establishing non-transducibility results. We prove that for most reduced parameters p^↓ (including reduced bandwidth), the family of classes of bounded p^↓ is closed under first-order transductions. We then answer a question of [BKW '26] by showing that the 3-dimensional grids have unbounded reduced bandwidth. As the class of planar graphs (or any class of bounded genus) has bounded reduced bandwidth [BKW '26], this reproves a recent result [GPP, LICS '25; HJ, LICS '25] that planar graphs do not first-order transduce the 3-dimensional grids.
Cite as
Édouard Bonnet, Yeonsu Chang, Julien Duron, Colin Geniet, and O-joung Kwon. Moderately Beyond Clique-Width: Reduced Component Max-Leaf and Related Parameters. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bonnet_et_al:LIPIcs.ESA.2026.50,
author = {Bonnet, \'{E}douard and Chang, Yeonsu and Duron, Julien and Geniet, Colin and Kwon, O-joung},
title = {{Moderately Beyond Clique-Width: Reduced Component Max-Leaf and Related Parameters}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {50:1--50:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.50},
URN = {urn:nbn:de:0030-drops-271864},
doi = {10.4230/LIPIcs.ESA.2026.50},
annote = {Keywords: Structural graph theory, reduced parameter, component max-leaf}
}
Document
Near-Optimal and Efficient Encoding for Two-Dimensional Range Minimum Queries
Abstract
We consider the 2D RMQ encoding problem: given an m× n array of mn elements over a total order, encode it such that, for any query rectangle, the position of its maximum element can be reported without accessing the original array. For m ≤ n, it is known how to encode the array in 𝒪(mn min{m, log n}) bits with 𝒪(1)-time queries [Brodal et al., Algorithmica 2012], and also how to obtain an asymptotically optimal encoding consisting of 𝒪(mn log m) bits [Brodal et al., ESA 2013]. However, the latter approach does not prove any guarantee on the query time, and it appears to be inherently sequential: it requires scanning the whole encoding to answer a query. We design a different encoding that uses near-optimal space while allowing for efficient queries. More concretely, for every parameter κ ∈ [1, log log n], our encoding uses 𝒪(κ mn(log m + log log n)) bits and answers 2D RMQ queries in 𝒪(log^{1/κ} n) time.
Cite as
Paweł Gawrychowski, Adam Górkiewicz, and Srinivasa Rao Satti. Near-Optimal and Efficient Encoding for Two-Dimensional Range Minimum Queries. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 51:1-51:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gawrychowski_et_al:LIPIcs.ESA.2026.51,
author = {Gawrychowski, Pawe{\l} and G\'{o}rkiewicz, Adam and Satti, Srinivasa Rao},
title = {{Near-Optimal and Efficient Encoding for Two-Dimensional Range Minimum Queries}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {51:1--51:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.51},
URN = {urn:nbn:de:0030-drops-271874},
doi = {10.4230/LIPIcs.ESA.2026.51},
annote = {Keywords: Encoding, Range Minimum Queries, Compact structure}
}
Document
A Dynamic (1+ε)-Spanner for Disk Intersection Graphs
Abstract
We maintain a (1+ε)-spanner over the disk intersection graph of a dynamic set of disks. We restrict all disks to have their diameter in [4,Ψ] for some fixed and known Ψ. The resulting (1+ε)-spanner has size O(n ε^{-2} log Ψ log(ε^{-1})), where n is the present number of disks.
We develop a novel use of persistent data structures to dynamically maintain our (1+ε)-spanner. Our approach requires O(ε^{-2} n log⁴n log Ψ) space and has an O((Ψ/ε)² log⁴n log²Ψ log²(ε^{-1})) expected amortised update time. For constant ε and Ψ, this spanner has near-linear size, uses near-linear space and has polylogarithmic update time. Furthermore, we observe that for any ε < 1, our spanner also serves as a connectivity data structure. With a slight adaptation of our techniques, this leads to better bounds for dynamically supporting connectivity queries in a disk intersection graph. In particular, we improve the space usage when compared to the dynamic data structure of (Baumann et al., DCG'24), replacing the linear dependency on Ψ by a polylogarithmic dependency. Finally, we generalise our results to d-dimensional hypercubes.
Cite as
Sarita de Berg, Ivor van der Hoog, Eva Rotenberg, Johanne Müller Vistisen, and Sampson Wong. A Dynamic (1+ε)-Spanner for Disk Intersection Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 52:1-52:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{deberg_et_al:LIPIcs.ESA.2026.52,
author = {de Berg, Sarita and van der Hoog, Ivor and Rotenberg, Eva and Vistisen, Johanne M\"{u}ller and Wong, Sampson},
title = {{A Dynamic (1+\epsilon)-Spanner for Disk Intersection Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {52:1--52:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.52},
URN = {urn:nbn:de:0030-drops-271880},
doi = {10.4230/LIPIcs.ESA.2026.52},
annote = {Keywords: intersection graphs, dynamic data structures, spanners}
}
Document
Faster Exponential Algorithms for Multi-Machine Scheduling Problems
Abstract
Minimizing the weighted completion times (P ‖ Σ w_j C_j) and weighted number of tardy jobs (P ‖ Σ w_j U_j) on multiple identical machines are two classical NP-hard scheduling problems. As shown by Lenté et al. (2014), both problems can be solved in time 𝒪^⋆(3ⁿ). In this paper, we improve these bounds to 𝒪(2.755ⁿ) and 𝒪^⋆(2ⁿ), respectively. Our algorithm for P ‖ Σ w_j C_j exploits the meet-in-the-middle paradigm and an efficient data structure answering linear programming queries. Additionally, when the number of machines is at most 6, we show that the running time for P ‖ Σ w_j C_j can further be improved.
Both scheduling problems are generalizations of the classical Bin Packing problem, which can be solved in 𝒪^⋆(2ⁿ) time. Improving this running time is an important open question. We show that, when assuming the Asymptotic Rank Conjecture (ARC), Bin Packing can be solved in time 𝒪((2-ε)ⁿ) for some ε > 0. Our algorithm makes use of two main ingredients: the recent 𝒪((2-ε)ⁿ)-time algorithm of Nederlof et al. [SICOMP'23] for Bin Packing when the number of bins is a fixed constant, and the 𝒪((2-ε)ⁿ)-time algorithm of Björklund et al. [SODA'25] for special instances of the 3-way Partitioning problem when assuming ARC.
Cite as
Anubhav Dhar, Anita Dürr, Ahmed Ghazy, Jakob Greilhuber, and Karol Węgrzycki. Faster Exponential Algorithms for Multi-Machine Scheduling Problems. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 53:1-53:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dhar_et_al:LIPIcs.ESA.2026.53,
author = {Dhar, Anubhav and D\"{u}rr, Anita and Ghazy, Ahmed and Greilhuber, Jakob and W\k{e}grzycki, Karol},
title = {{Faster Exponential Algorithms for Multi-Machine Scheduling Problems}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {53:1--53:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.53},
URN = {urn:nbn:de:0030-drops-271894},
doi = {10.4230/LIPIcs.ESA.2026.53},
annote = {Keywords: Scheduling, exact algorithms, exponential-time algorithms}
}
Document
Online Demand Strip Packing
Abstract
In the Demand Strip Packing problem (DSP), we are given a finite set of tasks, each characterized by a specific duration and energy demand. These tasks need to be scheduled non-preemptively within a given time frame while minimizing the peak demand: the maximum amount of energy consumed by the tasks being executed at any point in time. We are the first to consider the online variant of the problem, where tasks are revealed to an algorithm one by one in a list. Upon arrival, each task must be assigned an irrevocable starting time before the next task in the list is revealed. As usual in online optimization, we evaluate the performance of online algorithms using competitive analysis. We give a strictly 4.263-competitive algorithm for Online DSP, which is stronger than the respective bound of 6.479 for the related problem Online Strip Packing. Additionally, we prove a lower bound of 1.812 on the competitive ratio of any online algorithm for DSP and, thus, clearly separate Online DSP from Online Minimum Peak Appointment Scheduling (MPAS), a special case of Online DSP, for which a strictly 5/3-competitive algorithm is known.
Cite as
Sebastian Bruchhold, Franziska Eberle, Georgios Moneftsis, Malin Rau, and Albert Vesterlund. Online Demand Strip Packing. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 54:1-54:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bruchhold_et_al:LIPIcs.ESA.2026.54,
author = {Bruchhold, Sebastian and Eberle, Franziska and Moneftsis, Georgios and Rau, Malin and Vesterlund, Albert},
title = {{Online Demand Strip Packing}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {54:1--54:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.54},
URN = {urn:nbn:de:0030-drops-271909},
doi = {10.4230/LIPIcs.ESA.2026.54},
annote = {Keywords: Online Demand Strip Packing, competitive analysis, scheduling, packing}
}
Document
Multi Choice Min Prophet
Abstract
The prophet inequality is a fundamental problem in optimal stopping theory. Given n independent variables drawn from known distributions, a player observes values sequentially and must decide irrevocably whether to stop and accept the current value or continue. The goal is to select a single element while maximizing the ratio between the value chosen and that of the maximum value in the sequence. In this paper, we study the minimization counterpart, often termed the min prophet or cost prophet inequality. Unlike the maximization setting, where simple threshold algorithms achieve half of the prophet’s value, the minimization setting is significantly harder, with an exponential lower bound even for i.i.d. variables.
We study a multi-choice relaxation in which the algorithm may select multiple variables and gets to choose the best amongst them (the minimum amongst those selected). Our goal is to minimize the expected number of selections while achieving a constant competitive ratio. For adversarial order, we show that a constant competitive ratio requires a nearly linear number of choices in expectation, ergo, Ω(n/ln n). In contrast, we show that for the prophet secretary model (random order) one can attain constant competitiveness while requiring only an exponentially smaller expected number of choices i.e. O(ln n). We give a refined analysis and define M to be the ratio of the minimum expected value of any single variable to the expected minimum value of all variables (the prophet’s value) and present an algorithm that achieves a constant competitive ratio with O(min{ln ln M, ln n}) choices in expectation for the prophet secretary. We show that this is tight up to low order log factors even for the special case of the i.i.d. model. Specifically, the lower bound on the expected number of choices for any constant competitive algorithm is Ω(min{ln ln M/ln ln ln M, ln n/ln ln n}). We also show that if we insist on a deterministic bound on the number of choices then every constant competitive algorithm requires n choices. This holds even in the i.i.d. setting and shows that to achieve a constant competitive algorithm there is an exponential gap between the lower bound on the deterministic number of choices and the upper bound on the expected number of choices.
Finally, we consider a variant where both the algorithm and the adversary choose r values and pay their sum, this is the minimization multi unit version. We extend our techniques to the multi-unit variant for i.i.d. variables, achieving a constant competitive ratio with a small expected number of choices.
Cite as
Yossi Azar, Itamar Biran, and Amos Fiat. Multi Choice Min Prophet. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 55:1-55:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{azar_et_al:LIPIcs.ESA.2026.55,
author = {Azar, Yossi and Biran, Itamar and Fiat, Amos},
title = {{Multi Choice Min Prophet}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {55:1--55:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.55},
URN = {urn:nbn:de:0030-drops-271917},
doi = {10.4230/LIPIcs.ESA.2026.55},
annote = {Keywords: Prophet inequalities, Min Cost, Multiple choice, Multi Unit}
}
Document
Linear-Time Vertex-Connectivity for Graphs of Bounded Genus
Abstract
We provide a new linear-time algorithm for determining the vertex-connectivity of graphs with bounded genus. This generalizes and streamlines a linear-time algorithm for graphs with bounded crossing number which was recently obtained by Biedl, Bose and Murali [ESA 2024]. Compared to applying the even more recent fixed parameter linear-time algorithm for deciding bounded vertex-connectivity announced by Korhonen [STOC 2025] to graphs of bounded genus,our algorithm is far simpler, its correctness easier to establish, and it makes use of geometric ideas, as is natural for surface-embedded graphs.
Cite as
Sergio Cabello, Alexander Dobler, Gašper Fijavž, Thekla Hamm, and Mirko H. Wagner. Linear-Time Vertex-Connectivity for Graphs of Bounded Genus. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cabello_et_al:LIPIcs.ESA.2026.56,
author = {Cabello, Sergio and Dobler, Alexander and Fijav\v{z}, Ga\v{s}per and Hamm, Thekla and Wagner, Mirko H.},
title = {{Linear-Time Vertex-Connectivity for Graphs of Bounded Genus}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {56:1--56:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.56},
URN = {urn:nbn:de:0030-drops-271923},
doi = {10.4230/LIPIcs.ESA.2026.56},
annote = {Keywords: vertex-connectivity, graphs on surfaces, genus of a graph}
}
Document
Dynamic Matroids: Base Packing and Covering
Abstract
In this paper, we consider dynamic matroids, where elements can be inserted to or deleted from the ground set over time. The independent sets change to reflect the current ground set. As matroids are central to the study of many combinatorial optimization problems, it is a natural next step to also consider them in a dynamic setting. The study of dynamic matroids has the potential to generalize several dynamic graph problems, including, but not limited to, arboricity and maximum bipartite matching. We contribute by providing efficient algorithms for some fundamental matroid questions.
In particular, we study the most basic question of maintaining a base dynamically, providing an essential building block for future algorithms. We further utilize this result and consider the elementary problems of base packing and base covering. We provide a deterministic algorithm that maintains a (1± ε)-approximation of the base packing number Φ in O(Φ ⋅ poly(log n, ε^{-1})) queries per update. Similarly, we provide a deterministic algorithm that maintains a (1± ε)-approximation of the base covering number β in O(β ⋅ poly(log n, ε^{-1})) queries per update. Moreover, we give an algorithm that maintains a (1± ε)-approximation of the base covering number β in O(poly(log n, ε^{-1})) queries per update against an oblivious adversary.
These results are obtained by exploring the relationship between base collections, a generalization of tree-packings, and base packing and covering respectively. We provide structural theorems to formalize these connections, and show how they lead to simple dynamic algorithms.
Cite as
Tijn de Vos and Mara Grilnberger. Dynamic Matroids: Base Packing and Covering. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 57:1-57:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{devos_et_al:LIPIcs.ESA.2026.57,
author = {de Vos, Tijn and Grilnberger, Mara},
title = {{Dynamic Matroids: Base Packing and Covering}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {57:1--57:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.57},
URN = {urn:nbn:de:0030-drops-271931},
doi = {10.4230/LIPIcs.ESA.2026.57},
annote = {Keywords: matroid, minimum weight base, base packing, base covering, dynamic}
}
Document
Nearly Optimal Internal Dictionary Matching
Abstract
We study the internal dictionary matching (IDM) problem where a dictionary 𝒟 containing d substrings of a text T over a linearly sortable alphabet is given, and each query concerns the occurrences of patterns in 𝒟 in another substring of T. We propose a novel O(n)-sized data structure named Basic Substring Structure (BASS) where n is the length of the text T. With BASS, we are able to handle all types of queries in the IDM problem in nearly optimal query and preprocessing time. Specifically, our results include:
- The first algorithm that answers the CountDistinct query in Õ(1) time with Õ(n+d) preprocessing, where we need to compute the number of distinct patterns that exist in T[l,r]. Previously, the best result was Õ(m) time per query after Õ(n²/m+d) or Õ(nd/m+d) preprocessing, where m is a chosen parameter.
- Faster algorithms for two other types of internal queries. We improve the runtime for (1) Occurrence counting (Count) queries to O(log n/log log n) time per query with O(n+d√{log n}) preprocessing from O(log² n/log log n) time per query with O(nlog n/log log n + d log^{3/2} n) preprocessing. (2) Distinct pattern reporting (ReportDistinct) queries to O(1+|output|) time per query from O(log n+|output|) per query.
In addition, we match the optimal runtime in the remaining two types of queries, pattern existence (Exists), and occurrence reporting (Report). We also show that BASS is more generally applicable to other internal query problems.
Cite as
Jingbang Chen, Jiangqi Dai, Qiuyang Mang, Qingyu Shi, and Tingqiang Xu. Nearly Optimal Internal Dictionary Matching. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 58:1-58:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.ESA.2026.58,
author = {Chen, Jingbang and Dai, Jiangqi and Mang, Qiuyang and Shi, Qingyu and Xu, Tingqiang},
title = {{Nearly Optimal Internal Dictionary Matching}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {58:1--58:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.58},
URN = {urn:nbn:de:0030-drops-271944},
doi = {10.4230/LIPIcs.ESA.2026.58},
annote = {Keywords: Internal dictionary matching, pattern matching, string algorithms, data structures, substring queries}
}
Document
Revisiting Diameter in Directed Graphs
Abstract
The reachability diameter (ReachDiam) of a directed graph is the maximum distance over all pairs u,v where v is reachable from u. This notion is present in the definition of shortcut sets, and the name was recently coined in that context by Haeupler, Jiang, and Saranurak [SOSA 2026]. While this is a very natural notion of diameter in directed graphs, and especially DAGs, it is so far not computationally explored. Other definitions of diameter in directed graphs are either trivial (infinite) in graphs that are not strongly connected (e.g., the classical definition) or are non-trivial only in highly restrictive graph classes (e.g., Min-Diameter).
We initiate the problem of computing the (approximate) reachability diameter from a fine-grained complexity point of view. Under certain fine-grained assumptions, we prove that there is no algorithm in time 𝒪(n^{ω - ε}) that gives any approximation of ReachDiam in weighted graphs. Similarly, there is no algorithm with better than 2-approximation for unweighted graphs in this time. To supplement this, we provide algorithmic upper bounds that lead to additive approximation of ReachDiam for unweighted graphs. Hence, we establish a strong separation between the weighted and unweighted cases, which makes this type of diameter different in nature than other known notions.
Considering the hardness in general weighted graphs, we also study special graph classes and get small constant approximations for DAGs with bounded width or graphs with bounded treewidth. Interestingly, our techniques also lead to exact hopsets with hopbound 2 for bounded treewidth graphs. This and some of our upper bounds for general graphs show technical connections between approximating ReachDiam and computing shortcut sets and hopsets.
Cite as
Ben Bals, Joakim Blikstad, Daniel Dadush, Yasamin Nazari, and Jonas Schmidt. Revisiting Diameter in Directed Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 59:1-59:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bals_et_al:LIPIcs.ESA.2026.59,
author = {Bals, Ben and Blikstad, Joakim and Dadush, Daniel and Nazari, Yasamin and Schmidt, Jonas},
title = {{Revisiting Diameter in Directed Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {59:1--59:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.59},
URN = {urn:nbn:de:0030-drops-271959},
doi = {10.4230/LIPIcs.ESA.2026.59},
annote = {Keywords: Graph algorithms, Diameter, Fine-grained complexity, Shortcut sets}
}
Document
Optimal Enumeration of Eulerian Trails in Directed Graphs
Abstract
The BEST theorem, due to de Bruijn, van Aardenne-Ehrenfest, Smith, and Tutte, is a classical tool from graph theory that links the Eulerian trails in a directed graph G = (V,E) with the arborescences in G. In particular, one can use the BEST theorem to count the Eulerian trails in G in polynomial time. For enumerating the Eulerian trails in G, one could naturally resort to first enumerating the arborescences in G and then exploiting the insight of the BEST theorem to enumerate the Eulerian trails in G: every arborescence in G corresponds to at least one Eulerian trail in G. For over two decades, the fastest algorithm for enumerating arborescences in G took 𝒪(m log n + n + z_A log²n) time, where n = |V|, m = |E|, and z_A is the number of arborescences in G [Uno, ISAAC 1998]. Since Uno’s algorithm does not lead to an optimal enumeration of Eulerian trails in directed graphs, we were motivated to develop a direct algorithm for this problem.
Our central contribution is a remarkably simple algorithm to directly enumerate the z_T Eulerian trails in G in the optimal 𝒪(m + z_T) time. As a consequence, our result improves on an implementation of the BEST theorem for counting Eulerian trails in G when z_T = o(n²), and also unconditionally improves the combinatorial 𝒪(m⋅z_T)-time algorithm of Conte et al. [TKDD 2026] for the same task. Moreover, we show that, with some care, our algorithm can be extended to enumerate Eulerian trails in directed multigraphs in optimal time, enabling applications in bioinformatics and data privacy.
Cite as
Ben Bals, Solon P. Pissis, and Matei Tinca. Optimal Enumeration of Eulerian Trails in Directed Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 60:1-60:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bals_et_al:LIPIcs.ESA.2026.60,
author = {Bals, Ben and Pissis, Solon P. and Tinca, Matei},
title = {{Optimal Enumeration of Eulerian Trails in Directed Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {60:1--60:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.60},
URN = {urn:nbn:de:0030-drops-271960},
doi = {10.4230/LIPIcs.ESA.2026.60},
annote = {Keywords: graph algorithms, enumeration algorithms, Eulerian trails, Eulerian paths}
}
Document
Real-Weighted Diameter and Eccentricities of Minor-Free and Bounded VC-Dimension Graphs in Truly Subquadratic Time
Abstract
We present the first truly subquadratic time algorithm to compute diameter and eccentricities in real-weighted directed graphs with constant distance VC-dimension and strongly sublinear-sized balanced separators. For real-weighted K_h-minor-free digraphs, this runs in O(n^{2-1/(2h-2)} polylog(n)) time.
Prior to this work, truly subquadratic time computation of diameter was only known for real-weighted planar graphs, while extensions to broader classes like minor-free graphs were restricted to unweighted settings. In particular, existing algorithms that use VC-dimension [Ducoffe, Habib, Viennot; SICOMP 2022] [Le, Wulff-Nilsen; SODA 2024] [Chan, Chang, Gao, Le, Kisfaludi-Bak, Zheng; FOCS 2025] work with small integer weights, but do not naturally generalize to real weights. We overcome this barrier by introducing a randomized search-to-decision reduction, demonstrating that VC-dimension is a sufficiently powerful tool in the real-weighted regime.
Cite as
Da Wei Zheng. Real-Weighted Diameter and Eccentricities of Minor-Free and Bounded VC-Dimension Graphs in Truly Subquadratic Time. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{zheng:LIPIcs.ESA.2026.61,
author = {Zheng, Da Wei},
title = {{Real-Weighted Diameter and Eccentricities of Minor-Free and Bounded VC-Dimension Graphs in Truly Subquadratic Time}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {61:1--61:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.61},
URN = {urn:nbn:de:0030-drops-271973},
doi = {10.4230/LIPIcs.ESA.2026.61},
annote = {Keywords: Diameter, eccentricities, minor-free graphs, real weights, VC-dimension}
}
Document
Warm-Starting All-Pairs Shortest Paths with Predictions
Abstract
One of the three key hypotheses of fine-grained complexity asserts that computing All-Pairs Shortest Paths (APSP) requires cubic time, up to subpolynomial factors, in the worst case. We initiate the study of APSP in the paradigm of algorithms with predictions, also known as learning-augmented algorithms. We propose an APSP algorithm that takes as additional input a prediction (e.g., given by a model learned from similar instances seen in the past) consisting of sets of vertices causing the shortest detour for each pair of vertices. The algorithm runs in time 𝒪(n^{2.83} + η n), where η denotes the prediction error defined as the number of pairs of vertices for which, informally speaking, the prediction was not sufficient to compute and certify optimality of the shortest path length. This is already subcubic when the prediction error is (polynomially) smaller than its maximum possible values n², i.e., whenever the prediction is at least slightly better than terrible.
We build on the co-nondeterministic algorithm for the Exact Triangle problem by Chan, Vassilevska Williams, and Xu (STOC 2023), essentially enabling this algorithm to detect mistakes in the nondeterministic certificate and recover from them.
Our result constitutes the first necessary step towards designing learning-augmented algorithms for problems with known fine-grained lower bounds conditioned on the APSP Hypothesis.
Cite as
Adam Polak and Jonas Schmidt. Warm-Starting All-Pairs Shortest Paths with Predictions. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 62:1-62:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{polak_et_al:LIPIcs.ESA.2026.62,
author = {Polak, Adam and Schmidt, Jonas},
title = {{Warm-Starting All-Pairs Shortest Paths with Predictions}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {62:1--62:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.62},
URN = {urn:nbn:de:0030-drops-271981},
doi = {10.4230/LIPIcs.ESA.2026.62},
annote = {Keywords: Algorithms with predictions, APSP, fine-grained complexity, graph algorithms}
}
Document
Partition Rank and Algebraic Circuit Lower Bounds
Abstract
Strassen’s theory of bilinear complexity provides a mathematical characterization of the arithmetic complexity of primitives such as matrix multiplication via the rank of tensors. However, the connection to tensor rank is known to break down in higher degrees of multilinearity.
In this work, we highlight an unexplored connection between a generalized notion of tensor rank, which can be defined in Naslund’s framework of partition ranks (JCTA 2020), and multiplicative complexity. These partition ranks allow us to control the multiplicative complexity, and thus arithmetic complexity, in any constant degree of multilinearity from below, while recovering Strassen’s seminal characterization in the bilinear case. This enables novel potential applications of the rank-based approaches to problems in fine-grained algorithms and complexity, such as the hyperclique conjecture of Lincoln-Williams-Vassilevska Williams (SODA 2018). Moreover, we exhibit connections to established notions of rank, such as tensor slice rank (in the sense of Tao and Sawin), as well as its symmetric variant. For computing the latter symmetric variant, we point out a simple NP-hardness proof, contrasting the rather involved NP-hardness proof for ordinary, non-symmetric tensor slice rank by Bläser et al. (SODA 2021).
Cite as
Cornelius Brand, Petteri Kaski, and Jiaheng Wang. Partition Rank and Algebraic Circuit Lower Bounds. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brand_et_al:LIPIcs.ESA.2026.63,
author = {Brand, Cornelius and Kaski, Petteri and Wang, Jiaheng},
title = {{Partition Rank and Algebraic Circuit Lower Bounds}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {63:1--63:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.63},
URN = {urn:nbn:de:0030-drops-271994},
doi = {10.4230/LIPIcs.ESA.2026.63},
annote = {Keywords: Polynomials, tensors, arithmetic circuits, rank, multiplicative complexity, strength}
}
Document
Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs
Abstract
A recent paper by the authors (ITCS'26) initiates the study of the Triangle Detection problem in graphs avoiding a fixed pattern H as a subgraph and proposes a dichotomy hypothesis characterizing which patterns H make the Triangle Detection problem easier in H-free graphs than in general graphs. In this work, we demonstrate that this hypothesis is, in fact, equivalent to analogous hypotheses in two broader settings that a priori seem significantly more challenging: induced H-free graphs and colored H-free graphs.
Our main contribution is a reduction from the induced H-free case to the non-induced H^{+}-free case, where H^{+} preserves the structural properties of H that are relevant for the dichotomy, namely 3-colorability and triangle count. A similar reduction is given for the colored case.
A key technical ingredient is a self-reduction to Unique Triangle Detection that preserves the induced H-freeness property, via a new color-coding-like reduction.
Cite as
Amir Abboud, Ron Safier, and Nathan Wallheimer. Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 64:1-64:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{abboud_et_al:LIPIcs.ESA.2026.64,
author = {Abboud, Amir and Safier, Ron and Wallheimer, Nathan},
title = {{Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {64:1--64:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.64},
URN = {urn:nbn:de:0030-drops-272004},
doi = {10.4230/LIPIcs.ESA.2026.64},
annote = {Keywords: Fine-grained complexity, Triangle Detection, H-free graphs}
}
Document
Beyond Monotone Delays for Multi-Level Aggregation
Abstract
In the online Multi-Level Aggregation Problem (MLAP), requests arrive over time and are associated with nodes of a given weighted rooted tree of depth D. Each request must eventually be served by performing a service. Serving a request consists of selecting a rooted subtree that contains the request’s node, incurring a service cost equal to the total weight of the selected subtree. To reduce service costs, multiple requests may be served simultaneously by selecting a single rooted subtree that spans all of them. In addition, each request is associated with a penalty function that specifies the cost incurred when the request is served at a particular time. The objective is to minimize the total cost, consisting of both service costs and penalty costs.
Most previous work on MLAP assumes monotone non-decreasing penalty functions, commonly referred to as delay functions. Only very recent results consider penalty functions that initially decrease and subsequently increase, and even then only for the special cases of depths D = 1 and D = 2, namely the Joint Replenishment Problem (JRP).
In this work, we extend previous results in two ways. First, we allow arbitrary penalty functions, which may decrease and increase multiple times. Second, we study the general MLAP with arbitrary tree depth D under these arbitrary penalty functions. We present a randomized algorithm which is O(D log n log(nDW))-competitive, where W is the maximum service window among all penalty functions after normalizing the Lipschitz parameter of each penalty function to be 1 and normalizing the minimum positive edge weight incident to the root to be 1; and n is the number of requests. We note that our algorithm runs in polynomial-time, and even for D = 1 the problem admits hardness of approximation of Ω(log n) for polynomial time algorithms.
As mentioned above, prior to our work even for trees of depth D = 1,2, non-monotone penalty functions have been studied only in special cases of functions that decrease and increase only once. In contrast, for such trees we obtain O(log n log (nW))-competitive algorithms for arbitrary non-monotone penalty functions.
Cite as
Yossi Azar and Liad Iluz. Beyond Monotone Delays for Multi-Level Aggregation. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 65:1-65:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{azar_et_al:LIPIcs.ESA.2026.65,
author = {Azar, Yossi and Iluz, Liad},
title = {{Beyond Monotone Delays for Multi-Level Aggregation}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {65:1--65:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.65},
URN = {urn:nbn:de:0030-drops-272019},
doi = {10.4230/LIPIcs.ESA.2026.65},
annote = {Keywords: online algorithms, multi-level aggregation problem, arbitrary penalty functions, multicast problem}
}
Document
Improved Approximation Algorithms for n-Pairs Shortest Paths
Abstract
Let G = (V, E) be a graph with n = |V| nodes and m = |E| edges. The t-Pairs Shortest Paths problem, introduced by Cohen [FOCS'93; SICOMP'99], asks to approximate the distances between t prespecified pairs of vertices. Recently, this problem has received renewed attention, particularly in the case where t = Θ(n): the n-Pairs Shortest Paths problem. In this setting, new algorithms and conditional lower bounds have been developed by Dalirrooyfard, Jin, Vassilevska Williams, and Wein [FOCS'22], and Chechik, Hoch, and Lifshitz [SODA'25].
In this paper, we present the first algorithm for the n-Pairs Shortest Paths problem in weighted undirected graphs that achieves a (2 - α)k-approximation, for constant α > 0, that runs in Õ(mn^{1/k} + n^{1 + 2/k}) time. Specifically, we present a 1.622k-approximation, improving upon the (2k - 3)-approximation of Chechik, Hoch, and Lifshitz [SODA'25] for graphs that are not super sparse, which answers in the affirmative the open question posed by them. We also develop improved approximation algorithms with better tradeoffs for unweighted graphs and dense weighted graphs that improve upon the results of Dalirrooyfard et al. and Chechik, Hoch, and Lifshitz.
Our main technical contribution is the new heavy-edge technique. Using this technique, we transform an algorithm with an approximation guarantee that depends on W_{uv}, the weight of the heaviest edge on the shortest path between u and v, into an algorithm with purely multiplicative approximation that does not depend on W_{uv}.
Cite as
Avi Kadria, Liam Roditty, and Virginia Vassilevska Williams. Improved Approximation Algorithms for n-Pairs Shortest Paths. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 66:1-66:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kadria_et_al:LIPIcs.ESA.2026.66,
author = {Kadria, Avi and Roditty, Liam and Vassilevska Williams, Virginia},
title = {{Improved Approximation Algorithms for n-Pairs Shortest Paths}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {66:1--66:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.66},
URN = {urn:nbn:de:0030-drops-272028},
doi = {10.4230/LIPIcs.ESA.2026.66},
annote = {Keywords: Fine-grained complexity, Graph algorithm, Graph distances, n pairs shortest paths, all pairs shortest paths, distance oracles}
}
Document
Decomposing a Simple Polygon with Geodesic Unit-Balls
Abstract
We consider covering and partitioning a simple polygon into pieces which either have unit geodesic radius or unit geodesic diameter, using the 𝓁₂-metric for distances. There is no known method for finding an exact solution to these problems, even when the input size is constant, and the problem is known to be NP-hard in the case of polygons with holes. With this in mind, we instead devote our attention to developing simple approximation algorithms that run in polynomial time. For the radius problem, we present the first known approximation algorithms for both covering and partitioning, achieving a factor of 9. For the diameter problem, we are only able to give a positive result for the partition version of the problem, where we improve upon a complicated 72-approximation from Abrahamsen and Rasmussen [Mikkel Abrahamsen and Nichlas Langhoff Rasmussen, 2025], achieving a simple 15-approximation.
Cite as
Reilly Browne and Prahlad Narasimhan Kasthurirangan. Decomposing a Simple Polygon with Geodesic Unit-Balls. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{browne_et_al:LIPIcs.ESA.2026.67,
author = {Browne, Reilly and Kasthurirangan, Prahlad Narasimhan},
title = {{Decomposing a Simple Polygon with Geodesic Unit-Balls}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {67:1--67:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.67},
URN = {urn:nbn:de:0030-drops-272037},
doi = {10.4230/LIPIcs.ESA.2026.67},
annote = {Keywords: Covering, partitioning, polygon, k-center, constant factor approximation}
}
Document
Text Indexing: From Reporting to Counting
Abstract
We prove an elementary yet powerful combinatorial lemma: in any rooted tree with L leaves, the number of nodes whose depth is smaller than the number of their leaf descendants is at most L. For any string T of length n, a direct application of this lemma to the suffix trie of T yields that the number of substrings of T whose length is smaller than their number of occurrences in T is at most n. This combinatorial insight leads to space-efficient data structures with optimal query times for string counting problems via the following algorithmic framework: store the counts for the at most n "frequent" substrings of T in a preprocessing step, and use a reporting query to count for the "infrequent" substrings. Our framework acts as a convenient black box, lifting indexes with reporting time 𝒪(|P|+|Occ_T(P)|) to support counting queries in time 𝒪(|P|), where P is the queried pattern and Occ_T(P) is the set of occurrences of P in T. As applications, we show efficient indexes for consecutive occurrences, weighted sequences, strings with utilities, and non-overlapping occurrences.
Cite as
Ben Bals, Panagiotis Charalampopoulos, Oded Lachish, Solon P. Pissis, and Hilde Verbeek. Text Indexing: From Reporting to Counting. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 68:1-68:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bals_et_al:LIPIcs.ESA.2026.68,
author = {Bals, Ben and Charalampopoulos, Panagiotis and Lachish, Oded and Pissis, Solon P. and Verbeek, Hilde},
title = {{Text Indexing: From Reporting to Counting}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {68:1--68:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.68},
URN = {urn:nbn:de:0030-drops-272046},
doi = {10.4230/LIPIcs.ESA.2026.68},
annote = {Keywords: text indexing, data structures, string algorithms, string processing}
}
Document
Parameterized Approximation of Rectangle Stabbing
Abstract
In the Rectangle Stabbing problem, input is a set R of axis-parallel rectangles and a set L of axis-parallel lines in the plane. The task is to find a minimum size set L^* ⊆ L such that for every rectangle R ∈ R there is a line 𝓁 ∈ L^* such that 𝓁 intersects R. Gaur et al. [Journal of Algorithms, 2002] gave a polynomial time 2-approximation algorithm, while Dom et al. [WALCOM 2009] and Giannopoulos et al. [EuroCG 2009] independently showed that, assuming FPT ≠ W[1], there is no algorithm with running time f(k)(|L||R|)^O(1) that determines whether there exists an optimal solution with at most k lines. We give the first parameterized approximation algorithm for the problem with a ratio better than 2. In particular we give an algorithm that given R, L, and an integer k runs in time k^O(k)(|L||R|)^O(1) and either correctly concludes that there does not exist a solution with at most k lines, or produces a solution with at most 7k/4 lines. We complement our algorithm by showing that unless FPT = W[1], the Rectangle Stabbing problem does not admit a (5/4-ε)-approximation algorithm running in f(k)(|L||R|)^O(1) time for any function f and ε > 0.
Cite as
Huairui Chu, Ajaykrishnan E S, Daniel Lokshtanov, Anikait Mundhra, Thomas Schibler, Xiaoyang Xu, and Jie Xue. Parameterized Approximation of Rectangle Stabbing. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 69:1-69:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chu_et_al:LIPIcs.ESA.2026.69,
author = {Chu, Huairui and E S, Ajaykrishnan and Lokshtanov, Daniel and Mundhra, Anikait and Schibler, Thomas and Xu, Xiaoyang and Xue, Jie},
title = {{Parameterized Approximation of Rectangle Stabbing}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {69:1--69:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.69},
URN = {urn:nbn:de:0030-drops-272056},
doi = {10.4230/LIPIcs.ESA.2026.69},
annote = {Keywords: rectangle stabbing, parameterized algorithms, approximation algorithms, computational geometry, parameterized approximation complexity, geometric hitting set, lower bounds}
}
Document
Streaming with Catalytic Memory
Abstract
We introduce a streaming model that uses both catalytic and regular memory. In this model, we show how to exactly compute the frequency moments using a logarithmic number of bits of regular memory and a polynomial number of bits of catalytic memory. More generally, we show how to compute arbitrary polynomials of the item frequencies exactly within the same space bounds. As an application, we obtain catalytic streaming algorithms that exactly compute the number of distinct elements in a stream, count the number of triangles (or any other small subgraph) in a graph whose edges arrive in a stream, and identify heavy hitters.
Our algorithms for frequency moments perform a constant number of passes over the stream, and for polynomial evaluation, we require one more pass than the degree of the polynomial. In particular, for the second moment, we perform three passes over the stream. By relating our catalytic streaming model to the catalytic communication model introduced in [Pyne et al., 2025], we show that catalytic memory is not useful for any one pass streaming algorithms. For lower bounds on multi pass streaming algorithms, the impossibility results of [Pyne et al., 2025] are not strong enough. However, using a different technique, we show that computing the second frequency moment cannot be achieved by a two pass catalytic streaming algorithm that satisfies certain natural assumptions.
Cite as
Tamara Kaplan, Nimrod Kaplan, and Haim Kaplan. Streaming with Catalytic Memory. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 70:1-70:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kaplan_et_al:LIPIcs.ESA.2026.70,
author = {Kaplan, Tamara and Kaplan, Nimrod and Kaplan, Haim},
title = {{Streaming with Catalytic Memory}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {70:1--70:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.70},
URN = {urn:nbn:de:0030-drops-272060},
doi = {10.4230/LIPIcs.ESA.2026.70},
annote = {Keywords: Catalytic memory, streaming algorithms, frequency moments, space complexity, polynomial evaluation}
}
Document
On Estimating Operator Norm Distance, with Optimal Trace Distance Estimation When One State Is Pure
Abstract
We investigate the computational complexity of estimating the operator norm distance T_{∞}(ρ₀,ρ₁), defined via the operator norm ‖A‖_∞ := σ_max(A), where σ_max(A) is the largest singular value of A, given poly(n)-size state-preparation circuits of n-qubit quantum states ρ₀ and ρ₁. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states:
- When one state is pure, we establish an optimal quantum estimator using Θ(1/ε) queries to the state-preparation circuits. Consequently, for constant additive error, say ε = 1/5, our estimator runs in poly(n) time. Since the operator norm distance T_∞(|ψ⟩⟨ψ|,ρ) is exactly half of the trace distance T(|ψ⟩⟨ψ|,ρ), our result gives a rank-independent query complexity for estimating T_∞(|ψ⟩⟨ψ|,ρ) and T(|ψ⟩⟨ψ|,ρ), whereas the approaches due to van Apeldoorn, Cornelissen, Gilyén, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with rank(ρ), which can be exp(n) in general. In addition, our query complexity matches the optimal bound when both states are pure by Wang (TIT 2024).
- For general quantum states, we also provide a quantum estimator using Õ(1/ε^{3/2}) queries to the state-preparation circuits, which shows that the corresponding promise problem is BQP-complete and improves the QMA upper bound sketched by Liu and Wang (ESA 2025). Together with an Ω(1/ε) quantum query complexity lower bound, this leaves only square-root room for improvement. The key intuition behind our estimators is that, when one state is pure, the pure state |ψ⟩ has overlap at least 1/2 with the top unit eigenvector of |ψ⟩⟨ψ|-ρ, reflecting a structural feature specific to the operator norm distance.
Cite as
Yupan Liu, Qisheng Wang, and Zhan Yu. On Estimating Operator Norm Distance, with Optimal Trace Distance Estimation When One State Is Pure. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 71:1-71:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{liu_et_al:LIPIcs.ESA.2026.71,
author = {Liu, Yupan and Wang, Qisheng and Yu, Zhan},
title = {{On Estimating Operator Norm Distance, with Optimal Trace Distance Estimation When One State Is Pure}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {71:1--71:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.71},
URN = {urn:nbn:de:0030-drops-272076},
doi = {10.4230/LIPIcs.ESA.2026.71},
annote = {Keywords: Quantum state testing, Operator norm distance, Trace distance}
}
Document
On the Communication Complexity of Maximum Matching and Negative-Weight Shortest Paths
Abstract
We revisit several fundamental graph problems in the deterministic two-party communication model. Our main contributions include:
- We give a new Õ(n^{3/2})-bit protocol for computing a maximum matching in general graphs. While the same upper bound can be obtained by simulating the classic algorithms of Micali-Vazirani [Silvio Micali and Vijay V. Vazirani, 1980] and Gabow [Harold N. Gabow, 2017], our protocol is conceptually simple and avoids the intricacies of finding a maximal set of shortest augmenting paths.
- We give a new Õ(n)-bit protocol for negative-cycle detection and negative-weight single-source shortest paths. Our protocol simplifies that of Blikstad et al. [Joakim Blikstad et al., 2022] by replacing a long chain of reductions with a more direct approach based on vertex potentials.
- We give a combinatorial Õ(n)-bit protocol for computing a maximum matching in bipartite graphs, obtained by reinterpreting the near-linear communication protocol of Blikstad et al. [Joakim Blikstad et al., 2022] through a discretized analysis. Together, these results provide simpler protocols for several basic graph problems. We hope they will inspire further advances on the communication complexity of a wide range of graph problems.
Cite as
Yu Cheng, Tianle Jiang, Pachara Sawettamalya, and Huacheng Yu. On the Communication Complexity of Maximum Matching and Negative-Weight Shortest Paths. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 72:1-72:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cheng_et_al:LIPIcs.ESA.2026.72,
author = {Cheng, Yu and Jiang, Tianle and Sawettamalya, Pachara and Yu, Huacheng},
title = {{On the Communication Complexity of Maximum Matching and Negative-Weight Shortest Paths}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {72:1--72:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.72},
URN = {urn:nbn:de:0030-drops-272086},
doi = {10.4230/LIPIcs.ESA.2026.72},
annote = {Keywords: Graph algorithms, communication complexity, maximum matching, negative-weight shortest paths}
}
Document
Answering Related Questions
Abstract
We introduce the meta-problem Sidestep(Π, dist, d) for a problem Π, a metric dist over its inputs, and a map d: ℕ → ℝ_+ ∪ {∞}. A solution to Sidestep(Π, dist, d) on an input I of Π is a pair (J, Π(J)) such that dist(I,J) ⩽ d(|I|) and Π(J) is a correct answer to Π on input J. This formalizes the notion of answering a related question (or sidestepping the question), for which we give some motivations, and compare it with the adjacent concepts of smoothed analysis, certified algorithms, planted problems, modification problems, and approximation algorithms. Informally, we call hardness radius the "largest" d such that Sidestep(Π, dist, d) is NP-hard. This framework calls for establishing the hardness radius of problems Π of interest for the relevant distances dist.
We exemplify it with graph problems and two distances dist_∆ and dist_e (the edge edit distance) such that dist_∆(G,H) (resp. dist_e(G,H)) is the maximum degree (resp. number of edges) of the symmetric difference of G and H if these graphs are on the same vertex set, and +∞ otherwise. Thus when solving Sidestep(Π, dist_d, d) (resp. Sidestep(Π, dist_e, d)) on an n-vertex input G, acceptably close graphs H are obtained by XORing G with a graph of maximum degree at most d(n) (resp. having at most d(n) edges). We show that the decision problems Independent Set, Clique, Vertex Cover, Coloring, Clique Cover have hardness radius n^{1/2-o(1)} for dist_∆, and n^{4/3-o(1)} for dist_e, that Hamiltonian Cycle (or Hamiltonian Path) has hardness radius 0 for dist_∆, and somewhere between n^{1/2-o(1)} and n/3 for dist_e, and that Dominating Set has hardness radius n^{1-o(1)} for dist_e. We leave several open questions.
Cite as
Édouard Bonnet. Answering Related Questions. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bonnet:LIPIcs.ESA.2026.73,
author = {Bonnet, \'{E}douard},
title = {{Answering Related Questions}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {73:1--73:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.73},
URN = {urn:nbn:de:0030-drops-272095},
doi = {10.4230/LIPIcs.ESA.2026.73},
annote = {Keywords: Sidestep problem, hardness radius, edge edit distance, degree-bounded edge edits, robust reductions, classical graph problems, NP-hardness}
}
Document
Pure Nash Equilibria in Graphical Games of Bounded Width Revisited
Abstract
We revisit the complexity of deciding whether a graphical game admits a pure Nash equilibrium (PNE) parameterized by standard measures of the input graph, such as treewidth. The natural dynamic programming algorithm for this problem has parameter dependence α^{(Δ+1)tw} where α is the maximum number of strategies available to each player, each player’s utility depends on at most Δ other players, and the input graph has width tw. Our first contribution is to point out that an algorithm by Thomas and van Leeuwen [Algorithmica 2015] claiming to improve this dependence to α^O(tw) is flawed and, more strongly, such an algorithm would imply that FPT=W[1].
We then set out to pinpoint the fine-grained complexity of this problem with respect to standard parameters and show that the natural DP algorithm is not optimal, as the problem can be solved with dependence α^{⌊2Δ/3+1⌋tw}, α^{⌊Δ/2}+1⌋pw} , and α^{ctw}, where pw,ctw are the pathwidth and cutwidth of the input respectively. Our main algorithmic tool is a tightening of the relationship between the width of a graph G, its maximum degree, and the width of G², which may be of independent interest. Complementing these results, we show that our algorithms for pathwidth and cutwidth are likely to be optimal, as improving them is equivalent to falsifying the pw-SETH.
Cite as
Michael Lampis and Yiren Lu. Pure Nash Equilibria in Graphical Games of Bounded Width Revisited. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 74:1-74:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lampis_et_al:LIPIcs.ESA.2026.74,
author = {Lampis, Michael and Lu, Yiren},
title = {{Pure Nash Equilibria in Graphical Games of Bounded Width Revisited}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {74:1--74:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.74},
URN = {urn:nbn:de:0030-drops-272108},
doi = {10.4230/LIPIcs.ESA.2026.74},
annote = {Keywords: Graphical games, Pure Nash equilibria, Pathwidth, Treewidth}
}
Document
The Presort Hierarchy for Geometric Problems
Abstract
Many fundamental problems in computational geometry admit no algorithm running in o(n log n) time for n planar input points, via classical reductions from sorting. Prominent examples include the computation of convex hulls, quadtrees, onion layer decompositions, Euclidean minimum spanning trees, KD-trees, Voronoi diagrams, and decremental closest-pair.
A classical result shows that, given n points sorted along a single direction, the convex hull can be constructed in linear time. Subsequent works established that for all of the other above problems, this information does not suffice. In 1989, Aggarwal, Guibas, Saxe, and Shor asked: Under which conditions can a Voronoi diagram be computed in o(n log n) time? Since then, the question of whether sorting along two directions enables a o(n log n)-time algorithm for such problems has remained open and has been repeatedly mentioned in the literature.
In this paper, we introduce the Presort Hierarchy: A problem is 1-Presortable if, given a sorting along one axis, it permits a (possibly randomised) o(n log n)-time algorithm. It is 2-Presortable if sortings along both axes suffice. It is Presort-Hard otherwise. Our main result is that quadtrees, and by extension Delaunay triangulations, Voronoi diagrams, and Euclidean minimum spanning trees, are 2-Presortable: we present an algorithm with expected running time O(n √{log n}). This addresses the longstanding open problem posed by Aggarwal, Guibas, Saxe, and Shor (albeit randomised). We complement this result by showing that some of the other above geometric problems are also 2-Presortable or Presort-Hard.
Cite as
Ivor van der Hoog, Eva Rotenberg, Jack Spalding-Jamieson, and Lasse Wulf. The Presort Hierarchy for Geometric Problems. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 75:1-75:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vanderhoog_et_al:LIPIcs.ESA.2026.75,
author = {van der Hoog, Ivor and Rotenberg, Eva and Spalding-Jamieson, Jack and Wulf, Lasse},
title = {{The Presort Hierarchy for Geometric Problems}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {75:1--75:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.75},
URN = {urn:nbn:de:0030-drops-272111},
doi = {10.4230/LIPIcs.ESA.2026.75},
annote = {Keywords: Planar geometry, data structures, algebraic decision trees, algorithms with advice}
}
Document
A More Versatile Model for Enumerative Kernelization: A Case Study for Vertex Cover
Abstract
Enumerative kernelization is a relatively recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms, with two main models being proposed. The first, known as enum-kernels and due to Creignou et al. [Theory Comput. Syst., 2017], was too permissive, leading to constant-sized kernels for every problem solvable with FPT-delay. To remedy this, Golovach et al. [J. Comput. Syst. Sci., 2022] proposed the polynomial-delay enumeration kernelization model that, while addressing the shortcoming of the previous one, appears to be too strict, which we believe is a central reason for the slow development that the area has enjoyed so far. In this paper, we propose a new model for enumeration kernels, which we have called polynomial-delay (PD) kernels. It is more flexible than Golovach et al.’s kernels while still preserving their qualities; informally, it allows us to ignore "bad" solutions of the compressed instance when producing the solution set of the input instance, but still requires that the "good" solutions are lifted with polynomial-delay. After discussing the main properties of our model, we design a generic framework for vertex-subset problems to adapt decision kernels into PD kernels of the same size. We showcase our model’s increased versatility and the expressive power of our framework on the Enum Vertex Cover problem, where we want to list all vertex covers of size at most k of a given graph. In particular, we manage to generalize the kernelization dichotomy by Bougeret et al. [SIAM J. Discrete Math., 2022] about the existence of polynomial kernels for Vertex Cover parameterized by the vertex deletion distance to a minor-closed graph class, as well as the solution size and feedback vertex number parameterizations. The second one, in particular, is significantly simpler than the kernel designed by Bougeret et al. [IPEC, 2025], requiring only a few lines for its lifting algorithm. Beyond our framework, we also show how to generalize to the enumeration setting the kernel of Bougeret et al. [Algorithmica, 2019] for the vertex-deletion distance to c-treedepth.
Cite as
Marin Bougeret, Guilherme C. M. Gomes, and Ignasi Sau. A More Versatile Model for Enumerative Kernelization: A Case Study for Vertex Cover. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 76:1-76:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bougeret_et_al:LIPIcs.ESA.2026.76,
author = {Bougeret, Marin and C. M. Gomes, Guilherme and Sau, Ignasi},
title = {{A More Versatile Model for Enumerative Kernelization: A Case Study for Vertex Cover}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {76:1--76:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.76},
URN = {urn:nbn:de:0030-drops-272122},
doi = {10.4230/LIPIcs.ESA.2026.76},
annote = {Keywords: Kernelization, Enumeration, Vertex Cover, Structural parameterization}
}
Document
Tree-Independence Number of P₅-Free Graphs with No Large Bicliques
Abstract
The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique K_{𝓁,𝓁} forces tree-independence number at least 𝓁. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers t and 𝓁, {P_t,K_{𝓁,𝓁}}-free graphs have bounded tree-independence number. We prove this conjecture for t = 5 by showing that every {P₅,K_{𝓁,𝓁}}-free graph has tree-independence number at most 4𝓁. We also obtain related bounds for the weaker parameter of α-degeneracy.
Cite as
Václav Blažej, J. Pascal Gollin, Tomáš Hons, Tomáš Masařík, Martin Milanič, Paweł Rzążewski, Ondřej Suchý, and Alexandra Wesolek. Tree-Independence Number of P₅-Free Graphs with No Large Bicliques. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 77:1-77:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blazej_et_al:LIPIcs.ESA.2026.77,
author = {Bla\v{z}ej, V\'{a}clav and Gollin, J. Pascal and Hons, Tom\'{a}\v{s} and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Milani\v{c}, Martin and Rz\k{a}\.{z}ewski, Pawe{\l} and Such\'{y}, Ond\v{r}ej and Wesolek, Alexandra},
title = {{Tree-Independence Number of P₅-Free Graphs with No Large Bicliques}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {77:1--77:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.77},
URN = {urn:nbn:de:0030-drops-272138},
doi = {10.4230/LIPIcs.ESA.2026.77},
annote = {Keywords: tree-independence number, independence degeneracy, independence treewidth, P₅-free graphs}
}
Document
Learning-Augmented Online Sorting and TSP
Abstract
The online sorting problem is a natural online analog of classical sorting: n elements arrive one by one and must be placed irrevocably into an array of size n so as to minimize the sum of absolute differences between consecutive elements. Recent work by Aamand et al. [SODA 2023] and Abrahamsen et al. [ESA 2024] showed that the optimal competitive ratio for this problem is Θ(√n), even when randomization is allowed. Bertram [ESA 2025] extended this bound to the online traveling salesman problem (TSP), of which online sorting is a special case on the line metric. These polynomial bounds raise the question of whether additional information can lead to improved performance. In this paper, we initiate the study of online sorting and TSP in the framework of machine-learned predictions. We characterize the exact tradeoff between consistency and robustness for online sorting with predictions, and prove a surprising lower bound showing that robustness is not lossless in this setting. This phenomenon sets online sorting apart from most previously studied online problems with predictions. We extend our results to online TSP with predictions on general metric spaces, where the same consistency-robustness tradeoff persists. Finally, we present a sharp contrast in the case of online TSP on the uniform metric. While Abrahamsen et al. gave an O(log n)-competitive algorithm without predictions for the uniform metric, we show that predictions enable an algorithm that is simultaneously O(1)-consistent and O(log n)-robust. We further extend this result to the setting of multiple predictions.
Cite as
Ioana O. Bercea, Gerth Stølting Brodal, John Iacono, László Kozma, and Debmalya Panigrahi. Learning-Augmented Online Sorting and TSP. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 78:1-78:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bercea_et_al:LIPIcs.ESA.2026.78,
author = {Bercea, Ioana O. and Brodal, Gerth St{\o}lting and Iacono, John and Kozma, L\'{a}szl\'{o} and Panigrahi, Debmalya},
title = {{Learning-Augmented Online Sorting and TSP}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {78:1--78:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.78},
URN = {urn:nbn:de:0030-drops-272145},
doi = {10.4230/LIPIcs.ESA.2026.78},
annote = {Keywords: Online sorting, online traveling salesman problem, learned predictions}
}
Document
Unimodal-Cost k-Median on a Line
Abstract
Given n piecewise-linear unimodal functions f_1,… ,f_n:ℝ → ℝ and an integer 1 ≤ k ≤ n, the Unimodal-Cost k-Median problem asks for k real numbers y_1,… ,y_k minimizing ∑_{i=1}^n min_{1 ≤ r ≤ k} f_i(y_r). Let m be the number of breakpoints: the total number of affine-piece endpoint occurrences plus one occurrence at a chosen minimizer of each function. We give an exact algorithm running in O((m+nlog n)log m ⋅ min{k, log m√{klog m}, log m⋅ 2^O(√{log k log log m})}) . The first term inside the minimum comes from a direct k-stage dynamic program. The other two use the minimum-weight k-link path algorithms of Aggarwal et al. [Aggarwal et al., 1994] and Schieber [Schieber, 1998] for Monge costs, replacing their O(1) edge-weight access by batched access to the implicit transition costs.
Cite as
Yike Chen and Chao Xu. Unimodal-Cost k-Median on a Line. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.ESA.2026.79,
author = {Chen, Yike and Xu, Chao},
title = {{Unimodal-Cost k-Median on a Line}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {79:1--79:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.79},
URN = {urn:nbn:de:0030-drops-272153},
doi = {10.4230/LIPIcs.ESA.2026.79},
annote = {Keywords: unimodal-cost k-median, Monge, k-link path, dynamic programming, total monotonicity}
}
Document
Theoretical Analysis of Byte-Pair Encoding
Abstract
Byte-Pair Encoding (BPE) is a widely used method for subword tokenization, with origins in grammar-based text compression. It is employed in a variety of language processing tasks such as machine translation or large language model (LLM) pretraining, to create a token dictionary of a prescribed size. Most evaluations of BPE to date are empirical, and the reasons for its good practical performance are not well understood.
In this paper we focus on the optimization problem underlying BPE: finding a pair encoding that achieves optimal compression utility. We show that this problem is APX-complete, indicating that it is unlikely to admit a polynomial-time approximation scheme. This answers, in a stronger form, a question recently raised by Zouhar et al. [Zouhar et al., 2023].
On the positive side, we show that BPE approximates the compression utility of the optimal pair encoding to a worst-case factor between 0.333 and 0.625. Our results aim to explain the ongoing success of BPE and are, to our knowledge, the first rigorous guarantees on its compression utility that hold for all inputs.
Cite as
László Kozma and Johannes Voderholzer. Theoretical Analysis of Byte-Pair Encoding. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 80:1-80:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kozma_et_al:LIPIcs.ESA.2026.80,
author = {Kozma, L\'{a}szl\'{o} and Voderholzer, Johannes},
title = {{Theoretical Analysis of Byte-Pair Encoding}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {80:1--80:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.80},
URN = {urn:nbn:de:0030-drops-272166},
doi = {10.4230/LIPIcs.ESA.2026.80},
annote = {Keywords: byte pair encoding, approximation, hardness, tokenization}
}
Document
Primal-Dual Online Algorithms for the Parking Permit Problem
Abstract
The Parking Permit Problem (PPP), first studied by Meyerson, is a classic online problem generalizing the ski rental problem. We re-examine the PPP using the primal-dual scheme, obtaining simple algorithms with superior performance guarantees. Unlike previous work, which relied on reductions that degraded competitive ratios, we work with the problem’s structure directly. We also provide near-matching lower bounds. Using the primal-dual framework, we find the PPP’s deterministic competitive ratio exactly, and the randomized competitive ratio within an additive constant.
Cite as
Christian Coester and Alex Turoczy. Primal-Dual Online Algorithms for the Parking Permit Problem. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 81:1-81:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{coester_et_al:LIPIcs.ESA.2026.81,
author = {Coester, Christian and Turoczy, Alex},
title = {{Primal-Dual Online Algorithms for the Parking Permit Problem}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {81:1--81:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.81},
URN = {urn:nbn:de:0030-drops-272171},
doi = {10.4230/LIPIcs.ESA.2026.81},
annote = {Keywords: Online Algorithms, Competitive Analysis, Primal-Dual Algorithms, Parking Permit Problem}
}
Document
Tighter Bounds for Weighted and Unweighted Shortest Cycle Approximation
Abstract
We study the problem of approximating the length of a shortest cycle in a given graph, known as the girth of the graph. The state-of-the-art approximation algorithms for unweighted graphs by Kadria et al. [SODA'22] and Roditty and Trabelsi [arXiv'25] achieve the following trade-off: for every integer k ≥ 2, there is an Õ(n^{1+2/k}) time algorithm that achieves a (2k/3)-approximation for the girth in unweighted n-node graphs. The first result of this paper is to achieve the same trade-off for m-edge, n-node graphs with non-negative real edge weights: a 2k/3-approximation algorithm running in Õ(m+n^{1+2/k}) time. The dependence on m is unavoidable in weighted graphs. Our result improves on the work of Kadria et al. [SODA'23] and Ducoffe [ICALP'19 and SIDMA'21], who were only able to achieve such a trade-off for some values of k. We also prove new fine-grained lower bounds for girth approximation and related problems in unweighted graphs.
Cite as
Avi Kadria, Liam Roditty, and Virginia Vassilevska Williams. Tighter Bounds for Weighted and Unweighted Shortest Cycle Approximation. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 82:1-82:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kadria_et_al:LIPIcs.ESA.2026.82,
author = {Kadria, Avi and Roditty, Liam and Vassilevska Williams, Virginia},
title = {{Tighter Bounds for Weighted and Unweighted Shortest Cycle Approximation}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {82:1--82:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.82},
URN = {urn:nbn:de:0030-drops-272180},
doi = {10.4230/LIPIcs.ESA.2026.82},
annote = {Keywords: Fine-grained complexity, Graph algorithms, shortest cycle, girth approximations}
}
Document
A Computer-Assisted Proof of the Optimal Density Bound for Pinwheel Covering
Abstract
In the covering version of the pinwheel scheduling problem, a daily task must be assigned to agents under the constraint that agent i can perform the task at most once in any a_i-day interval. In this paper, we determine the optimal constant α^* = 1.264… such that every instance with ∑_i 1/a_i ≥ α^* is schedulable. This resolves an open problem posed by Kawamura and Soejima (2020). Our proof combines Kawamura’s (2026) techniques for the packing version with new mathematical insights to reduce the analysis to a finite set of instances, which are then verified through an exhaustive computer-aided search that draws on ideas from Gąsieniec, Smith, and Wild (2022). The same result was obtained independently by Mishra (2026).
Cite as
Akitoshi Kawamura and Yusuke Kobayashi. A Computer-Assisted Proof of the Optimal Density Bound for Pinwheel Covering. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 83:1-83:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kawamura_et_al:LIPIcs.ESA.2026.83,
author = {Kawamura, Akitoshi and Kobayashi, Yusuke},
title = {{A Computer-Assisted Proof of the Optimal Density Bound for Pinwheel Covering}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {83:1--83:7},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.83},
URN = {urn:nbn:de:0030-drops-272199},
doi = {10.4230/LIPIcs.ESA.2026.83},
annote = {Keywords: pinwheel scheduling, pinwheel covering, density threshold}
}
Document
Strategyproof Mechanisms Without Money for 2-Exchange Systems
Abstract
We study a mechanism design problem in which a ground set of items E is distributed among self-interested agents. The existence of an item is private information of the agent owning it. A mechanism takes as input a set of reported items together with their intrinsic weights and returns a feasible set of items. A mechanism is strategyproof if no agent can increase the total weight of their items in the solution by withholding a subset of their items from the mechanism. It is α-approximate if the total weight of items selected is at least an 1/α-fraction of the total weight of an optimal solution.
Our main result is a 6.018-approximate strategyproof mechanism for feasibility constraints defined by a 2-exchange system. This class of independence systems is defined by a combinatorial exchange condition and includes, for example, b-matchings in general graphs, intersections of strongly base-orderable matroids, and unit interval scheduling. This constant approximation generalizes and improves over a logarithmic approximation for matchings. We also obtain a strategyproof mechanism with logarithmic approximation for generalized assignment instances, where items represent compatibilities between jobs and machines. While prior work focused on the special case in which each agent controls a single job, we provide the first logarithmic approximation guarantee for the general setting in which agents may control multiple jobs. We finally provide improved approximation guarantees for matching instances with binary weights, beating the 2-approximation given by a simple greedy mechanism for any finite number of agents and showing a strict separation between deterministic and randomized mechanisms for the case of two agents.
Cite as
Javier Cembrano, Max Klimm, Martin Knaack, and Arturo Merino. Strategyproof Mechanisms Without Money for 2-Exchange Systems. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 84:1-84:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cembrano_et_al:LIPIcs.ESA.2026.84,
author = {Cembrano, Javier and Klimm, Max and Knaack, Martin and Merino, Arturo},
title = {{Strategyproof Mechanisms Without Money for 2-Exchange Systems}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {84:1--84:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.84},
URN = {urn:nbn:de:0030-drops-272202},
doi = {10.4230/LIPIcs.ESA.2026.84},
annote = {Keywords: mechanism design without money, strategyproof mechanisms, approximation algorithms, 2-exchange systems, matchings}
}
Document
Nearly Instance Optimal Sparse Matrix Approximation from Matrix-Vector Products
Abstract
A large body of work studies the problem of learning an approximation to an implicit matrix 𝐀 ∈ ℝ^{m×n} that is only accessible implicitly via matrix-vector product queries (matvec queries) of the form 𝐱 → 𝐀𝐱 or 𝐱 → 𝐀^T𝐱. Of particular interest are methods that learn a near-optimal approximation with a fixed sparsity pattern. For example, we might want to learn a near-optimal diagonal, banded, or arrow-head approximation to an implicit matrix 𝐀.
Naturally, the number of matvec queries required to solve this problem depends on the sparsity pattern, which can be encoded as a binary matrix 𝐒 ∈ {0,1}^{m× n}. The query complexity of previous algorithms scales with quantities like the total number of ones in 𝐒, its maximum column/row sparsity, or the chromatic number of its "conflict graph". These quantities are incomparable: for a given 𝐒, parameterizing by one might yield lower query complexity than another.
In this work, we unify and tighten these prior results by providing a nearly sharp characterization of the matvec query complexity of sparse matrix approximation. Generalizing a definition from graph algorithms, let the degeneracy, degen(𝐒), denote the smallest number k so that, if we iteratively delete all rows and columns of 𝐒 with ≤ k ones, we are left with an empty matrix. We show that a near-optimal approximation to 𝐀 with sparsity pattern 𝐒 can be learned with Õ(degen(𝐒)) matrix-vector product queries, and Ω(degen(𝐒)) queries are necessary, for any sparsity pattern 𝐒. Moreover, unlike prior work based on graph coloring, all of our methods run in polynomial time.
Cite as
Christopher Musco and Indu Ramesh. Nearly Instance Optimal Sparse Matrix Approximation from Matrix-Vector Products. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 85:1-85:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{musco_et_al:LIPIcs.ESA.2026.85,
author = {Musco, Christopher and Ramesh, Indu},
title = {{Nearly Instance Optimal Sparse Matrix Approximation from Matrix-Vector Products}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {85:1--85:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.85},
URN = {urn:nbn:de:0030-drops-272213},
doi = {10.4230/LIPIcs.ESA.2026.85},
annote = {Keywords: Matrix learning, sparse approximation, implicit methods}
}
Document
Sublinear Algorithms for Estimating Single-Linkage Clustering Costs
Abstract
Single-linkage clustering (SLC) is a fundamental method for hierarchical data analysis. In the distance setting, a k-clustering produced by SLC can be obtained by computing a minimum spanning tree (MST) and deleting its k-1 heaviest edges. This naturally induces a cost profile for the SLC hierarchy: for each k ∈ [n], we define cost_k to be the weight of the resulting k-component spanning forest, equivalently, the minimum total weight of any spanning forest with exactly k connected components. The corresponding SLC cost profile is (cost_1,…,cost_n), and the scalar quantity cost(G) = ∑_{k=1}^n cost_k is the area under this profile.
We study the problem of approximating these quantities in sublinear time. We assume that the input is a weighted graph G of average degree d with edge weights in {1,… ,W}, accessed through adjacency-list queries; missing edges are treated as having infinite distance. Our main result is a sampling-based algorithm that outputs a succinct sketch of the entire SLC cost profile in the distance setting. The algorithm runs in Õ(d√W/ε³) time and returns a sketch from which one can derive estimates (cost̂_1,…,cost̂_n) satisfying ∑_{k=1}^n |cost̂_k-cost_k| ≤ ε cost(G).
Thus, we obtain an 𝓁₁ approximation to the full profile whose error is at most an ε-fraction of the area under the true profile. In particular, this yields a (1±ε)-approximation to cost(G) within the same running time. We also prove a nearly matching lower bound of Ω(d√W/ε²) queries for estimating cost(G).
We further extend our results to the similarity setting, where SLC is defined via a maximum spanning tree. In this case, we obtain algorithms with running time Õ(dW/ε³) for both the profile and the total cost, together with a nearly matching lower bound of Ω(dW/ε²) queries. These bounds reveal a genuine separation between the distance and similarity settings. Finally, we extend our algorithms to metric spaces, where we obtain Õ(n/ε⁷)-query algorithms for both distance and similarity metrics, and we complement our theory with experimental validation.
Cite as
Pan Peng, Christian Sohler, and Yi Xu. Sublinear Algorithms for Estimating Single-Linkage Clustering Costs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 86:1-86:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{peng_et_al:LIPIcs.ESA.2026.86,
author = {Peng, Pan and Sohler, Christian and Xu, Yi},
title = {{Sublinear Algorithms for Estimating Single-Linkage Clustering Costs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {86:1--86:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.86},
URN = {urn:nbn:de:0030-drops-272227},
doi = {10.4230/LIPIcs.ESA.2026.86},
annote = {Keywords: Sublinear Algorithms, Single-linkage Clustering, Approximation Algorithms, Minimum Spanning Tree}
}
Document
Kernelization for H-Packing Revisited
Abstract
H-Packing asks whether a graph G contains k vertex-disjoint copies of a fixed pattern graph H. Via the standard reduction to d-Set Packing, one obtains generic kernels with O(k^{|V(H)|-1}) vertices and O(k^|V(H)|) edges. We revisit the question of beating these bounds for specific patterns H.
Our main results concern subdivided stars. Let S_{d₁,d₂} denote the subdivided star with d₁ branches of length 1 and d₂ branches of length 2. We obtain kernels with O(k²) vertices and O(k³) edges for P₅ = S_{0,2}, for S_{1,2}, and for every S_{d₁,1}, kernels with O(k⁴) vertices and O(k⁶) edges for every fixed S_{d₁,d₂} with d₁ ≥ 1, and a kernel with O(k²) vertices and O(k⁴) edges for the paw. Our proofs proceed in two steps. First, we reduce to instances in which all but a small part of the graph is independent, or in which the graph has a small vertex cover. Second, we reduce the independent side by keeping only a bounded number of witness vertices for each subset of the small part.
On the negative side, we prove a lower bound for the line S_{0,d}. For every d ≥ 3 and every ε > 0, S_{0,d}-Packing does not admit a compression of size O(k^{d-ε}) unless NP ⊆ coNP/poly. Thus, deleting a single vertex from the pattern may, surprisingly, make kernelization provably harder, showing that compressibility of H-Packing is not monotone under taking induced subgraphs.
Cite as
Tomohiro Koana and Soh Kumabe. Kernelization for H-Packing Revisited. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 87:1-87:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{koana_et_al:LIPIcs.ESA.2026.87,
author = {Koana, Tomohiro and Kumabe, Soh},
title = {{Kernelization for H-Packing Revisited}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {87:1--87:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.87},
URN = {urn:nbn:de:0030-drops-272239},
doi = {10.4230/LIPIcs.ESA.2026.87},
annote = {Keywords: Parameterized Complexity, Kernelization, H-Packing}
}
Document
Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces
Abstract
In this paper, we study the problem of constructing a (1/ε)-well-separated pair decomposition (WSPD) for a point set of size n in the Massively Parallel Computation (MPC) model, where multiple machines work in parallel and communicate in synchronous rounds. We present an O(1)-round MPC algorithm that constructs a O(1/ε)-WSPD of size (1/ε)^O(ddim) ⋅ Õ(n) for point sets in a metric space of a constant doubling dimension ddim, with high probability, using (1/ε)^O(ddim) ⋅ Õ(n) total space and O(n^δ) space per machine for a constant δ ∈ (0,1). In the d-dimensional Euclidean space, we can improve the size of the WSPD and the total space to (1/ε)^O(d) n. This improves the best-known algorithm [FOCS'93] for computing a WSPD which requires O(log n) rounds and works only in Euclidean spaces. As a consequence, the following problems can be solved in O(1) rounds in the MPC model: computing a (1+ε)-spanner, a (1-ε)-approximation of the diameter, the closest pair, and the k-nearest neighbors (k-NN). While our k-NN algorithm is specific to Euclidean space, the other three problems can be solved in both Euclidean and doubling metric spaces.
Cite as
Eunjin Oh and Hyeonjun Shin. Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 88:1-88:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{oh_et_al:LIPIcs.ESA.2026.88,
author = {Oh, Eunjin and Shin, Hyeonjun},
title = {{Fully Scalable MPC Algorithms for WSPD in Doubling and Euclidean Spaces}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {88:1--88:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.88},
URN = {urn:nbn:de:0030-drops-272244},
doi = {10.4230/LIPIcs.ESA.2026.88},
annote = {Keywords: MPC model, parallel algorithms, Computational Geometry, well-separated pair decomposition, doubling metric spaces}
}
Document
FPT Algorithms over Linear Delta-Matroids with Applications
Abstract
Matroids, particularly linear matroids, have been a powerful tool for applications in parameterized complexity, both for algorithms and kernelization. In particular, they have been instrumental in speeding up or replacing dynamic programming. Delta-matroids are a generalization of matroids that further encapsulate structures such as non-maximum matchings in general graphs and various path-packing and topological structures. There is also a notion of linear delta-matroids (represented by skew-symmetric matrices) which carries significant expressive power and enables powerful algorithms. We investigate parameterized complexity aspects of problems defined over linear delta-matroids, or with delta-matroid constraints. Our initial analysis of basic intersection and packing problems reveals a different complexity landscape compared to the more familiar matroid case. In particular, there is a stark contrast in complexity between the cardinality parameter k and the rank parameter r. For example, finding an intersection of size k of three linear delta-matroids is W[1]-hard when parameterized by k, while far more general problems (such as finding a set packing of size k that is feasible in a given linear delta-matroid) are FPT when parameterized by the rank r of the delta-matroid. In fact, we extend the recent determinantal sieving procedure of Eiben, Koana, and Wahlström (TheoretiCS 2025) into a process that sieves a given polynomial for a monomial whose support is feasible in a given linear delta-matroid, parameterized by r. This is a direct generalization of determinantal sieving.
Second, we investigate a curious class of problems that turns out to be FPT parameterized by k, even on delta-matroids of unbounded rank. We begin with Delta-matroid Triangle Cover - find a feasible set of size k that can be covered by a vertex-disjoint packing of triangles (i.e., sets of size 3) out of a given triangle collection. For example, this allows us to find, in a graph, a packing of K₃’s and K₂’s with the maximum possible number of edges, parameterized above the matching number of the graph (note that this problem is NP-hard, unlike the problem of finding such a packing covering a maximum number of vertices, which is tractable). As applications, we resolve the FPT status of Cluster Subgraph and Strong Triadic Closure parameterized above the matching number.
Cite as
Eduard Eiben, Tomohiro Koana, and Magnus Wahlström. FPT Algorithms over Linear Delta-Matroids with Applications. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 89:1-89:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{eiben_et_al:LIPIcs.ESA.2026.89,
author = {Eiben, Eduard and Koana, Tomohiro and Wahlstr\"{o}m, Magnus},
title = {{FPT Algorithms over Linear Delta-Matroids with Applications}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {89:1--89:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.89},
URN = {urn:nbn:de:0030-drops-272251},
doi = {10.4230/LIPIcs.ESA.2026.89},
annote = {Keywords: Delta-matroids, parameterized algorithms, triangle packing}
}
Document
Approximate Single Source Dual Fault Tolerant Distance Oracle
Abstract
We are given an undirected weighted graph G with n vertices and m edges, edge weights in [1, W], and a designated source vertex s. We design a single source dual fault tolerant distance oracle for G. Given a destination vertex t and a set F of at most two faulty edges, the oracle returns a (1 + O(ε))-approximation of the weight of the shortest path from the source s to t avoiding F. Our oracle uses Õ(n√n) space and has Õ(1) query time.
Prior to our result, single source single fault tolerant oracles were known to return a (1+ε) approximation of the weight of the shortest path using Õ(n) space and O(1) query time. However, extending these approaches to multiple faults remained an open problem. Indeed, all (1+ε)-approximate distance oracles that handle multiple faults require Ω(n²) space. We break this bound by presenting the first dual fault tolerant distance oracle with o(n²) space.
Cite as
Koustav Das and Manoj Gupta. Approximate Single Source Dual Fault Tolerant Distance Oracle. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 90:1-90:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{das_et_al:LIPIcs.ESA.2026.90,
author = {Das, Koustav and Gupta, Manoj},
title = {{Approximate Single Source Dual Fault Tolerant Distance Oracle}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {90:1--90:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.90},
URN = {urn:nbn:de:0030-drops-272263},
doi = {10.4230/LIPIcs.ESA.2026.90},
annote = {Keywords: Fault-tolerant distance oracle, Shortest paths, Graph algorithms, Dual failures, Approximate distance oracle}
}
Document
Reducing CMSO to Unbreakable Graphs Cannot Be Computable
Abstract
Lokshtanov, Ramanujan, Saurabh, and Zehavi [ICALP 2018] proved that for any CMSO formula ϕ, testing ϕ on arbitrary graphs can be reduced to testing it on (q,k)-unbreakable graphs for appropriate parameters. Their proof is non-constructive, and they ask whether it can be made constructive. We prove that this is impossible: specifically, the parameter q cannot be a computable function of ϕ.
Cite as
Colin Geniet and Roohani Sharma. Reducing CMSO to Unbreakable Graphs Cannot Be Computable. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 91:1-91:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{geniet_et_al:LIPIcs.ESA.2026.91,
author = {Geniet, Colin and Sharma, Roohani},
title = {{Reducing CMSO to Unbreakable Graphs Cannot Be Computable}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {91:1--91:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.91},
URN = {urn:nbn:de:0030-drops-272277},
doi = {10.4230/LIPIcs.ESA.2026.91},
annote = {Keywords: Unbreakability, CMSO, Uncomputability, Meta-theorem}
}
Document
Compact Representations of Pattern-Avoiding Permutations
Abstract
Pattern-avoiding permutations are a central object of study in both combinatorics and theoretical computer science. In this paper we design a data structure that can store any size-n permutation τ that avoids an arbitrary (and unknown) fixed pattern π in the asymptotically optimal 𝒪(n lg{s_π}) bits, where s_π is the Stanley-Wilf limit of π. Our data structure supports τ(i) and τ^{-1}(i) queries in 𝒪(1) time, sidestepping the lower bounds that hold for general permutations. Comparable results were previously known only in more restricted cases, e.g., when τ is separable, which means avoiding the patterns 2413 and 3142.
We also extend our data structure to support more complex geometric queries on pattern-avoiding permutations (or planar point sets) such as rectangle range counting in 𝒪(lg lg n) time. This result circumvents the lower bound of Ω(lg{n}/lg lg{n}) by Pătraşcu (STOC 2007) for the general case. For bounded treewidth permutation classes (which include the above-mentioned separable class), we further reduce the space overhead to a lower order additive term, making our data structure succinct. This extends and improves results of Chakraborty et al. (ISAAC 2024) obtained for separable permutations via different techniques. All our data structures can be constructed in linear time.
Cite as
László Kozma and Michal Opler. Compact Representations of Pattern-Avoiding Permutations. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 92:1-92:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kozma_et_al:LIPIcs.ESA.2026.92,
author = {Kozma, L\'{a}szl\'{o} and Opler, Michal},
title = {{Compact Representations of Pattern-Avoiding Permutations}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {92:1--92:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.92},
URN = {urn:nbn:de:0030-drops-272285},
doi = {10.4230/LIPIcs.ESA.2026.92},
annote = {Keywords: pattern-avoiding permutations, compact data structures, succinct data structures, range counting, twin-width}
}
Document
An Approximation Algorithm for 2-Vertex-Connectivity via Cycle-Restricted 2-Edge-Covers
Abstract
In the 2-Vertex-Connected Spanning Subgraph problem (2-VCSS), we are given an undirected graph G, and the objective is to find a 2-vertex-connected spanning subgraph S of G with the minimum number of edges. In the context of survivable network design, 2-VCSS is one of the most fundamental and well-studied problems. There has been active research on improving the approximation ratio of algorithms, and the current best ratio is 4/3, achieved by Bosch-Calvo, Grandoni, and Jabal Ameli. In this paper, we improve the approximation ratio to 95/72+ε (< 1.32). The key idea in our algorithm is to introduce a 2-edge-cover without certain cycle components, and use it as an initial solution.
Cite as
Yusuke Kobayashi and Takashi Noguchi. An Approximation Algorithm for 2-Vertex-Connectivity via Cycle-Restricted 2-Edge-Covers. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 93:1-93:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kobayashi_et_al:LIPIcs.ESA.2026.93,
author = {Kobayashi, Yusuke and Noguchi, Takashi},
title = {{An Approximation Algorithm for 2-Vertex-Connectivity via Cycle-Restricted 2-Edge-Covers}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {93:1--93:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.93},
URN = {urn:nbn:de:0030-drops-272293},
doi = {10.4230/LIPIcs.ESA.2026.93},
annote = {Keywords: approximation algorithm, survivable network design, minimum 2-vertex-connected spanning subgraph, triangle-free 2-matching}
}
Document
DAG Covers for Structured Graphs: The Steiner Point Effect
Abstract
Given a weighted digraph G, a (t,g,μ)-DAG cover is a collection of g dominating DAGs D_1,… ,D_g such that all distances are approximately preserved: for every pair (u,v) of vertices, min_id_{D_i}(u,v) ≤ t⋅ d_G(u,v), and the total number of non-G edges is bounded by |(∪_i D_i)⧵ G| ≤ μ. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of Steiner DAG cover, where the DAGs are allowed to contain Steiner points.
We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a (1,2,Õ(n⋅tw))-Steiner DAG cover. For planar digraphs we provide a (1+ε,2,Õ_ε(n))-Steiner DAG cover.
We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth 1 with stretch t < 2 and sub-quadratic number of extra edges requires Ω(log n) DAGs.
Cite as
Sujoy Bhore, Hsien-Chih Chang, Jonathan Conroy, Arnold Filtser, Eunjin Oh, Nicole Wein, and Da Wei Zheng. DAG Covers for Structured Graphs: The Steiner Point Effect. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 94:1-94:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.ESA.2026.94,
author = {Bhore, Sujoy and Chang, Hsien-Chih and Conroy, Jonathan and Filtser, Arnold and Oh, Eunjin and Wein, Nicole and Zheng, Da Wei},
title = {{DAG Covers for Structured Graphs: The Steiner Point Effect}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {94:1--94:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.94},
URN = {urn:nbn:de:0030-drops-272306},
doi = {10.4230/LIPIcs.ESA.2026.94},
annote = {Keywords: Directed graphs, DAG (directed acyclic graphs), distortion, metric embeddings, planar graph, treewidth}
}
Document
Fast Metric Decompositions in High Dimension
Abstract
Metric decompositions are a fundamental tool in the design of algorithms involving distances. We study fast algorithms for sampling from probabilistic metric decompositions of n-point sets in 𝓁_∞ and 𝓁₂ spaces of high dimension d. For 𝓁_∞, we design a padded-decomposition algorithm that runs in time Õ(nd²), which is near-linear in n, and achieves padding parameter Õ(log n). Our algorithm constructs a new sparse neighborhood cover that is based on geometric properties of 𝓁_∞ [Indyk, JCSS'01], and utilizes recent reductions between covers and decompositions [Conroy and Filtser, STOC'25]. For 𝓁₂, we design a separating-decomposition algorithm that achieves near optimal separation Õ(√{log n}) in almost-linear time n^{1+o(1)}. Our bounds improve over known algorithms with similar running time by a factor Ω(√{log n}), and the techniques have additional applications to spanners and nearest-neighbor search.
Cite as
Robert Krauthgamer, Asaf Petruschka, and Nir Petruschka. Fast Metric Decompositions in High Dimension. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 95:1-95:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{krauthgamer_et_al:LIPIcs.ESA.2026.95,
author = {Krauthgamer, Robert and Petruschka, Asaf and Petruschka, Nir},
title = {{Fast Metric Decompositions in High Dimension}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {95:1--95:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.95},
URN = {urn:nbn:de:0030-drops-272315},
doi = {10.4230/LIPIcs.ESA.2026.95},
annote = {Keywords: Metric Spaces, Separating Decomposition, Padded Decomposition}
}
Document
Time Series Decomposition Using the Fréchet Distance
Abstract
In this paper, we introduce a new data analysis problem that aims to decompose a set of univariate time series into a small set of k base curves of length at most l such that the sum of Fréchet distances of the time series to a "Fréchet combination" of the base curves is minimized. Here, a Fréchet combination allows to combine individually scaled base curves using a k-dimensional traversal. We call the problem of finding a set of optimal base curves the Fréchet decomposition problem and we consider two variants: (a) the base curves can be arbitrary curves of bounded length and (b) the curves come from a given finite set of candidate curves. We think of the Fréchet decomposition problem as a Fréchet variant of principal component analysis. For the case of a single base curve we develop a (1+ε)-approximation algorithm for the Fréchet decomposition problem. Additionally we give an exact algorithm for the projection distance problem that asks to compute the distance of one given time series to a given set of k base curves. This allows us to design an exact algorithm for the Fréchet decomposition problem for general k when curves come from a fixed candidate set.
Cite as
Anne Driemel, Jan Höckendorff, Ioannis Psarros, and Christian Sohler. Time Series Decomposition Using the Fréchet Distance. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 96:1-96:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{driemel_et_al:LIPIcs.ESA.2026.96,
author = {Driemel, Anne and H\"{o}ckendorff, Jan and Psarros, Ioannis and Sohler, Christian},
title = {{Time Series Decomposition Using the Fr\'{e}chet Distance}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {96:1--96:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.96},
URN = {urn:nbn:de:0030-drops-272322},
doi = {10.4230/LIPIcs.ESA.2026.96},
annote = {Keywords: Time Series Analysis, Fr\'{e}chet Distance, Approximation Algorithms}
}
Document
The Price of Being Partial: Complexity of Partial Generalized Dominating Set on Bounded-Treewidth Graphs
Abstract
For fixed sets σ, ρ of non-negative integers, the (σ, ρ)-domination framework introduced by Telle [Nord. J. Comput. 1994] captures many classical graph problems. For a graph G, a (σ,ρ)-set is a set S of vertices such that for every v ∈ V(G), we have [(1)]
1) if v ∈ S, then |N(v) ∩ S| ∈ σ, and
2) if v ∉ S, then |N(v) ∩ S| ∈ ρ. Algorithms and lower bounds for the decision, optimization, and counting versions of finding (σ,ρ)-sets on bounded-treewidth graphs were systematically studied [van Rooij et al., ESA 2009][Focke et al., TALG 2025]. We initiate the study of a natural partial variant (σ,ρ)-MinParDomSet of the problem, in which the constraints given by σ, ρ need not be fulfilled for all vertices, but we want to find a set of size at most k that maximizes the number of vertices that are satisfied in the sense that they satisfy (1) and (2) above.
Our goal is to understand whether (σ,ρ)-MinParDomSet can be solved in the same running time as the nonpartial version, or whether it is strictly harder. Formally, we consider nonempty finite or simple cofinite sets σ and ρ (simple cofinite sets are of the form ℤ_{≥ c}), and we try to determine the smallest constant c_{σ,ρ} such that there is a c_{σ,ρ}^tw ⋅ n^O(1) time algorithm for the problem if a tree decomposition of width tw is given. We obtain matching upper and lower bounds on c_{σ,ρ} for every such fixed σ and ρ under the Primal Pathwidth Strong Exponential Time Hypothesis, and establish whether the partial problem is harder than the nonpartial variant. For some sets σ and ρ, the more general (σ,ρ)-MinParDomSet has the same complexity as the nonpartial special case (e.g., for Dominating Set), while for other choices, the partial version is significantly harder (e.g., for Perfect Code).
Cite as
Jakob Greilhuber and Dániel Marx. The Price of Being Partial: Complexity of Partial Generalized Dominating Set on Bounded-Treewidth Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 97:1-97:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{greilhuber_et_al:LIPIcs.ESA.2026.97,
author = {Greilhuber, Jakob and Marx, D\'{a}niel},
title = {{The Price of Being Partial: Complexity of Partial Generalized Dominating Set on Bounded-Treewidth Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {97:1--97:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.97},
URN = {urn:nbn:de:0030-drops-272330},
doi = {10.4230/LIPIcs.ESA.2026.97},
annote = {Keywords: Generalized Dominating Set, Partial Domination, Treewidth, Primal Pathwidth Strong Exponential Time Hypothesis}
}
Document
Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-η Deletion
Abstract
For a constant η ≥ 0, Pathwidth-η Deletion is the problem of deciding whether, for a given graph G and integer k, there is a set S ⊆ V(G) of size at most k such that the pathwidth of G - S is at most η. The problems Treewidth-η Deletion and Treedepth-η Deletion are defined similarly for the parameters treewidth and treedepth, respectively. A landmark result of Fomin et al. [FOCS, 2012] shows that, for any constant η, all three problems admit a kernel on O(k^c(η)) vertices, where c(η) is a constant depending on η.
Giannopoulou et al. [ACM TALG, 2017] show that, in some sense, this result is optimal for Treewidth-η Deletion: for η ≥ 2 and even when parameterizing by the size of a vertex cover M of the input graph, there is no kernel of size O(|M|^{(η+1)/2-ε}), for any ε > 0. Contrasting this result, they prove that Treedepth-η Deletion admits a uniform polynomial kernel, that is, a kernel of size O(k^c) for a constant c that is independent of η.
In comparison, the question whether Pathwidth-η Deletion admits a uniform polynomial kernel has been neglected in the literature. As treewidth and pathwidth tend to behave similarly, it is natural to expect that no uniform polynomial kernel exists when parameterizing by the size of a vertex cover. Surprisingly, we show this not to be the case. More concretely, we prove the existence of a uniform polynomial kernel for Pathwidth-η Deletion when parameterizing by
1) the solution size k plus the size of a set M such that G - M has bounded treedepth,
2) the (vertex-deletion) distance to pathwidth-1 graphs,
3) the distance to the class of graphs with treedepth at most η + 1.
This pinpoints a striking difference between Pathwidth-η Deletion and Treewidth-η Deletion and leads us to conjecture that Pathwidth-η Deletion admits a uniform polynomial kernel when parameterizing by the solution size k.
Cite as
Ahmed Ghazy, Jakob Greilhuber, Tim A. Hartmann, and Roohani Sharma. Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-η Deletion. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 98:1-98:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ghazy_et_al:LIPIcs.ESA.2026.98,
author = {Ghazy, Ahmed and Greilhuber, Jakob and Hartmann, Tim A. and Sharma, Roohani},
title = {{Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-\eta Deletion}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {98:1--98:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.98},
URN = {urn:nbn:de:0030-drops-272343},
doi = {10.4230/LIPIcs.ESA.2026.98},
annote = {Keywords: Uniform kernelization, pathwidth deletion, treedepth, elimination distance}
}
Document
Online Approximate Circular Pattern Matching in Small Space
Abstract
In approximate circular pattern matching the goal is to compute all approximate occurrences of all rotations of a pattern P in a text T. We study this problem under the two most fundamental string distance metrics, the Hamming distance and the edit distance, in the setting where the text arrives online and the available space is limited. Specifically, we wish to report each ending position j of an approximate occurrence before symbol T[j+1] arrives, using sublinear space on top of having read-only access to P and (the seen prefix of) T. For both variants, we present algorithms that use O(poly(k)) extra space and process each arriving symbol in O(poly(k)) time. Notably, with an overhead, our algorithms can be lifted to the asymmetric streaming setting, where we only have read-only access to the pattern for free and account for all extra space.
Cite as
Panagiotis Charalampopoulos, Taha El Ghazi, Jonas Ellert, Paweł Gawrychowski, and Tatiana Starikovskaya. Online Approximate Circular Pattern Matching in Small Space. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 99:1-99:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{charalampopoulos_et_al:LIPIcs.ESA.2026.99,
author = {Charalampopoulos, Panagiotis and El Ghazi, Taha and Ellert, Jonas and Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana},
title = {{Online Approximate Circular Pattern Matching in Small Space}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {99:1--99:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.99},
URN = {urn:nbn:de:0030-drops-272358},
doi = {10.4230/LIPIcs.ESA.2026.99},
annote = {Keywords: Small-space algorithms, approximate pattern matching, circular pattern matching}
}
Document
Graph Scheduling with Group Completion Times
Abstract
In the Graph Scheduling problem we schedule a given multiset of edges on discrete time steps, such that at each step the set of edges forms a matching. The goal is to minimize the sum of weighted group completion times, where a group is a set of edges and it completes when the last edge has been scheduled. Two popular variants of this problem are Coflow Scheduling and Data Migration. Our main result is extending a recent iterated rounding approach from Coflow Scheduling, roughly corresponding to the bipartite case, to the general Graph Scheduling problem. This yields an essentially tight (2+ε)-approximation for the asymptotic setting where OPT is assumed to be large. For this we rely on polyhedral techniques from general matching, namely odd-set inequalities, and graph theoretical results on edge colorings in multigraphs. The state-of-the-art approximation algorithm for Data Migration is a (1+ϕ)-approximation that improves when OPT is small. Taking the best of this and our main result, we obtain an improvement of the approximation rate for Data Migration in any regime.
Cite as
Lars Rohwedder and Leander Schnaars. Graph Scheduling with Group Completion Times. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 100:1-100:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rohwedder_et_al:LIPIcs.ESA.2026.100,
author = {Rohwedder, Lars and Schnaars, Leander},
title = {{Graph Scheduling with Group Completion Times}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {100:1--100:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.100},
URN = {urn:nbn:de:0030-drops-272369},
doi = {10.4230/LIPIcs.ESA.2026.100},
annote = {Keywords: Graph Scheduling, Approximation Algorithms, Iterated Rounding}
}
Document
Tight Better-Than-Worst-Case Bounds for Element Distinctness and Set Intersection
Abstract
The element distinctness problem takes as input a list I of n values from a totally ordered universe, where pairwise comparisons between values are allowed, and the goal is to decide whether I contains any duplicates. It is a well-studied problem with a classical worst-case Ω(n log n) comparison-based lower bound by Fredman [TCS'76]. At first glance, this lower bound appears to rule out any algorithm more efficient than the naive approach of sorting I and comparing adjacent elements. However, upon closer inspection, the Ω(n log n) bound is overly pessimistic. For instance, if I contains n/2 identical elements, a median-finding algorithm will, regardless of the input order, find a duplicate in linear time. This raises a natural question: Are there comparison-based lower bounds for element distinctness that are sensitive to the amount of duplicates in the input instance?
To address this question, we derive instance-specific lower bounds. For any input instance I, we represent the combinatorial structure of the duplicates in I by an undirected graph G(I) that connects identical elements. Each such graph G is a union of cliques, and we study algorithms by their worst-case running time over all inputs I' with G(I') ≅ G. We establish an adversarial lower bound showing that, for any deterministic algorithm 𝒜, there exists a graph G and an algorithm 𝒜' that, for all inputs I with G(I) ≅ G, is a factor O(log log n) faster than 𝒜. Consequently, no deterministic algorithm can be o(log log n)-competitive for all graphs G. We complement this with an O(log log n)-competitive deterministic algorithm, thereby obtaining tight bounds for element distinctness that go beyond classical worst-case analysis. Subsequently, we study the related problem of set intersection. We show that no deterministic set intersection algorithm can be o(log n)-competitive, and provide an O(log n)-competitive deterministic algorithm. We find it interesting and surprising to discover tight O(log log n)-competitive bounds for element distinctness. Moreover, we find the separation between element distinctness and the set intersection problem unexpected.
Cite as
Ivor van der Hoog, Eva Rotenberg, and Daniel Rutschmann. Tight Better-Than-Worst-Case Bounds for Element Distinctness and Set Intersection. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 101:1-101:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vanderhoog_et_al:LIPIcs.ESA.2026.101,
author = {van der Hoog, Ivor and Rotenberg, Eva and Rutschmann, Daniel},
title = {{Tight Better-Than-Worst-Case Bounds for Element Distinctness and Set Intersection}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {101:1--101:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.101},
URN = {urn:nbn:de:0030-drops-272372},
doi = {10.4230/LIPIcs.ESA.2026.101},
annote = {Keywords: Comparison-based analysis, set intersection, universal optimality}
}
Document
On Small Pair Decompositions for Point Sets
Abstract
We study the problem of computing a minimum-size Well-Separated Pair Decomposition (WSPD) of a given point set. We obtain the following results for the minimum-size WSPD: (1) a constant-factor approximation in doubling metrics, (2) a simple 3-approximation in ℝ, and (3) an NP-hardness proof in ℝ². We also provide an optimal output-sensitive runtime for the algorithm in doubling metrics and an implementation of the 3-approximation algorithm in ℝ. Furthermore, we introduce a new pair-decomposition for point sets in a metric space. It is defined using a relaxed requirement that, for all pairs {X,Y} in the decomposition, all the distances of pairs of points in X × Y are equal up to a factor in [1 ± ε]. Surprisingly, we show that in a general metric space, one can compute such a decomposition of size O(n/ε log n), which is dramatically smaller than the Θ(n²) bound for WSPDs. For a point set in ℝ^d, the bound improves to O(d n/ε log 1/ε).
Cite as
Kevin Buchin, Jacobus Conradi, Sariel Har-Peled, Antonia Kalb, Abhiruk Lahiri, Lukas Plätz, Carolin Rehs, and Sampson Wong. On Small Pair Decompositions for Point Sets. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 102:1-102:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{buchin_et_al:LIPIcs.ESA.2026.102,
author = {Buchin, Kevin and Conradi, Jacobus and Har-Peled, Sariel and Kalb, Antonia and Lahiri, Abhiruk and Pl\"{a}tz, Lukas and Rehs, Carolin and Wong, Sampson},
title = {{On Small Pair Decompositions for Point Sets}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {102:1--102:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.102},
URN = {urn:nbn:de:0030-drops-272380},
doi = {10.4230/LIPIcs.ESA.2026.102},
annote = {Keywords: Well-separated pair decomposition, Semi-separated pair decomposition, Approximation algorithms, Doubling metrics, Computational geometry}
}
Document
A Faster Polynomial-Space Algorithm for Hamiltonian Cycle Parameterized by Treedepth
Abstract
A large number of NP-hard graph problems can be solved in c^w n^𝒪(1) time and space when the input graph is provided together with a tree decomposition of width w. Moreover, assuming the Strong Exponential-Time Hypothesis (SETH) we have essentially matching lower bounds for many such problems. The main drawback of these results is that the corresponding dynamic programming algorithms use exponential space, which makes them infeasible for larger w, and there is some evidence that this cannot be avoided.
This motivates using somewhat more restrictive structure/decompositions of the graph to also get good (exponential) dependence on the corresponding parameter but use only polynomial space. A number of papers have contributed to this quest by studying problems relative to treedepth, and have obtained fast polynomial space algorithms, often matching the dependence on treewidth in the time bound. E.g., a number of connectivity problems could be solved by adapting the cut-and-count technique of Cygan et al. (FOCS 2011, TALG 2022) to treedepth, but this excluded well-known path and cycle problems such as Hamiltonian Cycle (Hegerfeld and Kratsch, STACS 2020).
Recently, Nederlof et al. (SIDMA 2023) showed how to solve Hamiltonian Cycle, and several related problems, in 5^τ n^𝒪(1) randomized time and polynomial space when provided with an elimination forest of depth τ. We present a faster (also randomized) algorithm, running in 4^τ n^𝒪(1) time and polynomial space, for the same set of problems. We use ordered pairs of what we call consistent matchings, rather than perfect matchings in an auxiliary graph, to get the improved time bound.
Cite as
Stefan Kratsch. A Faster Polynomial-Space Algorithm for Hamiltonian Cycle Parameterized by Treedepth. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 103:1-103:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kratsch:LIPIcs.ESA.2026.103,
author = {Kratsch, Stefan},
title = {{A Faster Polynomial-Space Algorithm for Hamiltonian Cycle Parameterized by Treedepth}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {103:1--103:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.103},
URN = {urn:nbn:de:0030-drops-272397},
doi = {10.4230/LIPIcs.ESA.2026.103},
annote = {Keywords: Parameterized algorithms, Hamiltonian cycle, polynomial space, treedepth}
}
Document
Faster Exponential-Time Approximate Counting via Bounded Self-Reductions
Abstract
We give faster exponential-time randomised approximation algorithms for counting problems where polynomial-time approximation is unavailable and exact exponential-time counting remains expensive. For general n-vertex graphs, our independent-set counter runs in O^{∗}(1.1869ⁿ) time, improving the previous O^*(1.2041ⁿ) general-graph bound. For n-variable #2-SAT, we obtain an O^*(1.2373ⁿ)-time approximation algorithm, narrowly below Wahlström’s currently cited O^*(1.2377ⁿ) variable-parameter exact bound.
The new algorithmic point is to take the square root after decomposition. For a single bounded unweighted self-reduction with f(x) positive leaves and recursion-compatible upper bound b(x), an enumerate-or-sample estimator gives an (ε,δ)-approximation in O^*(√{b(x)} ε^{-2}log(1/δ)) time. After preprocessing decomposes an input into many bounded cores, the combined estimator pays O^*(√{∑_i b_i(x_i)} ε^{-2} log (1/δ)) , rather than estimating the cores separately at cost ∑_i √{b_i(x_i)}.
The same conversion improves the bases for counting maximal cliques, minimal separators, and perfect matchings in subcubic graphs. Bounded unweighted self-reductions provide the formal language; at the level of counting classes, the resulting unweighted formulation has the same Karp closure as TotP. With explicit recursion-tree access, the framework yields black-box quantum speed-ups.
Cite as
Katie Clinch, Serge Gaspers, Simon Mackenzie, and Qi Wang. Faster Exponential-Time Approximate Counting via Bounded Self-Reductions. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 104:1-104:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{clinch_et_al:LIPIcs.ESA.2026.104,
author = {Clinch, Katie and Gaspers, Serge and Mackenzie, Simon and Wang, Qi},
title = {{Faster Exponential-Time Approximate Counting via Bounded Self-Reductions}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {104:1--104:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.104},
URN = {urn:nbn:de:0030-drops-272400},
doi = {10.4230/LIPIcs.ESA.2026.104},
annote = {Keywords: Approximate counting, exponential-time algorithms, randomised algorithms, #Independent-Set, #2-SAT, self-reducibility, TotP}
}
Document
String Matching in (Block) Graphs: A Full Classification by Walk Length
Abstract
We consider directed graphs in which the nodes are labeled with strings. A walk in such a graph naturally corresponds to the concatenation of the visited nodes' labels. These graphs are widely used in bioinformatics to compactly describe large collections of highly similar genomes. Given such a graph G = (V,E) and a pattern of length m, we seek a walk whose corresponding string has an occurrence of the pattern. We call this the SMLG problem. Amir et al. [J. Algorithms, 2000] showed that SMLG can be solved in 𝒪(m |E| + N) time, where N is the total length of all node labels. Equi et al. [ACM Trans. Algorithms, 2023] showed that this is essentially optimal (under SETH).
The existing lower bound assumes that the sought walk is of length Θ(|V|). Thus, we might be able to bypass this lower bound by restricting the walk length to b-1, which naturally reduces to having as input a directed graph whose set of nodes is partitioned into b blocks. Then, we seek a walk in this graph that starts in the first block and ends in the last block. We call this the b-SMBG problem. Equi et al. [Algorithmica, 2023] showed that, if we impose no restriction on b, the existing algorithm of Amir et al. is essentially optimal for b-SMBG (again under SETH). We provide a more fine-grained classification that essentially settles the complexity of b-SMBG parameterized by b:
1) For b = 2, Pissis [SOSA 2025] already provided a simple 𝒪(m + |E|+N)-time algorithm.
2) We design a new 𝒪̃(m + |E| + N)-time algorithm for b = 3. As a direct implication of this result, the SMLG problem for b ≤ 3 (walks of length at most 2) also admits near-linear-time complexity.
3) There is no 𝒪((m |E|)^{1-ε} + N)-time combinatorial algorithm, for any b ≥ 4 and ε > 0.
4) There is an algorithm working in 𝒪(max(|V|, m)^ω+N) time, where ω is the matrix multiplication exponent, which is conditionally optimal for graphs with b ≥ 4 blocks.
5) Under SETH, no 𝒪((m |E|)^{1-ε} + N)-time algorithm exists, for any b = ω(log |V|) and ε > 0.
Although our motivation is primarily of a theoretical nature, we stress that our algorithms are simple to implement. As such, they may contribute to practical advancements in applications where the SMLG problem is an important primitive, such as in the analysis of pangenome graphs.
Cite as
Sebastian Angrick, Ben Bals, Paweł Gawrychowski, Solon P. Pissis, and Yuki Yonemoto. String Matching in (Block) Graphs: A Full Classification by Walk Length. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 105:1-105:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{angrick_et_al:LIPIcs.ESA.2026.105,
author = {Angrick, Sebastian and Bals, Ben and Gawrychowski, Pawe{\l} and Pissis, Solon P. and Yonemoto, Yuki},
title = {{String Matching in (Block) Graphs: A Full Classification by Walk Length}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {105:1--105:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.105},
URN = {urn:nbn:de:0030-drops-272417},
doi = {10.4230/LIPIcs.ESA.2026.105},
annote = {Keywords: string algorithms, pattern matching, lower bounds, fine-grained complexity}
}
Document
Instance Optimal and Universally Optimal Bounds for Imprecise Pareto Fronts
Abstract
In the imprecise geometry model, the input is a family of regions F = (R₁, R₂, …,R_n), each containing a point p_i ∈ R_i. The task is then to compute some function of the points p₁,p₂,… p_n, in our case an implicit representation of their Pareto front. To this end, one may query a region R_i to retrieve its contained point p_i ∈ R_i. In this model, efficiency is interpreted in two ways: minimizing (i) the number of retrievals, and (ii) the computation time both for preprocessing, and the execution of the query stage, i.e. for computing which points to query and constructing the output.
We present an algorithm to construct (an implicit representation of) the Pareto front for possibly overlapping rectangles, that is instance-optimal with respect to the number of retrievals. This means that for every fixed input (F, P), there is no algorithm that retrieves asymptotically fewer regions to compute the output. This is a strong algorithmic quality, as it means that our algorithm is competitive even to clairvoyant algorithms which only have to verify the correctness of a correct guess. In terms of algorithmic running time, instance-optimality is provably unobtainable. We instead present an algorithm which is within a log n-factor of instance optimality. This generalizes earlier results which assumed the regions to not overlap, at only a minor cost in running time.
For unit squares, we present an algorithm that is not only instance optimal in the number of retrievals, but also universally optimal in terms of running time. This means that for any fixed set of regions F, no algorithm has a better worst-case running time for all possible point sets P. Thus, this work presents the first universally optimal algorithm for overlapping planar input. Compared to previous work, our result improves the degree to which the input regions may overlap, the preprocessing time, the number of retrievals, and the running time.
Cite as
Sarita de Berg, Nynne Maria Foldager Bække, Frida Astrup Eriksen, Ivor van der Hoog, Eva Rotenberg, and Daniel Rutschmann. Instance Optimal and Universally Optimal Bounds for Imprecise Pareto Fronts. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 106:1-106:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{deberg_et_al:LIPIcs.ESA.2026.106,
author = {de Berg, Sarita and B{\ae}kke, Nynne Maria Foldager and Eriksen, Frida Astrup and van der Hoog, Ivor and Rotenberg, Eva and Rutschmann, Daniel},
title = {{Instance Optimal and Universally Optimal Bounds for Imprecise Pareto Fronts}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {106:1--106:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.106},
URN = {urn:nbn:de:0030-drops-272426},
doi = {10.4230/LIPIcs.ESA.2026.106},
annote = {Keywords: Pareto front, imprecise geometry, instance optimality, universal optimality, preprocessing model, partial information}
}
Document
Revisiting Graph Modification via Disk Scaling: From One Radius to Interval-Based Radii
Abstract
For a fixed graph class Π, the goal of Π-Modification is to transform an input graph G into a graph H ∈ Π using at most k modifications. Vertex and edge deletions are common operations, and their (parameterized) complexity for various Π is well-studied. Classic graph modification operations such as edge deletion do not consider the geometric nature of intersection graphs such as (unit) disk graphs. This led Fomin et al. [ITCS' 25] to introduce scaling as a geometric graph modification operation for unit disk graphs: For a given radius r, each modified disk will be rescaled to radius r.
In this paper, we generalize their model by allowing rescaled disks to choose a radius within a given interval [r_min, r_max] and study the (parameterized) complexity (with respect to k) of the corresponding problem Π-Scaling. We show that Π-Scaling is in XP for every graph class Π that can be recognized in polynomial time. Furthermore, we show that Π-Scaling: (1) is NP-hard and FPT for cluster graphs, (2) can be solved in polynomial time for complete graphs, and (3) is W[1]-hard for connected graphs. In particular, (1) and (2) answer open questions of Fomin et al. and (3) generalizes the hardness result for their variant where the set of scalable disks is restricted.
Cite as
Thomas Depian and Frank Sommer. Revisiting Graph Modification via Disk Scaling: From One Radius to Interval-Based Radii. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 107:1-107:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{depian_et_al:LIPIcs.ESA.2026.107,
author = {Depian, Thomas and Sommer, Frank},
title = {{Revisiting Graph Modification via Disk Scaling: From One Radius to Interval-Based Radii}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {107:1--107:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.107},
URN = {urn:nbn:de:0030-drops-272431},
doi = {10.4230/LIPIcs.ESA.2026.107},
annote = {Keywords: NP-hardness, Parameterized Complexity, Unit Disk Graphs, Cluster Graphs, Connected Graphs}
}
Document
Maximum Coverage k-Antichains and Chains: A Greedy Approach
Abstract
Given an acyclic digraph G = (V,E) and a positive integer k, the problem of Maximum Coverage k-Antichains (resp. Chains) denoted as MA-k (resp. MC-k) asks to find k sets of pairwise unreachable vertices, known as antichains (resp. k subsequences of paths, known as chains), maximizing the number α_k (resp. β_k) of vertices covered by these antichains (resp. chains). While MC-k was solved in almost optimal |E|^{1+o(1)} time [Kogan and Parter, ICALP'22], the fastest algorithms for MA-k are a (k|E|)^{1+o(1)}-time solution and a |E|^{1+o(1)}-time 1/2 approximation [Kogan and Parter, ESA'24].
We simplify and improve previous results. Specifically, we obtain the following for MA-k:
- An algorithm running in |E|^{1+o(1)} time, and an algorithm running in parameterized near-linear Õ(α_k |E|) time. Our algorithms are simple solutions exploiting a paths-based proof of the Greene-Kleitman theorems leveraged by the greedy algorithm for set cover as well as recent advances in fast algorithms for flows and shortest paths.
- An approximation algorithm running in parameterized linear time O(α₁²|V| + (α₁+k)|E|) with approximation ratio of (1-1/e) > 0.63 > 1/2, beating the state-of-the-art 1/2 approximation. Our solution uses greedy for antichains and a simple strategy to amortize the cost of computing consecutive maximum antichains. Additionally, we obtain analogous results for MC-k as well as the corresponding dual problems derived from the Greene-Kleitman theorems, which might be of independent interest.
We complement these results with two examples (one for chains and one for antichains) showing that, for every k ≥ 2, greedy misses the tight 1/e portion of the optimal coverage for chains, and a 1/4 portion for antichains. We also show that greedy is a Ω(log{|V|}) factor away from minimality when required to cover all vertices: previously unknown for sets of chains or antichains.
Cite as
Manuel Cáceres, Andreas Grigorjew, Wanchote Po Jiamjitrak, and Alexandru I. Tomescu. Maximum Coverage k-Antichains and Chains: A Greedy Approach. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 108:1-108:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{caceres_et_al:LIPIcs.ESA.2026.108,
author = {C\'{a}ceres, Manuel and Grigorjew, Andreas and Jiamjitrak, Wanchote Po and Tomescu, Alexandru I.},
title = {{Maximum Coverage k-Antichains and Chains: A Greedy Approach}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {108:1--108:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.108},
URN = {urn:nbn:de:0030-drops-272442},
doi = {10.4230/LIPIcs.ESA.2026.108},
annote = {Keywords: Maximum coverage antichains, maximum coverage chains, directed acyclic graph, minimum cost flow, greedy set cover, parameterized algorithms, approximation algorithms}
}
Document
Dynamic Dominating Set in Uniformly Sparse Graphs
Abstract
In the dynamic minimum dominating set (MDS) problem, the goal is to efficiently maintain an approximate MDS in an n-vertex graph with vertex costs in [1/C,1] undergoing edge insertions and deletions. In STACS'19 [Niklas Hjuler et al., 2019] it was shown that an O(log n)-approximate MDS can be maintained in unweighted graphs with O(Δ ⋅ log n) update time, where Δ is an upper bound on the maximum degree throughout the update sequence, and in STOC'23 [Solomon and Uzrad, 2023] this was extended to weighted graphs and improves the approximation guarantee to (1+ε)ln Δ.
Is it possible to achieve poly(log n) update time without any dependence on Δ, for any nontrivial graph family? This basic question has remained open even in forests and even for unweighted instances. The arboricity α = α(G) of a graph G is the minimum number of edge-disjoint forests whose union is G, and is a standard measure of sparsity. While α is bounded by Δ in any graph, various real-world graph families exhibit a significant gap between α and Δ.
In this work, we show that one can maintain an O(α)-approximate MDS with update time O(α⋅log(Cn)), for dynamic graphs whose arboricity is bounded by α throughout the update sequence. This replaces the dependence on Δ in prior update bounds with α, while also improving the approximation guarantee for bounded-arboricity graphs. In particular, for any graph family of constant arboricity, such as planar graphs, bounded treewidth graphs, and more generally graphs excluding a fixed minor, our algorithm gives an O(1)-approximation with O(log (Cn)) update time. To achieve this result, our algorithm departs from prior greedy-based approaches, relying instead on the primal-dual framework and new structural insights specific to bounded arboricity graphs.
Cite as
Anton Bukov and Shay Solomon. Dynamic Dominating Set in Uniformly Sparse Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 109:1-109:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bukov_et_al:LIPIcs.ESA.2026.109,
author = {Bukov, Anton and Solomon, Shay},
title = {{Dynamic Dominating Set in Uniformly Sparse Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {109:1--109:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.109},
URN = {urn:nbn:de:0030-drops-272455},
doi = {10.4230/LIPIcs.ESA.2026.109},
annote = {Keywords: dynamic algorithms, minimum dominating set, arboricity, sparse graphs, approximation algorithms, primal-dual methods}
}
Document
Log-Concavity and Tunneling: Quantum Adiabatic Algorithm for Convex Functions (With a Spike)
Abstract
Quantum tunneling is expected to provide a computational speedup in quantum computing, a phenomenon that Adiabatic Quantum Optimization (AQO) aims to leverage. While some academic proofs of concept have been studied, such as the "Hamming weight with a spike" (HWS) problem, the algorithmic gains of this effect remain underexplored. In this work we extend the analysis underlying HWS to more general potentials.
In the first half of the work, we establish (discrete) log-concavity of the ground state as a key structural property in this context. We devise a framework for establishing log-concavity of the ground state for a large family of discrete, 1-dimensional Schrödinger operators. The family includes convex potentials, but also certain potentials with local minima. In the convex case, this provides a discrete version of a continuous result by Brascamp and Lieb [Brascamp and Lieb, 1976]. We demonstrate the utility of our result by establishing new spectral gap bounds, going beyond related results by Jarret and Jordan [Jarret and Jordan, 2014] for convex potentials.
In the second half of the work, we use our results on log-concavity to extend the perturbative analysis of HWS by Reichardt [Reichardt, 2004] to the larger family of potentials with log-concave ground state. As a concrete instantiation, we use our result to extend the HWS analysis from a linear potential (which is exactly solvable) to a quadratic potential (which is no longer solvable). Our result strongly suggests the broader applicability of tunneling to convex potentials with spikes.
Cite as
Arthur Braida, Elie Bermot, and Simon Apers. Log-Concavity and Tunneling: Quantum Adiabatic Algorithm for Convex Functions (With a Spike). In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 110:1-110:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{braida_et_al:LIPIcs.ESA.2026.110,
author = {Braida, Arthur and Bermot, Elie and Apers, Simon},
title = {{Log-Concavity and Tunneling: Quantum Adiabatic Algorithm for Convex Functions (With a Spike)}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {110:1--110:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.110},
URN = {urn:nbn:de:0030-drops-272465},
doi = {10.4230/LIPIcs.ESA.2026.110},
annote = {Keywords: Quantum adiabatic computing, Convex optimization}
}
Document
A Strongly-Subquadratic (3+ε)-Approximation for the Fréchet Distance for Paths in Metric Spaces
Abstract
The Fréchet distance is a well-studied distance measure for paths in a metric space. It is mostly studied for paths in d-dimensional Euclidean space. Here, computing the Fréchet distance between two polylines takes time roughly quadratic in the number of vertices. Assuming the strong exponential time hypothesis (SETH), it cannot be approximated to within a factor less than 3 in strongly-subquadratic time. Recently, it was shown that for any ε > 0, there exists a randomized algorithm that can compute a (7+ε)-approximation in strongly-subquadratic expected time [Cheng, Huang, and Zhang; STOC'25]. For polylines with n and m vertices in a Euclidean space of constant dimension, where n ≥ m, their algorithm takes O(nm^{0.99} log(n/ε)) time in expectation.
We present a deterministic approximation algorithm that significantly improves upon the approximation factor and running time. Specifically, our algorithm computes a (3+ε)-approximation in O(nm^{2/3} log n ⋅ log (1/(ε) log n)) time. Our algorithm nearly matches the conditional lower bound on the approximation factor implied by SETH. For polylines in ℝ, we present a 3-approximation algorithm that runs in O(nm^{2/3} log^{5/3} n) time, and exactly matches the conditional lower bound.
For our results, we introduce a general strongly-subquadratic time 3-approximate decision algorithm. This algorithm makes no assumptions on the ambient metric space, and relies only on standard assumptions on the so-called free space of the input paths. Under some mild assumptions, our decision algorithm leads to a (3+ε)-approximation algorithm in general metric spaces. These assumptions hold automatically for polylines in any metric space (ℝ^d, L_p) with p ≥ 1.
Cite as
Thijs van der Horst and Tim Ophelders. A Strongly-Subquadratic (3+ε)-Approximation for the Fréchet Distance for Paths in Metric Spaces. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 111:1-111:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vanderhorst_et_al:LIPIcs.ESA.2026.111,
author = {van der Horst, Thijs and Ophelders, Tim},
title = {{A Strongly-Subquadratic (3+\epsilon)-Approximation for the Fr\'{e}chet Distance for Paths in Metric Spaces}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {111:1--111:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.111},
URN = {urn:nbn:de:0030-drops-272478},
doi = {10.4230/LIPIcs.ESA.2026.111},
annote = {Keywords: Fr\'{e}chet distance, path similarity, approximation algorithm}
}
Document
One-Exact Approximate Pareto Sets for APX-Hard Multiobjective Problems
Abstract
There are several frameworks to compute approximate Pareto sets for multiobjective optimization (MOO) problems. An approximate Pareto set that is even precise in one specific objective is called one-exact. In such frameworks, some auxiliary single-objective problem is considered, for which a problem-specific oracle is required. However, often these oracles are required to be a PTAS or even FPTAS. As such, these frameworks are only applicable to "simple" MOO problems that allow for such strong oracles to exist. They are inapplicable whenever the auxiliary problem is APX-hard.
We propose a general framework that, for a (possibly even non-constant) accuracy vector β = (β_2, … , β_d) and any ε > 0, computes polynomially sized, one-exact (1,(1 + ε)β_2,… ,(1 + ε)β_d)-Pareto sets for d-objective minimization problems. The framework is analogously applicable to maximization and mixed MOO problems. The running time is polynomial in the time required to solve our auxiliary problem β-RelaxedDualRestrict. Notably, these guarantees hold even if β-RelaxedDualRestrict is APX-hard. We further show that if β-RelaxedDualRestrict cannot be solved in polynomial time, then no (1,β_2, … ,β_d)-Pareto set can be computed in polynomial time. For biobjective problems, our framework even yields a (1,(1 + ε)β_2)-Pareto set of at most 𝒪(log β₂) times the size of the minimum-size one-exact (1,(1 + ε)β_2)-Pareto set. We show that this relative size guarantee is asymptotically tight. Further, we present techniques to obtain suitable oracles for β-RelaxedDualRestrict from existing (single-objective) approximation algorithms, including a general "re-randomization" method that may be of independent interest. Using these, we obtain new best approximation guarantees for several established MOO problems, including Spanner, Clique, TSP, Facility Location, and Set Cover problems.
Cite as
Fritz Bökler, Markus Chimani, and Henning Jasper. One-Exact Approximate Pareto Sets for APX-Hard Multiobjective Problems. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 112:1-112:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bokler_et_al:LIPIcs.ESA.2026.112,
author = {B\"{o}kler, Fritz and Chimani, Markus and Jasper, Henning},
title = {{One-Exact Approximate Pareto Sets for APX-Hard Multiobjective Problems}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {112:1--112:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.112},
URN = {urn:nbn:de:0030-drops-272485},
doi = {10.4230/LIPIcs.ESA.2026.112},
annote = {Keywords: multiobjective optimization, approximate Pareto sets, scalarization}
}
Document
On the Adversarial Robustness of Online Importance Sampling
Abstract
Online sampling algorithms, which irrevocably either keep or discard each stream element, have seen wide use in streaming due to their efficiency and simplicity. Braverman et al. [NeurIPS 2021] claimed that online importance-sampling algorithms, where elements are sampled proportionally to some notion of importance, succeed with high probability when their input stream is adaptively chosen by an adversary. Unfortunately, their results on importance sampling do not beat trivial bounds in many instances. Therefore, we reopen the question about the robustness of online importance sampling to adaptive inputs. This question was also addressed by Jiang, Peng and Weinstein [FOCS 2023] for the problem of 𝓁₂-subspace embedding.
We develop a unified framework for online importance sampling algorithms in adaptive streams. This framework offers two main advantages: first, it provides better bounds than prior work, and second, it unifies and simplifies the analysis of importance sampling algorithms across different problems. We then leverage the framework to provide algorithms for cut sparsification in hypergraphs and 𝓁_p-subspace embeddings in adaptive streams whose space complexity nearly matches the oblivious case (non-adaptive).
Cite as
Yotam Kenneth-Mordoch and Shay Sapir. On the Adversarial Robustness of Online Importance Sampling. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 113:1-113:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kennethmordoch_et_al:LIPIcs.ESA.2026.113,
author = {Kenneth-Mordoch, Yotam and Sapir, Shay},
title = {{On the Adversarial Robustness of Online Importance Sampling}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {113:1--113:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.113},
URN = {urn:nbn:de:0030-drops-272491},
doi = {10.4230/LIPIcs.ESA.2026.113},
annote = {Keywords: Importance sampling, Adversarial robustness, Streaming algorithms, Coresets, Cut sparsification, Subspace embedding}
}
Document
Edge Geography is XNLP-hard for Pathwidth and in XP for Tree-Partition Width
Abstract
Directed Edge Geography and Undirected Edge Geography are classical PSPACE-complete two-player graph games in which players alternately make moves along edges, deleting each one after use; the first player unable to move loses. We prove that both problems are XNLP-hard when parameterized by pathwidth, addressing a question raised by Bodlaender over 30 years ago. On the positive side, we observe that Directed Edge Geography is fixed-parameter tractable when parameterized by treewidth and maximum degree. We also prove that both problems are in XP on simple graphs when parameterized by tree-partition width. These results develop modern lower-bound and decomposition-based algorithmic methods for width-based questions in PSPACE-complete graph games.
Cite as
Thobias Kvalvik Høivik and Erlend Raa Vågset. Edge Geography is XNLP-hard for Pathwidth and in XP for Tree-Partition Width. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 114:1-114:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hoivik_et_al:LIPIcs.ESA.2026.114,
author = {H{\o}ivik, Thobias Kvalvik and V\r{a}gset, Erlend Raa},
title = {{Edge Geography is XNLP-hard for Pathwidth and in XP for Tree-Partition Width}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {114:1--114:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.114},
URN = {urn:nbn:de:0030-drops-272504},
doi = {10.4230/LIPIcs.ESA.2026.114},
annote = {Keywords: Geography games, graph games, pathwidth, parameterized complexity, XNLP-hardness, PSPACE-complete games, tree-partition width}
}
Document
Tight Bounds for Clique-Packing Parameterized by Clique-Width
Abstract
In the d-Clique Packing problem, given a graph G and an integer t, we need to decide whether G contains a set of t pairwise vertex-disjoint cliques of size d each. This generalizes Triangle Packing and it is NP-complete for all d ≥ 3. For each such d, we show how to solve the problem in n^𝒪(k^{d-1}) time where k is the clique-width of the graph (with a k-expression of G given in the input). We complement this by showing that, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem in n^o(k^{d-1}) time for any fixed d ≥ 3, already for the special case of seeking a partition into cliques of size d. Our proof also entails W[1]-hardness of d-Clique Packing (and d-Clique Partitioning) parameterized by clique-width for each d ≥ 3. Our work continues a series of results on ETH-tight bounds for fundamental graph problems started by Fomin et al. (SICOMP 2010+2014) who obtained tight bounds for Max-Cut and Edge Dominating Set.
Cite as
Narek Bojikian and Stefan Kratsch. Tight Bounds for Clique-Packing Parameterized by Clique-Width. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 115:1-115:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bojikian_et_al:LIPIcs.ESA.2026.115,
author = {Bojikian, Narek and Kratsch, Stefan},
title = {{Tight Bounds for Clique-Packing Parameterized by Clique-Width}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {115:1--115:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.115},
URN = {urn:nbn:de:0030-drops-272519},
doi = {10.4230/LIPIcs.ESA.2026.115},
annote = {Keywords: Parameterized complexity, triangle packing, clique packing, clique-width}
}
Document
Dimension Reduction for Curves: Simplified and Generalized
Abstract
We revisit random projections for reducing the dimension of high-dimensional polygonal curves. Drawing from the toolbox of randomized linear algebra, we give a considerably simplified proof of the known O(ε^{-2} log(nm)) bound on the target dimension of a random projection that preserves the continuous Fréchet distance of polygonal curves up to a factor (1±ε). Our proof is based on the concept of sparse oblivious subspace embeddings. While previous techniques were limited to the case of the Fréchet distance, our techniques are fairly general and extend to all possible distance measures that involve the maximum, a sum or an integral over Euclidean distances between pairs of points on both input curves. We define a generalized dissimilarity measure for curves that includes several popular measures such as Fréchet, q-DTW, Hausdorff, etc. as special cases and show that the same dimension reduction technique works for this generalized dissimilarity measure. Finally, we apply the same framework for dimension reduction to piecewise linear surfaces, after extending the distance measure suitably to such surfaces.
Cite as
Matthijs Ebbens, Jie Lu, and Alexander Munteanu. Dimension Reduction for Curves: Simplified and Generalized. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 116:1-116:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ebbens_et_al:LIPIcs.ESA.2026.116,
author = {Ebbens, Matthijs and Lu, Jie and Munteanu, Alexander},
title = {{Dimension Reduction for Curves: Simplified and Generalized}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {116:1--116:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.116},
URN = {urn:nbn:de:0030-drops-272521},
doi = {10.4230/LIPIcs.ESA.2026.116},
annote = {Keywords: dimension reduction, Fr\'{e}chet distance, dynamic time warping, polygonal curves, piecewise linear surfaces}
}
Document
Sparse Oriented Spanners in Metric Spaces
Abstract
Oriented spanners were presented at ESA'23 as an extension of the well-researched geometric spanners: Given a set P of points in a metric space and an oriented graph G, the oriented dilation of two points p,q ∈ P is the length of the shortest closed walk in G containing p and q divided by the minimum perimeter triangle of p and q. G is called a t-spanner, if the maximum dilation over all pairs of points in P is at most t.
This paper presents the first constructions of sparse oriented spanners for metric spaces beyond the Euclidean space. Given an orientation of the complete graph (i.e. a tournament) with dilation t on n points that satisfies an additional short-cycle property, we show how to extract a (t+ε)-spanner with 𝒪(k) edges in 𝒪(kn²+T(n)) time, for any metric space admitting a well-separated pair decomposition with k pairs computable in T(n) time. We supplement this with an improved construction of tournaments for metric point sets, obtaining dilation 5/3. This improves the previous bound of 2 and approaches the lower bound of 1.5. Combined, for n points in a metric space with constant doubling dimension d, this yields a (5/3 + ε)-spanner with (1/ε)^{𝒪(d)}n edges computable in (1/ε)^𝒪(d) n³ time using 𝒪(n²) space. This improves the dilation over the (2+ε)-spanner for Euclidean point sets presented at SoCG’25 while applying to more general metric spaces.
Moreover, we generalize the known (2+ε)-spanner to doubling spaces. In particular, an oriented (2+ε)-spanner with 𝒪(ε^{-d} n) edges can be constructed in (1/ε)^𝒪(d) n log n time using 𝒪(ε^{-d} n) space.
Since the oriented dilation can be dominated by one pair of points, we also consider the oriented average dilation, which is the sum over the oriented dilation of all pairs of points divided by the number of pairs. While oriented (1+ε)-spanners do not exist for every point set, we present an algorithm that computes a spanner with average dilation 1+ε for point sets in a metric space of constant doubling dimension d: More concretely, our algorithm computes an oriented spanner with average dilation at most 1 + 𝒪(1/s) + s^𝒪(d)/n with s^𝒪(d) n edges in s^𝒪(d) n log n time using s^𝒪(d) n space, where s is any sufficiently large number that may depend on n.
Cite as
Sujoy Bhore, Ahmad Biniaz, Kevin Buchin, Jean-Lou De Carufel, Antonia Kalb, Anil Maheshwari, Saeed Odak, Carolin Rehs, and Michiel Smid. Sparse Oriented Spanners in Metric Spaces. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 117:1-117:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.ESA.2026.117,
author = {Bhore, Sujoy and Biniaz, Ahmad and Buchin, Kevin and De Carufel, Jean-Lou and Kalb, Antonia and Maheshwari, Anil and Odak, Saeed and Rehs, Carolin and Smid, Michiel},
title = {{Sparse Oriented Spanners in Metric Spaces}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {117:1--117:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.117},
URN = {urn:nbn:de:0030-drops-272532},
doi = {10.4230/LIPIcs.ESA.2026.117},
annote = {Keywords: spanner, oriented graph, dilation, doubling dimension, orientation, tournament, well-separated pair decomposition}
}
Document
Exploiting Spanning Trees for Directed Acyclicity
Abstract
We study the weighted case of the Maximum Acyclic Subgraph (MAS) problem, where each edge of a given directed graph has a positive weight assigned, and the task is to find a maximum-weight acyclic edge set. The famous and well-studied random ordering lower bound guarantees the existence of an acyclic set that gives at least the half of the total edge weight.
The maximum spanning tree (MaxST) guarantee, which is the weight of a maximum-weight acyclic subgraph of the underlying undirected graph of G, is another natural lower bound for the weight of an acyclic subgraph. A solution of this weight dominates the random ordering solution on instances where MaxST spans the most of the total edge weight.
Our main contribution are two parameterized algorithms that find acyclic subgraphs of total weight larger than the weight of the MaxST of G. Both our algorithms find a solution of total weight at least MaxST(G)+k, for a given integer k ≥ 0, or report that it does not exist, and
- First of our algorithms runs in time 2^{k^𝒪(1)}⋅|ℐ|^𝒪(1) and works when all weights are integers;
- Our second algorithm handles rational weights not less than 1, and its running time is upper-bounded by n^{k^𝒪(1)}⋅|ℐ|^𝒪(1). This positive result is rather surprising since solving MAS above the random ordering lower bound is NP-hard in the same rational weights scenario, when k = 1.
Our findings unravel intricate connections between structure of MaxSTs and directed cycles, use perfect graph theorem to tackle rational weights, and raise graph-theoretic questions that are interesting on their own. Of another importance, this is one of the few examples of positive "above guarantee" results for a weighted problem on directed graphs, especially for rational weights.
Cite as
Sergei Khargeliia and Danil Sagunov. Exploiting Spanning Trees for Directed Acyclicity. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 118:1-118:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{khargeliia_et_al:LIPIcs.ESA.2026.118,
author = {Khargeliia, Sergei and Sagunov, Danil},
title = {{Exploiting Spanning Trees for Directed Acyclicity}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {118:1--118:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.118},
URN = {urn:nbn:de:0030-drops-272540},
doi = {10.4230/LIPIcs.ESA.2026.118},
annote = {Keywords: parameterized algorithms, spanning tree, above guarantee parameterizations, perfect graph theorem, feedback arc set, maximum acyclic subgraph}
}
Document
All-Pairs kth Mincuts: Combinatorial and Structural Results
Abstract
Let G be an undirected graph on a set V of n vertices. For any non-empty subset A⊊ V, cut defined by A is the ordered pair (A,V\A). Suppose each cut is assigned a value, which is any arbitrary real number. Let u,v ∈ V be any pair of vertices. A cut is said to be a (u,v)-cut if it separates u and v. A (u,v)-cut of the minimum value is called a (u,v)-mincut. A 2nd (u,v)-mincut is a (u,v)-cut of second minimum value. We can define k-th mincut accordingly. We present the following results for the all-pairs k-th mincuts.
(1) Distinct Values of all-pairs k-th Mincuts: There exist k spanning trees on V such that for any pair (u,v), the value of k-th (u,v)-mincut is equal to the capacity of an edge on the (u,v)-path in one of the k trees. We also show a matching lower bound of Ω(min{kn,n²}). Our result generalizes the well-known result by Gomory and Hu [JSIAM 1961] stating that there are at most n-1 distinct values of all-pairs mincuts.
(2) Ancestor Trees for all-pairs k-th Mincuts: In 1991, Cheng and Hu [AOR 1991] invented a rooted binary tree, called ancestor tree, whose leaves are the vertices of the graph and each internal node stores a cut with the following property. For any pair (u,v), the cut stored at their lowest common ancestor (LCA) is a (u,v)-mincut. We introduce a tree called gen-ancestor tree, that generalizes the ancestor tree for k-th mincuts, and achieve the following result. There exists a set of 𝒪(klog n) gen-ancestor trees such that, for any pair (u,v), a k-th (u,v)-mincut is stored at the LCA of u and v in at least one of these trees.
(3) Data Structures: We present the following data structures for all-pairs k-th mincuts. (i) There exists an 𝒪(nlog n) space data structure that can report the value of 2nd (u,v)-mincut in 𝒪(log n) time for any given pair (u,v). We generalize this data structure for k-th mincuts with a factor of k² in the space and query time. (ii) There exists an 𝒪(kn² log n) space data structure that can report a k-th (u,v)-mincut (A,V\A) in 𝒪(|A|) time for any given pair (u,v). For any constant k, the bounds stated above match, up to a logarithmic factor, the best-known bounds guaranteed by the data structure for all-pairs mincuts (Cheng and Hu [AOR 1991]).
Cite as
Surender Baswana and Anupam Roy. All-Pairs kth Mincuts: Combinatorial and Structural Results. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 119:1-119:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{baswana_et_al:LIPIcs.ESA.2026.119,
author = {Baswana, Surender and Roy, Anupam},
title = {{All-Pairs kth Mincuts: Combinatorial and Structural Results}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {119:1--119:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.119},
URN = {urn:nbn:de:0030-drops-272553},
doi = {10.4230/LIPIcs.ESA.2026.119},
annote = {Keywords: mincut, second mincut, kth mincut, suboptimal cuts, compact structure, all pairs, multi terminal cuts, generalization of Gomory Hu tree, ancestor tree, generalization of all pairs mincuts}
}
Document
On the Assadi–Liu–Tarjan Auction Algorithm for Bipartite Matching: Simplification, Alternative Analysis, and Hard Instance
Abstract
Assadi, Liu, and Tarjan [SOSA'21] gave an auction algorithm that outputs a (1-ε)-approximation to Maximum Matching in bipartite graphs. Their algorithm computes a sequence of O(1/(ε²)) maximal matchings in subgraphs of the input graph and can be implemented in the multi-pass streaming setting with O(1/(ε²)) passes in a straightforward manner, which constitutes the state-of-the-art pass/approximation trade-off result in the multi-pass streaming setting. Their analysis uses tools from combinatorial auctions and, at its heart, relies on a clever potential function argument. Their proof, however, provides only limited insight into the inner workings of the algorithm.
In this paper, we revisit the ALT-algorithm and present the following contributions:
1) Simplification. The ALT-algorithm is built upon a freezing mechanism where vertices on one side of the bipartition that have already been rematched Θ(1/(ε)) times over the course of the algorithm remain matched to their current partner forever. We show that this mechanism is in fact unnecessary, i.e., no special treatment of such vertices is needed. With the freezing mechanism removed, the parameter ε now solely determines the total number of iterations/maximal matching computations, which provides the option of adaptively refining ε as the algorithm runs.
2) Alternative Analysis. We give an alternative analysis of the algorithm that is based on augmenting paths. Beyond the auction-perspective of the algorithm as established by Assadi et al., our analysis allows for a reinterpretation as one that follows the traditional approach of searching for and eliminating augmenting paths. Our analysis also copes with the removal of the freezing mechanism in a natural way, whereas the analysis of Assadi et al. strictly depends on its use.
3) Hard Instance. We provide the first hard instance on which the algorithm requires Ω(1/(ε²)) iterations/maximal matching computations. The instance is a simple path graph, where we exhibit a cyclic behaviour that prevents fast progress. Hard instances for this algorithm therefore do not necessarily have to be dense.
Cite as
Christian Konrad, Kheeran K. Naidu, Archie Walton, and Eric Wang. On the Assadi–Liu–Tarjan Auction Algorithm for Bipartite Matching: Simplification, Alternative Analysis, and Hard Instance. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 120:1-120:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{konrad_et_al:LIPIcs.ESA.2026.120,
author = {Konrad, Christian and Naidu, Kheeran K. and Walton, Archie and Wang, Eric},
title = {{On the Assadi–Liu–Tarjan Auction Algorithm for Bipartite Matching: Simplification, Alternative Analysis, and Hard Instance}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {120:1--120:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.120},
URN = {urn:nbn:de:0030-drops-272562},
doi = {10.4230/LIPIcs.ESA.2026.120},
annote = {Keywords: Maximum Bipartite Matching, Augmenting Paths, Auction Algorithm, Approximation}
}
Document
A Simple Algorithmic Framework for Disambiguation of Finite Automata
Abstract
We study the task of disambiguation of finite state automata, namely, converting an automaton into an equivalent, unambiguous one. We do this by developing a novel and simple algorithmic framework that generalizes the subset construction for determinization, and that satisfies some desirable properties: (1) it preserves the original automaton if it was already unambiguous, (2) it computes the successor states on-the-fly and (3) computes each new state in polynomial time - this last point is crucial as it guarantees that the running time is polynomial in the size of the output automaton. Then, we show how to apply this framework for partial disambiguation: by changing the criterion that builds the new states, we develop algorithms for different levels of ambiguity, namely, finitely ambiguous, and polynomially ambiguous automata. These algorithms also satisfy condition (1) for their respective levels, and also (2) and (3). Finally, we show that the disambiguation framework can easily be extended to other models of automata like weighted automata.
Cite as
Mauricio Cari, Martín Muñoz, and Cristian Riveros. A Simple Algorithmic Framework for Disambiguation of Finite Automata. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 121:1-121:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cari_et_al:LIPIcs.ESA.2026.121,
author = {Cari, Mauricio and Mu\~{n}oz, Mart{\'\i}n and Riveros, Cristian},
title = {{A Simple Algorithmic Framework for Disambiguation of Finite Automata}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {121:1--121:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.121},
URN = {urn:nbn:de:0030-drops-272573},
doi = {10.4230/LIPIcs.ESA.2026.121},
annote = {Keywords: Algorithmic automata theory, unambiguous automata models, degree of ambiguity, determinization, disambiguation, weighted automata}
}
Document
Maximizing Reachability via Shifting of Temporal Paths
Abstract
We examine the problem of maximizing the reachability of a given source in temporal graphs that are given as the union of k temporal paths, i.e., every given path is a sequence of edges with strictly increasing labels that denote availability in time. This type of temporal graphs represent train networks. We consider shifting operations on the labels of the paths that maintain their temporal continuity. This means that we can move the availability of a temporal edge later or earlier in time, and propagate the shifts to all other affected edges of the path in order to preserve its temporal connectivity. We study the parameterized complexity of the problem with respect to the number of paths k, and the total budget b, where b is the maximum number of shifts we are allowed to perform. Our results reveal that fixed parameter tractability can be achieved (1) when parameterized both by k and b, and (2) when parameterized by k, and b is unlimited. In almost every other case, e.g., parameterized by a single parameter or parameterized by k, while having a bound on b, we establish intractability lower bounds that are matched by XP algorithms.
Cite as
Argyrios Deligkas, Michelle Döring, Eduard Eiben, George Skretas, and Georg Tennigkeit. Maximizing Reachability via Shifting of Temporal Paths. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 122:1-122:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{deligkas_et_al:LIPIcs.ESA.2026.122,
author = {Deligkas, Argyrios and D\"{o}ring, Michelle and Eiben, Eduard and Skretas, George and Tennigkeit, Georg},
title = {{Maximizing Reachability via Shifting of Temporal Paths}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {122:1--122:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.122},
URN = {urn:nbn:de:0030-drops-272580},
doi = {10.4230/LIPIcs.ESA.2026.122},
annote = {Keywords: temporal graphs, temporal path number, path graph, reachability, train system, network optimization, parameterized complexity}
}
Document
Fast Algorithms for Continuous Optimal Transport Between Histograms
Abstract
We give the first (relative) (1+ε)-approximation algorithm for the continuous optimal transport (OT) problem between two axis-aligned histograms in the plane, each of which is represented as a piecewise-constant function over a rectangular subdivision. Our algorithm runs in nearly quadratic time in the complexity of the histograms in the worst case. In contrast to the discrete and semi-discrete optimal transport problems, which always admit linear-size OT plans, we demonstrate that there is a quadratic lower bound on the complexity of even a (1+ε)-approximate OT plan in the continuous setting in the worst case. This suggests that the runtime of our approximation algorithm nearly matches the worst-case lower-bound complexity of an explicit (1+ε)-approximate OT plan between two histograms. We additionally provide near-linear time algorithms with either weaker approximation guarantees or restrictions on the input histograms.
Cite as
Pankaj K. Agarwal, Sharath Raghvendra, and Keegan Yao. Fast Algorithms for Continuous Optimal Transport Between Histograms. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 123:1-123:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{agarwal_et_al:LIPIcs.ESA.2026.123,
author = {Agarwal, Pankaj K. and Raghvendra, Sharath and Yao, Keegan},
title = {{Fast Algorithms for Continuous Optimal Transport Between Histograms}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {123:1--123:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.123},
URN = {urn:nbn:de:0030-drops-272595},
doi = {10.4230/LIPIcs.ESA.2026.123},
annote = {Keywords: Optimal transport, min-cost flow, minimum-weight matching, quadtrees, approximation algorithms, dynamic optimal transport algorithm}
}
Document
Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams
Abstract
Let S be a set of n points in R². Our goal is to preprocess S to efficiently compute the smallest enclosing disk of the points in S that lie inside an axis-aligned query rectangle. Previous data structures for this problem achieve a query time of O(log⁶ n) with O(n log² n) preprocessing time and space by lifting the points to 3D, dualizing them into polyhedra, and searching through their intersections. We present a significantly simpler approach, solely based on 2D geometric structures, specifically 2D farthest-point Voronoi diagrams. Our approach achieves a deterministic query time of O(log⁴ n) and, via randomization, an expected query time of O(log^{5/2} n log log n) with the same preprocessing bounds.
Cite as
Kevin Buchin, Mark Joachim Krallmann, and Frank Staals. Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 124:1-124:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{buchin_et_al:LIPIcs.ESA.2026.124,
author = {Buchin, Kevin and Krallmann, Mark Joachim and Staals, Frank},
title = {{Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {124:1--124:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.124},
URN = {urn:nbn:de:0030-drops-272608},
doi = {10.4230/LIPIcs.ESA.2026.124},
annote = {Keywords: Range searching, smallest enclosing disk, farthest point Voronoi diagram}
}
Document
Incongruity-Sensitive Access to Highly Compressed Strings
Abstract
Random access to highly compressed strings - represented by straight-line programs or Lempel-Ziv parses, for example - is a well-studied topic. Random access to such strings in strongly sublogarithmic time is impossible in the worst case, but previous authors have shown how to support faster access to specific characters and their neighbourhoods. In this paper we explore whether, since better compression can impede access, we can support faster access to less compressible substrings of highly compressed strings. We first show how, given a run-length compressed straight-line program (RLSLP) of size g_{rl} or a block tree of size L, we can build an O (g_{rl})-space or an O (L)-space data structure, respectively, that supports access to any character in time logarithmic in the length of the longest repeated substring containing that character. That is, the more "incongruous" a character is with respect to the characters around, the faster we can support access to it. We then prove a similar but more powerful and sophisticated result for parsings in which phrases' sources do not overlap much larger phrases, with the query time depending also on the number of phrases we must copy from their sources to obtain the queried character.
Cite as
Ferdinando Cicalese, Travis Gagie, Zsuzsanna Lipták, Gonzalo Navarro, Nicola Prezza, and Cristian Urbina. Incongruity-Sensitive Access to Highly Compressed Strings. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 125:1-125:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cicalese_et_al:LIPIcs.ESA.2026.125,
author = {Cicalese, Ferdinando and Gagie, Travis and Lipt\'{a}k, Zsuzsanna and Navarro, Gonzalo and Prezza, Nicola and Urbina, Cristian},
title = {{Incongruity-Sensitive Access to Highly Compressed Strings}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {125:1--125:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.125},
URN = {urn:nbn:de:0030-drops-272611},
doi = {10.4230/LIPIcs.ESA.2026.125},
annote = {Keywords: Data compression, parsing, straight-line program, random access, grammar compression, run-length grammar, longest repeated substring, distance-sensitive predecessor data structures}
}
Document
Separating Geodesic Structure and Product Structure
Abstract
The geodesic treewidth of a graph G is the smallest k for which there is a partition 𝒫 into geodesics such that G/𝒫 has treewidth k, where G/𝒫 is obtained from G by contracting each part of 𝒫. Based on this notion, row treewidth was developed and is defined for a graph G as the smallest k such that G ⊆ H ⊠ P for some graph H of treewidth k and a path P. Equivalently, the row treewidth of a graph G is the smallest k for which there is a partition 𝒫 into disjoint unions of geodesics that are aligned with respect to some layering such that G/𝒫 has treewidth k.
We separate the two notions by showing that bounded row treewidth does not imply bounded geodesic treewidth and by presenting a polynomial-time algorithm to decide whether a graph of treewidth 2 has geodesic treewidth 1, which is known to be NP-hard for row treewidth [Biedl, Eppstein, Ueckerdt, 2025]. More generally, we provide an algorithm to decide whether a given graph has geodesic treewidth at most d that is XP in the treewidth, whereas there is no such algorithm for row treewidth, unless P = NP [Biedl, Eppstein, Ueckerdt, 2025]. On the other hand, we show that computing the geodesic treewidth is NP-hard and that every graph with geodesic treewidth 1 has bounded row treewidth. Moreover, we improve the best known lower bound on the geodesic treewidth of planar graphs to 5.
Cite as
Laura Merker, Lena Scherzer, and Samuel Schneider. Separating Geodesic Structure and Product Structure. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 126:1-126:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{merker_et_al:LIPIcs.ESA.2026.126,
author = {Merker, Laura and Scherzer, Lena and Schneider, Samuel},
title = {{Separating Geodesic Structure and Product Structure}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {126:1--126:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.126},
URN = {urn:nbn:de:0030-drops-272626},
doi = {10.4230/LIPIcs.ESA.2026.126},
annote = {Keywords: product structure, row treewidth, geodesic structure, geodesic treewidth}
}
Document
Online Coloring for Graphs of Large Odd Girth
Abstract
We study the problem of online coloring for graphs with large odd girth. The best previously known algorithm uses O(n^{1/2}) colors, which was discovered by Kierstead in 1998. This algorithm works when the odd girth is 7 or more. In this paper, we provide the following: for every ε > 0, there exists a constant g' ∈ {3, 5, 7, …} such that graphs with odd girth at least g' can be deterministically colored online using O(n^ε) colors.
Cite as
Hirotaka Yoneda and Masataka Yoneda. Online Coloring for Graphs of Large Odd Girth. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 127:1-127:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{yoneda_et_al:LIPIcs.ESA.2026.127,
author = {Yoneda, Hirotaka and Yoneda, Masataka},
title = {{Online Coloring for Graphs of Large Odd Girth}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {127:1--127:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.127},
URN = {urn:nbn:de:0030-drops-272633},
doi = {10.4230/LIPIcs.ESA.2026.127},
annote = {Keywords: online coloring, online algorithms, graph theory, graph coloring, odd girth}
}
Document
Online Flow Time Minimization with Gradually Revealed Jobs
Abstract
We consider the problem of online preemptive scheduling on a single machine to minimize the total flow time. In clairvoyant scheduling, where job processing times are revealed upon arrival, the Shortest Remaining Processing Time (SRPT) algorithm is optimal. In practice, however, exact processing times are often unknown. At the opposite extreme, non-clairvoyant scheduling, in which processing times are revealed only upon completion, suffers from strong lower bounds on the competitive ratio. This motivates the study of intermediate information models. We introduce a new model in which processing times are revealed gradually during execution. Each job consists of a sequence of operations, and the processing time of an operation becomes known only after the preceding one completes. This models many scheduling scenarios that arise in computing systems.
Our main result is a deterministic O(m²)-competitive algorithm, where m is the maximum number of operations per job. More specifically, we prove a refined competitive ratio in O(m₁ ⋅ m₂), where m₁ and m₂ are instance-dependent parameters describing the operation size structure. Our algorithm and analysis build on recent advancements in robust flow time minimization (SODA '26), where jobs arrive with estimated sizes. However, in our setting we have no bounded estimate on a job’s processing time. Thus, we design a highly adaptive algorithm that gradually explores a job’s operations while working on them, and groups them into virtual chunks whose size can be well-estimated. This is a crucial ingredient of our result and requires a much more careful analysis compared to the robust setting. We also provide lower bounds showing that our bounds are essentially best possible. For the special case of scheduling with uniform obligatory tests, we show that SRPT at the operation level is 2-competitive, which is best possible.
Cite as
Alexander Lindermayr, Guido Schäfer, Jens Schlöter, and Leen Stougie. Online Flow Time Minimization with Gradually Revealed Jobs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 128:1-128:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lindermayr_et_al:LIPIcs.ESA.2026.128,
author = {Lindermayr, Alexander and Sch\"{a}fer, Guido and Schl\"{o}ter, Jens and Stougie, Leen},
title = {{Online Flow Time Minimization with Gradually Revealed Jobs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {128:1--128:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.128},
URN = {urn:nbn:de:0030-drops-272649},
doi = {10.4230/LIPIcs.ESA.2026.128},
annote = {Keywords: optimization, scheduling, online algorithms, competitive analysis, flow time, non-clairvoyance}
}
Document
Optimal Union Probability Interval Is NP-Hard
Abstract
A problem dating back to Boole [Laws of Thought, Walton & Maberly, 1854] is what can be computed about the probability of a finite union of events when given as input the probabilities of intersections of some of the events. The modern geometric study of the problem can be traced back to Hailperin [Amer. Math. Monthly 2 (1965) 343-359] who phrased the problem in the language of linear programming and generalized it to logical formulas of the events other than disjunction, heralding a substantial body of work in probabilistic logic [Nilsson, Artif. Intell. 28 (1986) 71-87], including the probabilistic satisfiability problem of Georgakopoulos, Kavvadis, and Papadimitriou [J. Complexity 4 (1988) 1-11], as well as fundamental connections to the geometry of metrics via cut and correlation polytopes [Deza and Laurent, Geometry of Cuts and Metrics, Springer, 1997] and to the study of marginal polytopes in graphical models of machine learning [Wainwright and Jordan, Found. Trends Mach. Learn. 1 (2008) 1-305]. This paper (i) describes the pertinent geometry of Boole’s problem via coordinate projections of an elementary polytope arising essentially from Hailperin’s linear program on the atoms of a Venn diagram, and (ii) shows that computing the optimal interval for the union probability is NP-hard, resolving an apparent gap in the literature highlighted by Pitowsky [Math. Programming 50 (1991) 395-414] and Boros et al. [Math. Oper. Res. 39 (2014) 1311-1329 and 51 (2026) 134-148].
Cite as
Petteri Kaski, Heikki Mannila, and Chandra Kanta Mohapatra. Optimal Union Probability Interval Is NP-Hard. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 129:1-129:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kaski_et_al:LIPIcs.ESA.2026.129,
author = {Kaski, Petteri and Mannila, Heikki and Mohapatra, Chandra Kanta},
title = {{Optimal Union Probability Interval Is NP-Hard}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {129:1--129:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.129},
URN = {urn:nbn:de:0030-drops-272650},
doi = {10.4230/LIPIcs.ESA.2026.129},
annote = {Keywords: combinatorial probability, computational complexity, correlation polytope, generalized Bonferroni inequalities, graphical models, inclusion-exclusion, intersection probability, linear programming, union polytope}
}
Document
Approximation Algorithms for the Traveling Thief Problem
Abstract
The Traveling Thief Problem (TTP) combines the Traveling Salesperson Problem with the Knapsack Problem. In this problem, a finite metric space is given, and at each location an item with some profit and weight is placed. An agent seeks to collect a subset of the items. To do so, the agent must decide which items to collect and to determine a cyclic tour visiting the corresponding locations. While collecting an item yields its profit as a reward, the agent’s speed decreases as more weight is picked up. The problem involves two competing objectives: maximizing the total profit of the collected items and minimizing the travel time of the tour.
While many heuristics and exact algorithms (with a non-polynomial running time) have been developed, no approximation algorithms are known for any variant of the TTP. We aim at computing an (α₁,α₂)-approximate Pareto set that, for every solution, contains another solution collecting at least a 1/(α₁) fraction of its profit while requiring at most α₂ times its travel time. Our main result is an algorithm that calculates a (9 + ε,9 + ε)-approximate Pareto set in polynomial time.
We also consider the setting in which the set of items to be collected is given in advance, so that the agent only has to compute a tour through the corresponding locations that minimizes the total travel time. This is the so-called Weighted TSP. For this setting, we present a (2e + ε)-approximation algorithm.
Cite as
Jan Eube, Kelin Luo, Heiko Röglin, and Sarah Sturm. Approximation Algorithms for the Traveling Thief Problem. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 130:1-130:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{eube_et_al:LIPIcs.ESA.2026.130,
author = {Eube, Jan and Luo, Kelin and R\"{o}glin, Heiko and Sturm, Sarah},
title = {{Approximation Algorithms for the Traveling Thief Problem}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {130:1--130:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.130},
URN = {urn:nbn:de:0030-drops-272668},
doi = {10.4230/LIPIcs.ESA.2026.130},
annote = {Keywords: Traveling Thief Problem, Traveling Salesperson Problem, Knapsack Problem, Approximation Algorithms, Bi-objective optimization}
}
Document
Characterizing Streaming Decidability of CSPs via Non-Redundancy
Abstract
We study the single-pass streaming complexity of deciding satisfiability of Constraint Satisfaction Problems (CSPs). A CSP is specified by a constraint language Γ, that is, a finite set of k-ary relations over the domain [q] = {0, … , q-1}. An instance of CSP(Γ) consists of m constraints over n variables x₁, …, x_n taking values in [q]. Each constraint C_i is of the form {R_i,(x_{i_1} + λ_{i_1}, …, x_{i_k} + λ_{i_k})}, where R_i ∈ Γ and λ_{i_1}, …, λ_{i_k} ∈ [q] are constants; it is satisfied if and only if (x_{i_1} + λ_{i_1}, …, x_{i_k} + λ_{i_k}) ∈ R_i, where addition is modulo q. In the streaming model, constraints arrive one by one, and the goal is to determine, using minimum memory, whether there exists an assignment satisfying all constraints.
For k-SAT, Vu (TCS 2024) proves an optimal Ω_k(n^k) space lower bound, while for general CSPs, Chou, Golovnev, Sudan, and Velusamy (JACM 2024) establish an Ω(n) lower bound; a complete characterization has remained open. We close this gap by showing that the single-pass streaming space complexity of CSP(Γ) is precisely governed by its non-redundancy, a structural parameter introduced by Bessiere, Carbonnel, and Katsirelos (AAAI 2020). The non-redundancy NRD_n(Γ) is the maximum number of constraints over n variables such that every constraint is non-redundant, i.e., omitting it strictly expands the set of satisfying assignments. We prove that the single-pass streaming complexity of CSP(Γ) is characterized, up to a logarithmic factor, by NRD_n(Γ). We also extend this characterization to positive Boolean CSPs, i.e., instances in which no additive shifts are applied, a class that includes graph 2-colorability (equivalently, bipartiteness) as a canonical example.
A key ingredient in our lower bound proof is a binary relation ℰ on the set of assignments [q]ⁿ, where (a, b) ∈ ℰ if every instance satisfied by a is also satisfied by b. While it is immediate that ℰ is reflexive and transitive, its symmetry, which would make it an equivalence relation, is non-trivial. We show that ℰ is an equivalence relation for general CSPs, and for positive Boolean CSPs when excluding the two constant assignments (0ⁿ and 1ⁿ). We believe this equivalence structure could be of independent interest.
Cite as
Amatya Sharma and Santhoshini Velusamy. Characterizing Streaming Decidability of CSPs via Non-Redundancy. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 131:1-131:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{sharma_et_al:LIPIcs.ESA.2026.131,
author = {Sharma, Amatya and Velusamy, Santhoshini},
title = {{Characterizing Streaming Decidability of CSPs via Non-Redundancy}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {131:1--131:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.131},
URN = {urn:nbn:de:0030-drops-272679},
doi = {10.4230/LIPIcs.ESA.2026.131},
annote = {Keywords: Constraint Satisfaction Problems, Streaming Algorithms, Satisfiability, Non-Redundancy}
}
Document
On Computing Minimum Wheeler DFA from Their Language
Abstract
Wheeler automata have recently emerged as a powerful generalization of the Burrows-Wheeler Transform, enabling optimal linear-time pattern matching on compressed labeled graphs - a task that is otherwise computationally hard. Consequently, when an automaton recognizes a Wheeler language (i.e., it is equivalent to some Wheeler automaton), computing its minimum equivalent Wheeler DFA is a powerful indexing strategy. This problem is particularly relevant in computational pangenomics, where pangenome graphs frequently recognize Wheeler languages.
However, constructing the minimum Wheeler DFA for a Wheeler language has remained a computational bottleneck. The problem is known to be PSPACE-hard for nondeterministic inputs. When the input is a DFA, state-of-the-art solutions forced a compromise: they were either fast but limited to acyclic DFAs (Alanko et al., SODA 2020) or capable of handling general topologies but prohibitively slow (D'Agostino et al., TCS 2023). In this work, we bridge this gap with the first algorithm solving the problem for general DFAs in near-optimal, linearithmic output-sensitive time. By matching the efficiency of acyclic-only solutions while retaining full generality, our approach improves upon the previous general solution by at least a quadratic factor. We demonstrate the practical impact of our algorithm on real-world pangenome graphs; our tool achieves a processing throughput of over 10⁵ transitions per second on a standard workstation, enabling the construction of a provably optimal pattern matching data structure in such applications.
Cite as
Ruben Becker, Davide Cenzato, Nicola Prezza, and Daniel Puttini. On Computing Minimum Wheeler DFA from Their Language. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 132:1-132:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{becker_et_al:LIPIcs.ESA.2026.132,
author = {Becker, Ruben and Cenzato, Davide and Prezza, Nicola and Puttini, Daniel},
title = {{On Computing Minimum Wheeler DFA from Their Language}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {132:1--132:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.132},
URN = {urn:nbn:de:0030-drops-272685},
doi = {10.4230/LIPIcs.ESA.2026.132},
annote = {Keywords: Wheeler Automata, Minimum DFA, Pangenomics, Pattern Matching}
}
Document
Beyond Trees: The Weighted Center Problem on Gromov Hyperbolic Graphs
Abstract
The Weighted Center problem takes as its input a graph G = (V,E) together with a profile π such that every vertex v is mapped to some nonnegative multiplicative weight π(v). Its output must be some vertex c minimizing max{π(v)d_G(c,v) : v ∈ V}. The classic Center problem corresponds to the case where π(v) = 1 for every vertex v. In the literature, various almost linear-time algorithms have been proposed for the Center problem on some well-structured classes of graphs. By contrast, similarly efficient algorithms for the Weighted Center problem have been scarce. We investigate how the Gromov hyperbolicity, alone or in combination with other metric and geometric properties on graphs, can be used in the design of exact and approximate almost linear-time algorithms for the Weighted Center problem. In particular, we derive almost optimal algorithms for the following well-studied classes of graphs: chordal graphs, distance-hereditary graphs (both in 𝒪(m) time), dually chordal graphs and chordal bipartite graphs (both in 𝒪(m log n) time).
Cite as
Guillaume Ducoffe. Beyond Trees: The Weighted Center Problem on Gromov Hyperbolic Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 133:1-133:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ducoffe:LIPIcs.ESA.2026.133,
author = {Ducoffe, Guillaume},
title = {{Beyond Trees: The Weighted Center Problem on Gromov Hyperbolic Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {133:1--133:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.133},
URN = {urn:nbn:de:0030-drops-272699},
doi = {10.4230/LIPIcs.ESA.2026.133},
annote = {Keywords: Weighted center problem, Gromov hyperbolicity, Chordal graphs and related graph classes, Planar graphs, Fine-grained complexity in P, Graph algorithms}
}
Document
Incremental Submodular Maximization: Better Than Greedy
Abstract
We consider submodular maximization under increasing cardinality constraint and ask for a good incremental solution, i.e., an ordering of the ground set such that each prefix of the ordering yields a good solution for its respective cardinality. A classical result in this setting is that the greedy algorithm achieves a competitive ratio, i.e., an approximation guarantee across all cardinalities, of e/(e-1) ≈ 1.582. No better general guarantee was previously known. We present an adaptive scaling algorithm achieving a competitive ratio of 1.373. We complement our result by a lower bound of 1.25 on the best possible deterministic competitive ratio for incremental submodular maximization.
Cite as
Marcin Bienkowski, Joakim Blikstad, Jarosław Byrka, Martín Costa, Yann Disser, and Annette Lutz. Incremental Submodular Maximization: Better Than Greedy. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 134:1-134:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bienkowski_et_al:LIPIcs.ESA.2026.134,
author = {Bienkowski, Marcin and Blikstad, Joakim and Byrka, Jaros{\l}aw and Costa, Mart{\'\i}n and Disser, Yann and Lutz, Annette},
title = {{Incremental Submodular Maximization: Better Than Greedy}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {134:1--134:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.134},
URN = {urn:nbn:de:0030-drops-272702},
doi = {10.4230/LIPIcs.ESA.2026.134},
annote = {Keywords: Submodular maximization, incremental optimization, competitive analysis}
}
Document
New Algorithms and Hardness Results for Connected Clustering
Abstract
Connected clustering denotes a family of constrained clustering problems in which we are given a distance metric and an undirected connectivity graph G that can be completely unrelated to the metric. The aim is to partition the n vertices into a given number k of clusters such that every cluster forms a connected subgraph of G and a given clustering objective gets minimized. The constraint that the clusters are connected has applications in many different fields, like for example community detection and geodesy.
So far, k-center and k-median have been studied in this setting. It has been shown that connected k-median is Ω(n^{1- ε})-hard to approximate which also carries over to the connected k-means problem, while for connected k-center it remained an open question whether one can find a constant approximation in polynomial time. We answer this question by providing an Ω(log^*(k))-hardness result for the problem. Given these hardness results, we study the problems on graphs with bounded treewidth. We provide exact algorithms that run in polynomial time if the treewidth w is a constant. Furthermore, we obtain constant approximation algorithms that run in FPT time with respect to the parameter max(w,k).
Additionally, we consider the min-sum-radii (MSR) and min-sum-diameter (MSD) objectives. We prove that on general graphs, connected MSR can be approximated with an approximation factor of (3 + ε) and connected MSD with an approximation factor of (4 + ε). The latter also directly improves the best known approximation guarantee for unconstrained MSD from (6 + ε) to (4 + ε). We complement this with a reduction showing that connected MSR is NP-hard to approximate with an approximation factor smaller than (4/3).
Cite as
Jan Eube and Heiko Röglin. New Algorithms and Hardness Results for Connected Clustering. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 135:1-135:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{eube_et_al:LIPIcs.ESA.2026.135,
author = {Eube, Jan and R\"{o}glin, Heiko},
title = {{New Algorithms and Hardness Results for Connected Clustering}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {135:1--135:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.135},
URN = {urn:nbn:de:0030-drops-272713},
doi = {10.4230/LIPIcs.ESA.2026.135},
annote = {Keywords: Clustering, Connectivity constraints, Approximation algorithms, Hardness of approximation}
}
Document
Approximation Algorithms for Machine Minimization
Abstract
In this paper we consider a classic scheduling problem known as Machine Minimization (MM). The input to MM is a set J of n jobs, where each job j has processing time p_j, release time r_j, and deadline d_j. The goal is to schedule the jobs to run non-preemptively on minimum number of machines such that each job is fully scheduled within its [r_j,d_j] interval and each machine runs at most one job at a time. This problem generalizes several NP-hard problems (e.g. the case of identical release time and deadlines reduces to the bin packing problem). Using Randomized Rounding [Prabhakar Raghavan and Clark D. Thompson, 1987] one can get an O((log n)/(log log n))-approximation. Chuzhoy et al. [Julia Chuzhoy and Paolo Codenotti, 2009] presented an algorithm that uses O(opt²) machines (i.e. O(opt)-approximation). Combined with the earlier work this yields an O(√{(log n)/(log log n)})-approximation and this remains the best known result for over 20 years. Even for when the ratio of largest to smallest processing time p_{max}/p_{min} is bounded, or the number of distinct processing times are bounded, there is no (universal) constant approximation. In this paper, we present a number of results. When p_max/p_min = ρ we present an algorithm that yields an asymptotic (2+ε)-approximation in time n^{O(ρ⁴/ε⁴)}. When the number of distinct processing times is c, we present a 2-approximation with run time n^{O(c²log³ n)}. If we have c distinct processing times and p_max/p_min = ρ we present an asymptotic (1+ε)-approximation that runs in time n^O(c²⋅ρ²⋅log³n/ε²).
Cite as
Mohsen Mohammadi and Mohammad R. Salavatipour. Approximation Algorithms for Machine Minimization. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 136:1-136:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mohammadi_et_al:LIPIcs.ESA.2026.136,
author = {Mohammadi, Mohsen and Salavatipour, Mohammad R.},
title = {{Approximation Algorithms for Machine Minimization}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {136:1--136:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.136},
URN = {urn:nbn:de:0030-drops-272720},
doi = {10.4230/LIPIcs.ESA.2026.136},
annote = {Keywords: Machine minimization, approximation algorithms, scheduling}
}
Document
Small Independent Sets Versus Small Separator in Geometric Intersection Graphs
Abstract
While most classical NP-hard graph problems cannot be solved in time 2^o(n) on general graphs under the Exponential Time Hypothesis (ETH), many exhibit the square-root phenomenon and admit optimal algorithms running in time 2^O(√n) on certain geometric intersection graphs, such as planar graphs or unit disk graphs. In 2018, de Berg et al. developed a general algorithmic framework for such problems on intersection graphs of similarly sized fat objects in ℝ^d, achieving running times of the form 2^O(n^{1-1/d}), along with matching lower bounds under ETH.
In this paper, we identify problems that do not exhibit the square-root phenomenon, yet still admit subexponential algorithms on intersection graphs of similarly sized fat objects in ℝ^d, for every fixed dimension d ⩾ 2. We introduce the notion of a weak square-root phenomenon: problems that can be solved in time 2^Õ(n^{1-1/(d+1)}), and for which matching lower bounds hold under ETH. We develop both an algorithmic framework and a corresponding lower bound framework. As concrete examples, we show that the problems 2-Subcoloring and Two Sets Cut-Uncut exhibit this behavior.
Our algorithms rely on a new win-win structural theorem, which can be informally stated as follows: every such graph admits a sublinear separator whose removal leaves connected components with sublinear independence number. To facilitate the design of these algorithms, we introduce a new graph parameter, the α-modulator number, which generalizes both the independence number and the vertex cover number.
Cite as
Malory Marin and Rémi Watrigant. Small Independent Sets Versus Small Separator in Geometric Intersection Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 137:1-137:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{marin_et_al:LIPIcs.ESA.2026.137,
author = {Marin, Malory and Watrigant, R\'{e}mi},
title = {{Small Independent Sets Versus Small Separator in Geometric Intersection Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {137:1--137:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.137},
URN = {urn:nbn:de:0030-drops-272734},
doi = {10.4230/LIPIcs.ESA.2026.137},
annote = {Keywords: Subexponential Algorithms, Unit Disk Graphs, 2-Subcoloring, Two-Sets Cut-Uncut}
}
Document
Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations
Abstract
Abstract Voronoi diagrams are defined in terms of a given system of planar bisecting curves satisfying some simple combinatorial properties. They offer a unifying framework for a wide range of concrete Voronoi instances on generalized sites and metrics. In this paper, we formulate higher-order abstract color Voronoi diagrams of a set S of n colored abstract sites, simultaneously considering all concrete instances under their umbrella. We prove that the number of vertices in the order-k abstract color Voronoi diagram is at most 4k(n-k)-2n, and present an iterative construction algorithm. The bound directly applies to a family of m disjoint simple polygons of total complexity n. For simple polygons the bound can further improve to O(min{k(n-k),(m-k)²n}). A critical ingredient of our proof is a combinatorial analysis on circular sequences of color permutations derived from the unbounded edges of these diagrams that is interesting in its own right.
Cite as
Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou. Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 138:1-138:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bae_et_al:LIPIcs.ESA.2026.138,
author = {Bae, Sang Won and Oliver, Nicolau and Papadopoulou, Evanthia},
title = {{Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {138:1--138:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.138},
URN = {urn:nbn:de:0030-drops-272748},
doi = {10.4230/LIPIcs.ESA.2026.138},
annote = {Keywords: higher-order Voronoi diagrams, abstract Voronoi diagrams, color Voronoi diagrams, circular sequences, allowable sequences, generalized sites, simple polygons}
}
Document
Non-Additive Discrepancy: Coverage Functions in a Beck-Fiala Setting
Abstract
Recent concurrent work by Dupré la Tour and Fujii and by Hollender, Manurangsi, Meka, and Suksompong [ITCS'26] introduced a generalization of classical discrepancy theory to non-additive functions, motivated by applications in fair division. As many classical techniques from discrepancy theory seem to fail in this setting, including linear algebraic methods like the Beck-Fiala Theorem [Discrete Appl. Math '81], it remains widely open whether comparable non-additive bounds can be achieved.
Towards a better understanding of non-additive discrepancy, we study coverage functions in a sparse setting comparable to the classical Beck-Fiala Theorem. Our setting generalizes the additive Beck-Fiala setting, rank functions of partition matroids, and edge coverage in graphs. More precisely, assuming each of the n items covers only t elements across all functions, we prove a constructive discrepancy bound that is polynomial in t, the number of colors k, and log n.
Cite as
Tatiana Rocha Avila, Lars Rohwedder, and Leo Wennmann. Non-Additive Discrepancy: Coverage Functions in a Beck-Fiala Setting. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 139:1-139:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{avila_et_al:LIPIcs.ESA.2026.139,
author = {Avila, Tatiana Rocha and Rohwedder, Lars and Wennmann, Leo},
title = {{Non-Additive Discrepancy: Coverage Functions in a Beck-Fiala Setting}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {139:1--139:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.139},
URN = {urn:nbn:de:0030-drops-272757},
doi = {10.4230/LIPIcs.ESA.2026.139},
annote = {Keywords: Combinatorial Optimization, Discrepancy Theory}
}
Document
Improved Bounds for Strategy Improvement Algorithms for Energy Games
Abstract
Strategy improvement is a natural and well-studied family of algorithms for solving various classes of stochastic and deterministic graph games. We present an improved upper bound of O(n 2ⁿ) on the number of iterations performed by the most natural, and most greedy, variant of the algorithm when applied to n-vertex Energy Games. We also obtain a similar upper bound of O(poly(n)⋅ 2ⁿ) on the expected number of iterations performed by Random-Edge, one of the most natural randomized variants of the algorithm. To the best of our knowledge, these are the first bounds for natural strategy-improvement algorithms on non-binary energy games that beat the trivial nⁿ = 2^{n log n} bound obtained by enumerating all strategies. The proof is based on a new adaptation of the layering technique of [Dorfman, Kaplan, Zwick, ICALP 2019].
Cite as
Dani Dorfman, Haim Kaplan, and Uri Zwick. Improved Bounds for Strategy Improvement Algorithms for Energy Games. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 140:1-140:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dorfman_et_al:LIPIcs.ESA.2026.140,
author = {Dorfman, Dani and Kaplan, Haim and Zwick, Uri},
title = {{Improved Bounds for Strategy Improvement Algorithms for Energy Games}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {140:1--140:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.140},
URN = {urn:nbn:de:0030-drops-272764},
doi = {10.4230/LIPIcs.ESA.2026.140},
annote = {Keywords: Graph Games, Energy Games, Strategy improvement}
}
Document
The Complexity of Stackelberg Pricing Games
Abstract
We consider Stackelberg pricing games, which are also known as bilevel pricing problems, or combinatorial price-setting problems. This family of problems consists of games between two players: the leader and the follower. There is a market that is partitioned into two parts: the part of the leader and the part of the leader’s competitors. The leader controls one part of the market and can freely set the prices for products. By contrast, the prices of the competitors' products are fixed and known in advance. The follower, then, needs to solve a combinatorial optimization problem in order to satisfy their own demands, while comparing the leader’s offers to the offers of the competitors. Therefore, the leader has to hit the intricate balance of making an attractive offer to the follower, while at the same time ensuring that their own profit is maximized.
Pferschy, Nicosia, Pacifici, and Schauer considered the Stackelberg pricing game where the follower solves a knapsack problem. They raised the question whether this problem is complete for the second level of the polynomial hierarchy, i.e., Σ^p₂-complete. The same conjecture was also made by Böhnlein, Schaudt, and Schauer. In this paper, we positively settle this conjecture. Moreover, we show that this result holds actually in a much broader context: The Stackelberg pricing game is Σ^p₂-complete for over 50 underlying problems whose decision versions are NP-complete, including most classics such as TSP, vertex cover, clique, subset sum, etc. This result falls in line of recent meta-theorems about higher complexity in the polynomial hierarchy by Grüne and Wulf.
Cite as
Christoph Grüne, Dorothee Henke, Eva Rotenberg, and Lasse Wulf. The Complexity of Stackelberg Pricing Games. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 141:1-141:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{grune_et_al:LIPIcs.ESA.2026.141,
author = {Gr\"{u}ne, Christoph and Henke, Dorothee and Rotenberg, Eva and Wulf, Lasse},
title = {{The Complexity of Stackelberg Pricing Games}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {141:1--141:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.141},
URN = {urn:nbn:de:0030-drops-272774},
doi = {10.4230/LIPIcs.ESA.2026.141},
annote = {Keywords: Bilevel pricing problems, Stackelberg games, bilevel optimization, polynomial hierarchy}
}
Document
Optimal Stochastic Online Sorting
Abstract
In the online sorting problem that was introduced by Aamand, Abrahamsen, Beretta and Kleist [SODA 2023], n items of real numbers arrive in an online fashion. Each item has to be placed irrevocably into an array of size n before the next item is revealed. The cost is the sum of the absolute differences between adjacent items. We study the stochastic online sorting problem where the items are sampled uniformly at random from the interval [0, 1], a problem first studied by Abrahamsen, Bercea, Klausen and Kozma [ESA 2024], who presented an algorithm achieving an expected competitive ratio of O((n log n)^{1/4}). Later Hu [SODA 2026] achieved an improved expected competitive ratio of log n ⋅ 2^O(log^* n). Hu also showed a lower bound of an expected competitive ratio of Ω(log n). In this paper, we present a simple algorithm achieving an expected competitive ratio of O(log n), thus settling the complexity. In the variant where the array has size ⌈(1 + ε) n⌉, our algorithm achieves an expected competitive ratio of O(1 + log ε^{-1}), improving the previous best known of O(1 + ε^{-1}) by Abrahamsen et al. We generalise the algorithm to higher dimensions (stochastic online Euclidean traveling salesman problem), where an expected competitive ratio of O(1) is achieved. This improves a previous competitive ratio of O(log² n) by Kalavas, Platanos and Tolias [STACS 2026].
Cite as
Daniel Anker Hermansen. Optimal Stochastic Online Sorting. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 142:1-142:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hermansen:LIPIcs.ESA.2026.142,
author = {Hermansen, Daniel Anker},
title = {{Optimal Stochastic Online Sorting}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {142:1--142:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.142},
URN = {urn:nbn:de:0030-drops-272780},
doi = {10.4230/LIPIcs.ESA.2026.142},
annote = {Keywords: Online algorithm, sorting, expected analysis}
}
Document
Hardness of Frequency-Related Queries on Compressed Strings
Abstract
Compressed indexing is a recent trend in the design of data structures that aims to support fundamental string queries in space proportional to the size of the data in compressed form. One of the most popular compression frameworks in this field is grammar compression. A length-n string T ∈ Σⁿ (where Σ is any finite set of size up to |Σ| = |T|^𝒪(1)) represented using a context-free grammar of size |G| can be augmented to support random access queries (given any i ∈ [1..n], return T[i]) in 𝒪(|G| log^𝒪(1) n) space and 𝒪(log^𝒪(1) n) time. Numerous other queries, including pattern matching, longest common extension, lexicographical predecessor/successor, Burrows-Wheeler Transform, suffix array, and even suffix tree queries, can also be supported within the same bounds.
Despite this progress, one fundamental class of queries has remained elusive: frequency-related queries, such as reporting the number of occurrences of a symbol c ∈ Σ in a substring T(b..e] (the so-called rank query), or simply checking whether c occurs in T(b..e] (the symbol occurrence query). To date, no fully general structure achieving 𝒪(|G| log^𝒪(1) n) space and 𝒪(log^𝒪(1) n) query time is known. In this work, we establish new conditional lower bounds for frequency-related problems:
- We prove that answering rank and symbol occurrence queries on grammar-compressed texts in polylogarithmic time using a 𝒪(|G| log^𝒪(1) n)-space structure that is constructible from the input grammar in 𝒪(|G| log^𝒪(1) n) time would imply an 𝒪(n² log^𝒪(1) n)-time algorithm for Boolean Matrix Multiplication (BMM), where the best known algorithms achieve 𝒪(n^{2.371339}) time. Our result is achieved using a more general lower bound for efficiently answering a batch of rank and symbol occurrence queries.
- We generalize the above result, showing that even LZ78-compressed strings cannot support efficient rank queries. Since LZ78 is provably weaker than grammar compression, this yields a stronger result: rank and symbol occurrence queries remain hard for a wider class of compressors. We further show that achieving even additive approximations of rank queries would imply faster BMM algorithms.
- After establishing hardness of rank and symbol occurrence queries, we consider a broader class of frequency-related queries and show that, under the popular Orthogonal Vectors (OV) conjecture, other problems, including range distinct counting and range mode frequency queries, also cannot be efficiently supported in compressed space.
In summary, we develop new techniques for reasoning about computation over compressed data, and establish tight connections between compressed indexing and long-standing problems in fine-grained complexity. This sheds new light on compressed indexing by isolating a new class of frequency-related queries whose complexity hinges on known hard problems.
Cite as
Rajat De and Dominik Kempa. Hardness of Frequency-Related Queries on Compressed Strings. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 143:1-143:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{de_et_al:LIPIcs.ESA.2026.143,
author = {De, Rajat and Kempa, Dominik},
title = {{Hardness of Frequency-Related Queries on Compressed Strings}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {143:1--143:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.143},
URN = {urn:nbn:de:0030-drops-272793},
doi = {10.4230/LIPIcs.ESA.2026.143},
annote = {Keywords: compressed indexing, grammar compression, Lempel-Ziv 78 compression, conditional lower bounds, rank queries, range distinct count queries, range mode frequency queries}
}
Document
A 0.8395-Approximation Algorithm for the EPR Problem
Abstract
We give an efficient 0.8395-approximation algorithm for the EPR Hamiltonian. Our improvement comes from a new nonlinear monogamy-of-entanglement bound on star graphs and a refined parameterization of a shallow quantum circuit from previous works. We also prove limitations showing that current methods cannot achieve substantially better approximation ratios, indicating that further progress will require fundamentally new techniques.
Cite as
Anuj Apte, Eunou Lee, Kunal Marwaha, Ojas Parekh, Lennart Sinjorgo, and James Sud. A 0.8395-Approximation Algorithm for the EPR Problem. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 144:1-144:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{apte_et_al:LIPIcs.ESA.2026.144,
author = {Apte, Anuj and Lee, Eunou and Marwaha, Kunal and Parekh, Ojas and Sinjorgo, Lennart and Sud, James},
title = {{A 0.8395-Approximation Algorithm for the EPR Problem}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {144:1--144:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.144},
URN = {urn:nbn:de:0030-drops-272806},
doi = {10.4230/LIPIcs.ESA.2026.144},
annote = {Keywords: Quantum computing, optimization, approximation algorithms}
}
Document
Hierarchical Spanners
Abstract
A hierarchical graph 𝒢 consists of a vertex set V(𝒢) and L pairwise disjoint edge sets E_1, … , E_L. Such a structure naturally defines a hierarchy of L unweighted graphs, where the 𝓁-th graph is G_𝓁 = (V, ⋃_{i=1}^𝓁 E_i).
In this paper, we initiate the study of hierarchical spanners, namely subgraphs of 𝒢 that approximately preserve distances among a given set of pairs of vertices in V(𝒢) at every level of the hierarchy. This notion generalizes classical spanners, and thus all known lower bounds extend to this setting; however, it is not clear whether the same size-stretch trade-offs can be achieved.
We investigate this question by devising both upper and lower bounds for hierarchical spanners under various types of stretch and pairs of vertices of interest whose approximate (or exact) distances are to be maintained. On the positive side, a trivial adaptation of the greedy construction yields (2k-1)-spanners of size O(n^{1+1/k}), matching the classical bounds. However, the non-hierarchical bounds do not extend to the hierarchical case when additive or nearly-additive spanners are considered. For instance, we prove that any β-additive single-pair hierarchical spanner must have size Ω (n √{n/(β+1)}) in the worst case. This bound is tight, as we provide a matching upper bound for every β ≥ 0, which in turn implies a O(n√n)-size single-pair hierarchical preserver.
Finally, we present additional positive results among which a 4-additive all-pairs hierarchical spanner of size Õ(n^{5/3}), an (essentially tight) single-source hierarchical (1+ε)-spanner of size Õ(n/ε), an all-pairs hierarchical spanner of size Õ(n√{n/(ε)}) achieving stretch (1+ε,2), for any constant value of ε > 0, and a subsetwise hierarchical preserver of size O(n √{n|S|}), where S ⊆ V(𝒢).
Cite as
Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Alessandro Straziota. Hierarchical Spanners. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 145:1-145:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bilo_et_al:LIPIcs.ESA.2026.145,
author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Straziota, Alessandro},
title = {{Hierarchical Spanners}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {145:1--145:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.145},
URN = {urn:nbn:de:0030-drops-272815},
doi = {10.4230/LIPIcs.ESA.2026.145},
annote = {Keywords: Graph spanners, Network sparsification, Hierarchical graphs}
}
Document
Towards the Recognition of Oriented Interval Graphs
Abstract
Oriented interval graphs, a recent generalization of interval graphs introduced by Gutowski et al. [GD 2022], are intersection graphs of intervals, each of which is oriented either left or right. Such a representation defines a mixed intersection graph: overlapping intervals with the same orientation define a (directed) arc; nested intervals (irrespective of the orientations of the intervals) and overlapping intervals of opposite orientations define an (undirected) edge. An oriented interval representation of a mixed graph G can be described combinatorially by the combination of (i) an orientation φ : V(G) → {-1,1} of all intervals, (ii) a clique ordering σ, and (iii) a set E_cont ⊆ E(G) of containment edges, which are represented by nested intervals. The non-trivial dependencies between these three ingredients make the recognition of oriented interval graphs a challenging problem.
In this paper, we take steps towards a general recognition algorithm by studying how orientation, clique ordering, and containment edges influence and restrict each other. We characterize the orientations that are consistent with a given set of containment edges as well as the clique orderings that are consistent with a given orientation. Based on these characterizations, we give linear-time algorithms for two constrained versions of the recognition problem where, in addition to the mixed input graph G, either the set of containment edges E_cont or the orientation φ is prescribed. This improves a quadratic-time algorithm of Gutowski et al. for the case that all vertices have the same orientation; an assumption that determines both the orientation and the containment edges. In particular, this also solves the recognition problem for oriented proper (or unit) interval graphs.
Cite as
Lukas P. Bachmann, Jiří Fiala, Miriam Münch, Ignaz Rutter, Peter Stumpf, and Alexander Wolff. Towards the Recognition of Oriented Interval Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 146:1-146:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bachmann_et_al:LIPIcs.ESA.2026.146,
author = {Bachmann, Lukas P. and Fiala, Ji\v{r}{\'\i} and M\"{u}nch, Miriam and Rutter, Ignaz and Stumpf, Peter and Wolff, Alexander},
title = {{Towards the Recognition of Oriented Interval Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {146:1--146:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.146},
URN = {urn:nbn:de:0030-drops-272825},
doi = {10.4230/LIPIcs.ESA.2026.146},
annote = {Keywords: Interval graphs, mixed graphs, oriented interval graphs, recognition}
}
Document
Strongly Polynomial Parallel Maximum Flow Revisited
Abstract
We study the maximum flow problem in directed networks with real capacities in the parallel setting. For a network with n vertices and m arcs, we show that a randomized parallel implementation of a variant of the strongly polynomial max-flow algorithm of Dadush, Orlin, Sidford, and Végh [Dadush et al., 2026] runs in Õ(mn) work and Õ(m) depth. This improves upon the previously described tradeoffs between work and depth for strongly polynomial parallel maximum flow algorithms: earlier Õ(n³)-work algorithms have Õ(n²) depth [Goldberg and Tarjan, 1988; Shiloach and Vishkin, 1982], while the known Õ(m)-depth approach uses Õ(mn³) work [Orlin, 1993].
Cite as
Adam Karczmarz and Paweł Pilarski. Strongly Polynomial Parallel Maximum Flow Revisited. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 147:1-147:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{karczmarz_et_al:LIPIcs.ESA.2026.147,
author = {Karczmarz, Adam and Pilarski, Pawe{\l}},
title = {{Strongly Polynomial Parallel Maximum Flow Revisited}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {147:1--147:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.147},
URN = {urn:nbn:de:0030-drops-272835},
doi = {10.4230/LIPIcs.ESA.2026.147},
annote = {Keywords: maximum flow, parallel algorithm, work-depth tradeoff, strongly polynomial}
}
Document
An FPT Algorithm for Diverse Minimum s-t Cuts
Abstract
We study the problem of finding a family of diverse minimum edge s-t cuts in a directed weighted graph G. Given integers k and d, the task is to decide whether G contains k minimum s-t cuts C_1, …, C_k such that for any i,j ∈ [k], the number of edges in the symmetric difference C_i △ C_j is at least d.
For d ∈ {1,2}, the problem corresponds to counting minimum s-t cuts in G, which is #P-complete [Provan and Ball, SICOMP 1983]. The problem is also known to be NP-complete already for k = 3 [de Berg, López Martínez, Spieksma, ISAAC 2024]. Our main result shows that the problem is fixed-parameter tractable (FPT) when parameterized by the combined parameter k + d.
The main ingredients of our FPT algorithm build on novel structural properties of diverse minimum s-t cuts and a non-trivial application of the flow-augmentation technique of Kim, Kratsch, Pilipczuk, and Wahlström [JACM 2025].
Cite as
Krishnan Dehaleesan, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach, and Laure Morelle. An FPT Algorithm for Diverse Minimum s-t Cuts. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 148:1-148:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dehaleesan_et_al:LIPIcs.ESA.2026.148,
author = {Dehaleesan, Krishnan and Drange, P\r{a}l Gr{\o}n\r{a}s and Fomin, Fedor V. and Golovach, Petr A. and Morelle, Laure},
title = {{An FPT Algorithm for Diverse Minimum s-t Cuts}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {148:1--148:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.148},
URN = {urn:nbn:de:0030-drops-272847},
doi = {10.4230/LIPIcs.ESA.2026.148},
annote = {Keywords: Minimum s-t-cut, diversity, parameterized complexity, color coding, flow-augmentation}
}
Document
Adaptive Sampling for Minimum-Norm k-Clustering
Abstract
In k-clustering problems, we are given a metric space (𝒞, d), and must choose a set S of k centers to open. Each client j ∈ 𝒞 incurs an assignment cost, which is the distance between j and center in S that it has been assigned to. In this work, we study the minimum-norm k-clustering problem, where we are given an arbitrary monotone symmetric norm f, and wish to open k centers so as to minimize f(assignment-cost vector). This is a powerful generalization, encompassing many classical k-clustering problems including the k-median, k-means, and k-center problems.
A simple and efficient algorithmic idea is that of adaptive sampling, wherein we randomly choose the location of the next center to open with probability proportional to its "cost" under the currently chosen set. While this has yielded fast algorithms for some k-clustering problem, little is known for settings without "min-sum" objectives.
We devise the first adaptive-sampling-based bicriteria constant-factor approximation algorithm for general minimum-norm k-clustering, vastly expanding the scope of problems handled by adaptive sampling. For the special case of Top_ℓ norms, which form a building block of monotone symmetric norms, we show that adaptive sampling yields an O(log k)-approximation algorithm.
Cite as
Haripriya Pulyassary and Chaitanya Swamy. Adaptive Sampling for Minimum-Norm k-Clustering. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 149:1-149:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{pulyassary_et_al:LIPIcs.ESA.2026.149,
author = {Pulyassary, Haripriya and Swamy, Chaitanya},
title = {{Adaptive Sampling for Minimum-Norm k-Clustering}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {149:1--149:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.149},
URN = {urn:nbn:de:0030-drops-272855},
doi = {10.4230/LIPIcs.ESA.2026.149},
annote = {Keywords: Approximation algorithms, Clustering, Adaptive sampling, Randomized algorithms, Minimum-norm k-clustering}
}
Document
Query Complexity of Hypergraph Connectivity and Learnability Using CUT Oracles
Abstract
We investigate the power of CUT queries to reveal the structure of unknown hypergraphs. While simple graphs allow for optimal O(n)-query connectivity algorithms, hypergraphs face a fundamental identifiability barrier in that distinct hypergraphs can share identical cut-profiles, making exact edge learning impossible in general, a primitive crucial in the graph connectivity algorithms.
We first present a zero-error randomized algorithm that identifies the connected components of any weighted hypergraph using O(n) expected queries, matching the Ω(n) lower bound. This approach bypasses the reconstruction barrier by introducing the notion of "independent families" - vertex subpartitions that do not share hyperedges - and iteratively coarsening them using auxiliary weighted graph connectivity techniques of [Liao and Chakrabarty, 2024].
Second, we demonstrate that the impossibility of exact learning depends on hyperedge parity. For hypergraphs all of whose hyperedges have even cardinality, we show that the structure is reconstructible using a Möbius transform on the CUT function to implement binary-search-style vertex identification. This yields deterministic algorithms for obtaining k-connectivity certificates for r-bounded even hypergraphs in Õ_r(kn) queries. Finally, we bypass parity and rank constraints for linear hypergraphs, achieving a subquadratic Õ(kn^1.5) query complexity for k-connectivity. This significantly improves upon the general Õ(n²) bound derived via symmetric submodular function minimization.
Cite as
Deeparnab Chakrabarty and Hang Liao. Query Complexity of Hypergraph Connectivity and Learnability Using CUT Oracles. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 150:1-150:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chakrabarty_et_al:LIPIcs.ESA.2026.150,
author = {Chakrabarty, Deeparnab and Liao, Hang},
title = {{Query Complexity of Hypergraph Connectivity and Learnability Using CUT Oracles}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {150:1--150:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.150},
URN = {urn:nbn:de:0030-drops-272860},
doi = {10.4230/LIPIcs.ESA.2026.150},
annote = {Keywords: Query Complexity, CUT queries, hypergraph connectivity}
}
Document
Indexing Integer Strings Using Local Difference Bounds
Abstract
Time series data can often be represented as a string T[1..n] over an integer alphabet Σ. A standard approach for indexing such strings is the compressed suffix tree, which supports efficient exact pattern matching using 𝒪(n log |Σ|) bits of space. However, when |Σ| is close to n, as is often the case for time-series data, this yields little to no space savings over the classical Θ(n log n)-bit suffix tree.
In this work, we study a different parameter that is often much smaller than |Σ|: the maximum absolute difference between consecutive values in T, denoted by Δ. Although representing T in 𝒪(n log Δ) bits is straightforward, supporting efficient pattern matching within this space bound remains challenging. By leveraging succinct data structure techniques, particularly the FM-index, we obtain a compressed index occupying n log Δ + 𝒪(n) bits. Given a query pattern P[1..m], the index answers counting queries in 𝒪(m log Δ) time and reporting queries in 𝒪(m log Δ + occ ⋅ log n ⋅ log Δ) time, where occ denotes the number of occurrences of P in T. In addition, the structure supports suffix array and inverse suffix array queries in 𝒪(log n ⋅ log Δ) time.
Our construction is conceptually simple. We show that the local-difference bound induces a strong form of locality in the Burrows-Wheeler Transform (BWT), allowing the transformed text to be decomposed into regions that each use only a small local alphabet of size 𝒪(Δ). This makes it possible to adapt the classical FM-index so that its space usage depends on Δ rather than the overall alphabet size, while still supporting efficient pattern matching queries.
Cite as
Daniel Gibney, Kaamil Kaka, and Sharma V. Thankachan. Indexing Integer Strings Using Local Difference Bounds. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 151:1-151:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gibney_et_al:LIPIcs.ESA.2026.151,
author = {Gibney, Daniel and Kaka, Kaamil and Thankachan, Sharma V.},
title = {{Indexing Integer Strings Using Local Difference Bounds}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {151:1--151:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.151},
URN = {urn:nbn:de:0030-drops-272870},
doi = {10.4230/LIPIcs.ESA.2026.151},
annote = {Keywords: String Algorithms, Pattern Matching, Suffix Trees, Compact Data Structures}
}
Document
On the Complexity of Multipacking
Abstract
A multipacking in an undirected graph G = (V, E) is a set M ⊆ V such that for every vertex v ∈ V and for every integer r ≥ 1, the ball of radius r around v contains at most r vertices of M, that is, there are at most r vertices in M at a distance at most r from v in G. The Multipacking problem asks whether a graph contains a multipacking of size at least k.
For more than a decade, it remained an open question whether the Multipacking problem is NP-complete or solvable in polynomial time, although the problem is known to be polynomial-time solvable for certain graph classes (e.g., strongly chordal graphs, grids, etc). Foucaud, Gras, Perez, and Sikora [Foucaud et al., 2021] [Algorithmica 2021] made a step towards solving the open question by showing that the Multipacking problem is NP-complete for directed graphs and W[1]-hard when parameterized by the solution size.
In this paper, we prove that the Multipacking problem is NP-complete on undirected graphs, which answers the open question. Moreover, the problem is W[2]-hard on undirected graphs when parameterized by the solution size. Furthermore, we show that the problem is NP-complete and W[2]-hard (parameterized by solution size) on chordal, bipartite, and claw-free graphs, and remains NP-complete on regular and CONV graphs (intersection graphs of convex sets in the plane). Additionally, the problem is NP-complete and W[2]-hard (parameterized by the solution size) on chordal ∩ 1/2-hyperbolic graphs, which is a superclass of strongly chordal graphs on which the problem is polynomial-time solvable. On the positive side, we present an exact exponential-time algorithm for the Multipacking problem on general graphs that breaks the 2ⁿ barrier, with running time O^*(1.58ⁿ), where n is the number of vertices.
Cite as
Sandip Das, Sk Samim Islam, and Daniel Lokshtanov. On the Complexity of Multipacking. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 152:1-152:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{das_et_al:LIPIcs.ESA.2026.152,
author = {Das, Sandip and Islam, Sk Samim and Lokshtanov, Daniel},
title = {{On the Complexity of Multipacking}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {152:1--152:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.152},
URN = {urn:nbn:de:0030-drops-272885},
doi = {10.4230/LIPIcs.ESA.2026.152},
annote = {Keywords: Multipacking, Complexity, Exact exponential algorithms}
}
Document
Covering Points with Rectangular Boundaries
Abstract
Geometric covering problems typically ask for a small family of geometric objects whose union contains all input points. In this paper we study a more rigid variant, boundary covering, where every point must lie on the boundary of at least one chosen object. Motivated by the framework of Langerman and Morin [Discret. Comput. Geom., 2005] for boundary covering by hyperspheres, we initiate a systematic study of boundary covering by axis-parallel rectangles in the plane.
We first consider the discrete setting, where the rectangles must be chosen from a given family. We define Boundary Covering with Discrete Axis-Parallel Rectangles (BCDAPR) as follows: given a point set P ⊆ ℝ², a collection ℛ of axis-parallel rectangles, and an integer k, decide whether P can be covered by the boundaries of at most k rectangles from ℛ. We prove that this discrete boundary-covering problem is W[1]-hard when parameterized by k.
This motivates the continuous variant, where we are allowed to place rectangles freely. We define Boundary Covering with Continuous Axis-Parallel Rectangles (BCCAPR) as follows: given a point set P ⊆ ℝ² and an integer k, decide whether P can be covered by the boundaries of at most k axis-parallel rectangles. In contrast to the discrete case, we show that BCCAPR is fixed-parameter tractable parameterized by k, with running time 2^𝒪(k log k) ⋅ n^𝒪(1), where n = |P|. Our results does a fine-grained structural analysis of how k rectangles can interact with the point set. On the hardness side, we show that moving from lines to slightly richer shapes already incurs intractability: we prove NP-completeness for boundary covering by axis-aligned L-shapes, and then lift it to NP-completeness of BCCAPR. For the algorithm we reduce BCCAPR to at most 2^𝒪(k log k) instances of Distinct Domain Monotone ,$-CSP, each solvable in polynomial time.
Cite as
Madhumita Kundu, Daniel Lokshtanov, Soumi Nandi, Saket Saurabh, and Kushal Singanporia. Covering Points with Rectangular Boundaries. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 153:1-153:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kundu_et_al:LIPIcs.ESA.2026.153,
author = {Kundu, Madhumita and Lokshtanov, Daniel and Nandi, Soumi and Saurabh, Saket and Singanporia, Kushal},
title = {{Covering Points with Rectangular Boundaries}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {153:1--153:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.153},
URN = {urn:nbn:de:0030-drops-272897},
doi = {10.4230/LIPIcs.ESA.2026.153},
annote = {Keywords: Geometric Covering, Axis-parallel Rectangles, W\lbrack1\rbrack and NP Hardness, Fixed Parameter Tractability, CSP}
}
Document
An LCA for Approximated MST in General Bounded-Degree Graphs
Abstract
We present a local computation algorithm (LCA) for constructing a connected spanning subgraph whose total weight is at most a (1+ε)-factor larger than that of a minimum spanning tree, in general bounded-degree graphs. Prior to our work, nontrivial LCAs for this problem, namely, algorithms with sublinear query complexity, were known only for the restricted graph family of minor-free graphs by Levi, Ron, and Rubinfeld (Algorithmica 2020).
The query complexity of our algorithm in terms of the number of vertices, n, is Õ(n^{2/3}). The best known lower bound for this problem is Ω(n^{1/2}).
Our approach consists of three conceptual layers. The first is a localized variant of Prim’s algorithm, which reconstructs, using only local queries, a large fraction of the edges of the minimum spanning tree. The resulting subgraph at this stage is disconnected. To address this, in the second layer, we partition the partially constructed forest into clusters of size Õ(n^{1/3}). To this end, we present a partition oracle, as introduced by Hassidim et al. (FOCS 2009), for trees whose query complexity is nearly optimal in terms of ε, the parameter that controls the number of edges in the boundary. In particular, its query complexity is Õ(d/ε), where d denotes the degree bound. In the third and last layer, we adapt the technique from Lenzen-Levi (ICALP 2018) for locally computing a sparse spanning subgraph and obtain an algorithm that locally identifies and adds a small number of carefully chosen edges in order to restore global connectivity. We show that the number of such additional edges is small, and consequently, their total weight contributes only a small amount to the overall cost, preserving the (1+ε)-approximation guarantee.
Cite as
Reut Levi, Moti Medina, and Daniel Prigan. An LCA for Approximated MST in General Bounded-Degree Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 154:1-154:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{levi_et_al:LIPIcs.ESA.2026.154,
author = {Levi, Reut and Medina, Moti and Prigan, Daniel},
title = {{An LCA for Approximated MST in General Bounded-Degree Graphs}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {154:1--154:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.154},
URN = {urn:nbn:de:0030-drops-272902},
doi = {10.4230/LIPIcs.ESA.2026.154},
annote = {Keywords: Locally Computable Algorithms, Sublinear Algorithms, Minimum Spanning Trees, Partition Oracles}
}
Document
A General Reduction from Near-Additive Emulators to Near-Exact Hopsets
Abstract
Graph emulators and hopsets are two fundamental concepts for distance approximation. For a given graph G, an (α,β)-emulator is a sparse graph on the same vertex set that preserves the distances of G up to a multiplicative stretch α and additive stretch β. In contrast, an (α,β)-hopset is a set of additional edges that, when added to G, ensures that distances can be approximated up to a multiplicative stretch α, using paths containing at most β edges. When α = 1+ε for arbitrarily small ε > 0, these structures are known as near-additive emulators and near-exact hopsets, respectively. Prior work showed that there is a remarkable similarity between the constructions and guarantees of these two objects. In their survey on this topic, Elkin and Neiman [Bull. EATCS 130, 2020] explicitly asked whether one can obtain a general reduction between near-additive emulators and near-exact hopsets. Following that, Kogan and Parter [FOCS, 2022] provided a general reduction from hopsets to emulators and spanners.
In this paper, we address the reverse direction and show that any construction for a near-additive emulator for undirected unweighted graphs can be leveraged as a black box to construct a hopset for an undirected weighted graph with comparable size, stretch, and a hopbound comparable to the emulator’s additive stretch. Specifically, we show that any algorithm that constructs a (1+ε',β)-emulator, with 0 ≤ ε' ≤ 1 and β ≥ 1, of size S_𝒜(n, ε',β), can be used to obtain a (1+ε, O(β²/ε² ln(n/ε)))-hopset of size O((S_𝒜(n + m β/ε², ε/294, β) 1/ε + n) ln(n/ε)), for any 0 < ε ≤ 1. Therefore, our reduction answers the question of Elkin and Neiman [Bull. EATCS 130, 2020] for sparse graphs and further advances the understanding of the formal connection between these two structures. Designing a reduction resulting in a hopset size that does not depend on m remains an intriguing open question.
Cite as
Julian Aeri, Sebastian Forster, and Mara Grilnberger. A General Reduction from Near-Additive Emulators to Near-Exact Hopsets. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 155:1-155:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aeri_et_al:LIPIcs.ESA.2026.155,
author = {Aeri, Julian and Forster, Sebastian and Grilnberger, Mara},
title = {{A General Reduction from Near-Additive Emulators to Near-Exact Hopsets}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {155:1--155:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.155},
URN = {urn:nbn:de:0030-drops-272919},
doi = {10.4230/LIPIcs.ESA.2026.155},
annote = {Keywords: hopsets, shortest paths, emulator-to-hopset-reduction}
}
Document
The Power of the Score Sequence of a Tournament
Abstract
What problems can one solve on a tournament if only its score sequence is known?
Tournaments are oriented complete graphs that form an extensively-studied class of directed graphs (digraphs), both from combinatorial and algorithmic perspectives. Over the years, researchers have identified multiple classical digraph problems that can be solved on a tournament from only its score sequence (indegree sequence). These problems include acyclicity testing and topological sorting [Chakrabarti, Ghosh, McGregor, and Vorotnikova; SODA'20], s,t-reachability, strong connectivity, and decomposition into strongly connected components (SCC) [Ghosh and Kuchlous; ESA'24], and vertex-ordering problems such as cutwidth and optimal linear arrangement [Barbero, Paul, and Pilipczuk; ICALP'17]. These prior works showed the sufficiency of the score sequence by designing distinct algorithms for the individual problems. In this work, we give a simple unified framework that solves all these problems using only indegrees and, in fact, completely characterises the class of problems that is determined by the indegree information: problems whose answers are invariant under cycle reversals.
As a byproduct of our results, we obtain algorithms for a variety of connectivity-based, cut-based, and vertex-ordering problems on tournaments and almost-tournaments in the streaming, the two-player communication, and the cut-query models of computation. Some of these algorithms match existing optimal bounds and others provide new bounds improving the state of the art. Specifically, our polynomial-time algorithms for almost-tournaments improve upon the exponential-time algorithms of Ghosh and Kuchlous and have much simpler analysis.
The said characterisation is a special case of a much more general result that we establish: for any arbitrary digraph, the knowledge of its skeleton (underlying undirected graph) and the vertex indegrees completely determines its properties that are invariant under cycle reversal. In particular, this gives us an O(n²)-cut-query algorithm to solve directed minimum cut on n-node graphs in polynomial time, a significant result that has been observed in the literature but not concretely stated in this form. Our results also unveil interesting general connections between two well-studied sublinear models for graph problems: semi-streaming and cut-query.
Cite as
Prantar Ghosh, Sahil Kuchlous, Shravan Mehra, and Sagnik Mukhopadhyay. The Power of the Score Sequence of a Tournament. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 156:1-156:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ghosh_et_al:LIPIcs.ESA.2026.156,
author = {Ghosh, Prantar and Kuchlous, Sahil and Mehra, Shravan and Mukhopadhyay, Sagnik},
title = {{The Power of the Score Sequence of a Tournament}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {156:1--156:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.156},
URN = {urn:nbn:de:0030-drops-272928},
doi = {10.4230/LIPIcs.ESA.2026.156},
annote = {Keywords: tournaments, score sequence, cycle reversal, streaming algorithms, graph connectivity, cut queries, min-cut}
}
Document
Robustifying Sparse Matrix Multiplication
Abstract
In the seminal sparse matrix multiplication problem the goal is to compute the product of two n × n matrices when the matrices are sparse, i.e., when the number of nonzeros in the input matrices m_in and/or the number of nonzeros in the output matrix m_out are much smaller than n². In this paper, we explore the generalized problem of (approximately) computing the k largest output entries, with an approximation error dependent solely on the smaller entries - from the viewpoint of sparse recovery, this can be seen as a robust variant of sparse matrix multiplication. Despite the substantial research dedicated to sparse matrix multiplication, almost no existing algorithms are robust in this sense. The one exception is Pagh’s algorithm in time Õ(m_in + nk) [ITCS '12], and it remained open whether other algorithms can be similarly made robust.
Our principal contribution is a black-box reduction from robust sparse matrix multiplication to conventional sparse matrix multiplication with only polylogarithmic overhead. Specifically, we show that any sparse matrix multiplication algorithm with running time T(n, m_in, m_out) can be transformed into a robust algorithm running in time Õ(T(n, m_in, k)). This reduction leverages an extensive toolkit from sparse recovery, and intriguingly, also involves solving a knapsack-type problem.
By plugging in the state-of-the-art algorithm for sparse matrix multiplication by Abboud, Bringmann, Fischer, and Künnemann [SODA'24], we achieve significantly improved bounds such as O((m_in + k)^1.346). Notably, in the regime where k ≥ m_in^1.762, our reduction culminates in an almost-optimal k^{1+o(1)}-time algorithm.
Cite as
Karl Bringmann, Nick Fischer, and Vasileios Nakos. Robustifying Sparse Matrix Multiplication. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 157:1-157:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bringmann_et_al:LIPIcs.ESA.2026.157,
author = {Bringmann, Karl and Fischer, Nick and Nakos, Vasileios},
title = {{Robustifying Sparse Matrix Multiplication}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {157:1--157:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.157},
URN = {urn:nbn:de:0030-drops-272932},
doi = {10.4230/LIPIcs.ESA.2026.157},
annote = {Keywords: matrix multiplication, robustness, sparse recovery, Knapsack}
}
Document
Online and Incremental Fractional Vertex Cover on Trees
Abstract
In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an 11/6 ≈ 1.83-competitive algorithm for trees in the more general edge arrival model.
In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.
Cite as
Júlia Baligács, Bartłomiej Bosek, Yann Disser, Andreas Emil Feldmann, Grzegorz Gutowski, Katarzyna Kępińska, Paweł Putra, and Anna Zych-Pawlewicz. Online and Incremental Fractional Vertex Cover on Trees. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 158:1-158:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{baligacs_et_al:LIPIcs.ESA.2026.158,
author = {Balig\'{a}cs, J\'{u}lia and Bosek, Bart{\l}omiej and Disser, Yann and Feldmann, Andreas Emil and Gutowski, Grzegorz and K\k{e}pi\'{n}ska, Katarzyna and Putra, Pawe{\l} and Zych-Pawlewicz, Anna},
title = {{Online and Incremental Fractional Vertex Cover on Trees}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {158:1--158:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.158},
URN = {urn:nbn:de:0030-drops-272943},
doi = {10.4230/LIPIcs.ESA.2026.158},
annote = {Keywords: fractional vertex cover, online algorithms, incremental algorithms, edge arrival model, trees, competitive analysis}
}
Document
Hardness of Multi-Agent Path Finding on Trees: A Unified Approach
Abstract
This paper presents a simple framework that settles the complexity of Multi-Agent Path Finding (MAPF) on trees across standard objectives - distance, makespan, and flowtime - for both labeled and colored variants. In MAPF, agents occupy the vertices of a graph and must move to target vertices without collisions while optimizing a given objective. In the labeled case, the agents are distinct and have respective targets; in the colored case, agents of the same color are interchangeable. While many MAPF variants are known to be intractable, several basic cases on trees have remained open. We prove NP-hardness on trees for both labeled and 2-colored MAPF under all three objectives. In particular, we resolve the classical Pebble Motion problem, where one pebble moves at a time to an adjacent empty vertex and the goal is to minimize the total number of moves. Despite being one of the most basic discrete motion models, its complexity on trees had remained open for several decades. Moreover, for colored Pebble Motion, we give the first hardness result on any graph class, already with two colors, which is tight.
All of these results are established through the hardness of Stack Rearrangement, itself posed as an open problem, which asks to optimally rearrange items stored in stacks, and which we also prove to be NP-hard. Notably, the connection to stacks yields hardness already on very simple trees - subdivided stars - across all problems. Together, these results reveal a common tractability barrier that permeates several fundamental motion models, thereby unifying and strengthening prior hardness results.
Cite as
Tzvika Geft. Hardness of Multi-Agent Path Finding on Trees: A Unified Approach. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 159:1-159:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{geft:LIPIcs.ESA.2026.159,
author = {Geft, Tzvika},
title = {{Hardness of Multi-Agent Path Finding on Trees: A Unified Approach}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {159:1--159:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.159},
URN = {urn:nbn:de:0030-drops-272957},
doi = {10.4230/LIPIcs.ESA.2026.159},
annote = {Keywords: Pebble Motion on Trees, Coordinated Motion Planning, Multi-Agent Path Finding, Stack Rearrangement, NP-Hardness}
}
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Shortest Path Map Equivalence Decompositions and Applications
Abstract
Given a polygonal domain 𝒫 in the plane, the shortest path map with respect to a point s, denoted by SPM(s), is the decomposition of 𝒫 into cells such that shortest paths from s to all points t in the same cell have the same vertex sequence. The shortest path map equivalence decomposition of 𝒫 is the decomposition of 𝒫 into cells so that SPM(s) is topologically equivalent for all points s in the same cell. In this paper, we prove new upper bounds on the combinatorial complexities of the SPM-equivalence decompositions under various settings, depending on whether s and/or t are restricted to be the boundary of 𝒫. We also propose new algorithms to compute these decompositions. Further, our results lead to new solutions to several other problems, including answering two-point shortest path queries in 𝒫, and computing geodesic diameter and center of 𝒫.
Cite as
Haitao Wang. Shortest Path Map Equivalence Decompositions and Applications. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 160:1-160:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wang:LIPIcs.ESA.2026.160,
author = {Wang, Haitao},
title = {{Shortest Path Map Equivalence Decompositions and Applications}},
booktitle = {34th Annual European Symposium on Algorithms (ESA 2026)},
pages = {160:1--160:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-445-1},
ISSN = {1868-8969},
year = {2026},
volume = {388},
editor = {Bille, Philip and Pettie, Seth and Storandt, Sabine},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.160},
URN = {urn:nbn:de:0030-drops-272966},
doi = {10.4230/LIPIcs.ESA.2026.160},
annote = {Keywords: shortest paths, shortest path maps, SPM-equivalent decompositions, two-point shortest path queries, geodesic diameter, geodesic center, polygononal domains}
}