Volume
LIPIcs, Volume 306
49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS)
Event
MFCS 2024, August 26-30, 2024, Bratislava, SlovakiaEditors
- Comenius University, Faculty of Mathematics, Physics, and Informatics, Bratislava, Slovakia
Publication Details
- published at: 2024-08-23
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-335-5
- DBLP: db/conf/mfcs/mfcs2024
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Complete Volume
LIPIcs, Volume 306, MFCS 2024, Complete Volume
Abstract
LIPIcs, Volume 306, MFCS 2024, Complete Volume
Cite as
49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 1-1362, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@Proceedings{kralovic_et_al:LIPIcs.MFCS.2024,
title = {{LIPIcs, Volume 306, MFCS 2024, Complete Volume}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {1--1362},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024},
URN = {urn:nbn:de:0030-drops-205555},
doi = {10.4230/LIPIcs.MFCS.2024},
annote = {Keywords: LIPIcs, Volume 306, MFCS 2024, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kralovic_et_al:LIPIcs.MFCS.2024.0,
author = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {0:i--0:xviii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.0},
URN = {urn:nbn:de:0030-drops-205568},
doi = {10.4230/LIPIcs.MFCS.2024.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Paper
On Key Parameters Affecting the Realizability of Degree Sequences (Invited Paper)
Abstract
Call a sequence d = (d_1,d_2, …, d_n) of positive integers graphic, planaric, outer-planaric, or forestic if it is the degree sequence of some arbitrary, planar, outer-planar, or cycle-free graph G, respectively. The two extreme classes of graphic and forestic sequences were given full characterizations. (The latter has a particularly simple criterion: d is forestic if and only if its volume, ∑ d ≡ ∑_i d_i, satisfies ∑ d ≤ 2n - 2.) In contrast, the problems of fully characterizing planaric and outer-planaric degree sequences are still open.
In this paper, we discuss the parameters affecting the realizability of degree sequences by restricted classes of sparse graph, including planar graphs, outerplanar graphs, and some of their subclasses (e.g., 2-trees and cactus graphs). A key parameter is the volume of the sequence d, namely, ∑ d which is twice the number of edges in the realizing graph. For planar graphs, for example, an obvious consequence of Euler’s theorem is that an n-element sequence d satisfying ∑ d > 4n-6 cannot be planaric. Hence, ∑ d ≤ 4n-6 is a necessary condition for d to be planaric. What about the opposite direction? Is there an upper bound on ∑ d that guarantees that if d is graphic then it is also planaric. Does the answer depend on additional parameters? The same questions apply also to sub-classes of the planar graphs.
A concrete example that is illustrated in the technical part of the paper is the class of outer-planaric degree sequences. Denoting the number of 1’s in d by ω₁, we show that for a graphic sequence d, if ω₁ = 0 then d is outer-planaric when ∑ d ≤ 3n-3, and if ω₁ > 0 then d is outer-planaric when ∑ d ≤ 3n-ω₁-2. Conversely, we show that there are graphic sequences that are not outer-planaric with ω₁ = 0 and ∑ d = 3n-2, as well as ones with ω₁ > 0 and ∑ d = 3n-ω₁-1.
Cite as
Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. On Key Parameters Affecting the Realizability of Degree Sequences (Invited Paper). In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{barnoy_et_al:LIPIcs.MFCS.2024.1,
author = {Bar-Noy, Amotz and B\"{o}hnlein, Toni and Peleg, David and Ran, Yingli and Rawitz, Dror},
title = {{On Key Parameters Affecting the Realizability of Degree Sequences}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {1:1--1:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.1},
URN = {urn:nbn:de:0030-drops-205573},
doi = {10.4230/LIPIcs.MFCS.2024.1},
annote = {Keywords: Degree Sequences, Graph Algorithms, Graph Realization, Outer-planar Graphs}
}
Document
Invited Paper
Challenges of the Reachability Problem in Infinite-State Systems (Invited Paper)
Abstract
The reachability problem is a central problem for various infinite state systems like automata with pushdown, with different kinds of counters or combinations thereof. Despite its centrality and decades of research the community still lacks a lot of answers for fundamental and basic questions of that type. I briefly describe my personal viewpoint on the current state of art and emphasise interesting directions, which are worth investigating in my opinion. I also formulate several easy to formulate and understand challenges, which might be pretty hard to solve but at the same time illustrate fundamental lack of our understanding in the area.
Cite as
Wojciech Czerwiński. Challenges of the Reachability Problem in Infinite-State Systems (Invited Paper). In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 2:1-2:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{czerwinski:LIPIcs.MFCS.2024.2,
author = {Czerwi\'{n}ski, Wojciech},
title = {{Challenges of the Reachability Problem in Infinite-State Systems}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {2:1--2:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.2},
URN = {urn:nbn:de:0030-drops-205582},
doi = {10.4230/LIPIcs.MFCS.2024.2},
annote = {Keywords: reachability problem, infinite-state systems, vector addition systems, pushdown}
}
Document
Invited Talk
On Low Complexity Colorings of Grids (Invited Talk)
Abstract
A d-dimensional configuration is a coloring of the infinite grid ℤ^d using a finite number of colors. For a finite subset D ⊆ ℤ^d, the D-patterns of a configuration are the patterns of shape D that appear in the configuration. A configuration is said to be admitted by these patterns. The number of distinct D-patterns in a configuration is a natural measure of its complexity. We focus on low complexity configurations, where the number of distinct D-patterns is at most |D|, the size of the shape. This framework includes the notorious open Nivat’s conjecture and the recently solved Periodic Tiling problem. We use algebraic tools to study the periodicity of low complexity configurations. In the two-dimensional case, if D ⊆ ℤ² is a rectangle or any convex shape, we establish an algorithm to determine if a given collection of |D| patterns admits any configuration. This is based on the fact that if the given patterns admit a configuration, then they admit a periodic configuration. We also demonstrate that a two-dimensional low complexity configuration must be periodic if it originates from the well-known Ledrappier subshift or from several other algebraically defined subshifts.
Cite as
Jarkko Kari. On Low Complexity Colorings of Grids (Invited Talk). In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 3:1-3:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kari:LIPIcs.MFCS.2024.3,
author = {Kari, Jarkko},
title = {{On Low Complexity Colorings of Grids}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {3:1--3:2},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.3},
URN = {urn:nbn:de:0030-drops-205590},
doi = {10.4230/LIPIcs.MFCS.2024.3},
annote = {Keywords: symbolic dynamics, Nivat’s conjecture, Periodic tiling problem, periodicity, low pattern complexity, annihilator}
}
Document
Invited Paper
From TCS to Learning Theory (Invited Paper)
Abstract
While machine learning theory and theoretical computer science are both based on a solid mathematical foundation, the two research communities have a smaller overlap than what the proximity of the fields warrant. In this invited abstract, I will argue that traditional theoretical computer scientists have much to offer the learning theory community and vice versa. I will make this argument by telling a personal story of how I broadened my research focus to encompass learning theory, and how my TCS background has been extremely useful in doing so. It is my hope that this personal account may inspire more TCS researchers to tackle the many elegant and important theoretical questions that learning theory has to offer.
Cite as
Kasper Green Larsen. From TCS to Learning Theory (Invited Paper). In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 4:1-4:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{larsen:LIPIcs.MFCS.2024.4,
author = {Larsen, Kasper Green},
title = {{From TCS to Learning Theory}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {4:1--4:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.4},
URN = {urn:nbn:de:0030-drops-205603},
doi = {10.4230/LIPIcs.MFCS.2024.4},
annote = {Keywords: Theoretical Computer Science, Learning Theory}
}
Document
Invited Talk
Fine-Grained Complexity of Program Analysis (Invited Talk)
Abstract
There is a well-known "cubic bottleneck" in program analysis and language theory: many program analysis problems can be solved in time cubic in the size of the input but, despite years of effort, there are no known sub-cubic algorithms. For example, context-free reachability (whether there is a path in a labeled graph that is labeled with a word from a context-free language), the emptiness problem for pushdown automata, and the recognition problem for two-way nondeterministic pushdown automata all belong to the cubic class. We survey the status of these problems through the lens of fine-grained complexity.
We study the related certification task: given an instance of any of these problems, are there small and efficiently checkable certificates for the existence and for the non-existence of a path? We show that, in both scenarios, there exist succinct certificates (O(n²) in the size of the problem) and these certificates can be checked in subcubic (matrix multiplication) time. Thus, all these problems lie in nondeterministic and co-nondeterministic subcubic time.
We also study a hierarchy of program analysis problems above the cubic bottleneck. A representative problem here is the recognition problem for two-way nondeterministic pushdown automata with k heads. We show fine-grained hardness results for this hierarchy.
We also discuss purely language-theoretic consequences of these results: for example, we obtain hardest languages accepted by two-way nondeterministic multihead pushdown automata, as well as separations between language classes.
(Joint work with A. R. Balasubramanian, Dmitry Chistikov, and Philipp Schepper.)
Cite as
Rupak Majumdar. Fine-Grained Complexity of Program Analysis (Invited Talk). In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, p. 5:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{majumdar:LIPIcs.MFCS.2024.5,
author = {Majumdar, Rupak},
title = {{Fine-Grained Complexity of Program Analysis}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {5:1--5:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.5},
URN = {urn:nbn:de:0030-drops-205619},
doi = {10.4230/LIPIcs.MFCS.2024.5},
annote = {Keywords: Fine-grained complexity, CFL reachability, 2NPDA recognition, PDA emptiness}
}
Document
Monotonicity of the Cops and Robber Game for Bounded Depth Treewidth
Abstract
We study a variation of the cops and robber game characterising treewidth, where in each round at most one cop may be placed and in each play at most q rounds are played, where q is a parameter of the game. We prove that if k cops have a winning strategy in this game, then k cops have a monotone winning strategy. As a corollary we obtain a new characterisation of bounded depth treewidth, and we give a positive answer to an open question by Fluck, Seppelt and Spitzer (2024), thus showing that graph classes of bounded depth treewidth are homomorphism distinguishing closed.
Our proof of monotonicity substantially reorganises a winning strategy by first transforming it into a pre-tree decomposition, which is inspired by decompositions of matroids, and then applying an intricate breadth-first "cleaning up" procedure along the pre-tree decomposition (which may temporarily lose the property of representing a strategy), in order to achieve monotonicity while controlling the number of rounds simultaneously across all branches of the decomposition via a vertex exchange argument. We believe this can be useful in future research.
Cite as
Isolde Adler and Eva Fluck. Monotonicity of the Cops and Robber Game for Bounded Depth Treewidth. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{adler_et_al:LIPIcs.MFCS.2024.6,
author = {Adler, Isolde and Fluck, Eva},
title = {{Monotonicity of the Cops and Robber Game for Bounded Depth Treewidth}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {6:1--6:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.6},
URN = {urn:nbn:de:0030-drops-205621},
doi = {10.4230/LIPIcs.MFCS.2024.6},
annote = {Keywords: tree decompositions, treewidth, treedepth, cops-and-robber game, monotonicity, homomorphism distinguishing closure}
}
Document
Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds
Abstract
The Polynomial-Time Hierarchy (PH) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to "quantum advantage" analyses for near-term quantum computers. Quantumly, however, despite the fact that at least four definitions of quantum PH exist, it has been challenging to prove analogues for these of even basic facts from PH. This work studies three quantum-verifier based generalizations of PH, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings (QCPH) and quantum mixed states (QPH) as proofs, and one of which is new to this work, utilizing quantum pure states (QPHpure) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for QCPH. Then, for our new class QPHpure, we show one-sided error reduction QPHpure, as well as the first bounds relating these quantum variants of PH, namely QCPH ⊆ QPHpure ⊆ EXP^PP.
Cite as
Avantika Agarwal, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph. Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{agarwal_et_al:LIPIcs.MFCS.2024.7,
author = {Agarwal, Avantika and Gharibian, Sevag and Koppula, Venkata and Rudolph, Dorian},
title = {{Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {7:1--7:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.7},
URN = {urn:nbn:de:0030-drops-205632},
doi = {10.4230/LIPIcs.MFCS.2024.7},
annote = {Keywords: Quantum complexity, polynomial hierarchy}
}
Document
Sublinear Time Shortest Path in Expander Graphs
Abstract
Computing a shortest path between two nodes in an undirected unweighted graph is among the most basic algorithmic tasks. Breadth first search solves this problem in linear time, which is clearly also a lower bound in the worst case. However, several works have shown how to solve this problem in sublinear time in expectation when the input graph is drawn from one of several classes of random graphs. In this work, we extend these results by giving sublinear time shortest path (and short path) algorithms for expander graphs. We thus identify a natural deterministic property of a graph (that is satisfied by typical random regular graphs) which suffices for sublinear time shortest paths. The algorithms are very simple, involving only bidirectional breadth first search and short random walks. We also complement our new algorithms by near-matching lower bounds.
Cite as
Noga Alon, Allan Grønlund, Søren Fuglede Jørgensen, and Kasper Green Larsen. Sublinear Time Shortest Path in Expander Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{alon_et_al:LIPIcs.MFCS.2024.8,
author = {Alon, Noga and Gr{\o}nlund, Allan and J{\o}rgensen, S{\o}ren Fuglede and Larsen, Kasper Green},
title = {{Sublinear Time Shortest Path in Expander Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {8:1--8:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.8},
URN = {urn:nbn:de:0030-drops-205646},
doi = {10.4230/LIPIcs.MFCS.2024.8},
annote = {Keywords: Shortest Path, Expanders, Breadth First Search, Graph Algorithms}
}
Document
Quantum Algorithms for Hopcroft’s Problem
Abstract
In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given n points and n lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in O(n^{4/3}) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity Õ(n^{5/6}). The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.
Cite as
Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs. Quantum Algorithms for Hopcroft’s Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{andrejevs_et_al:LIPIcs.MFCS.2024.9,
author = {Andrejevs, Vladimirs and Belovs, Aleksandrs and Vihrovs, Jevg\={e}nijs},
title = {{Quantum Algorithms for Hopcroft’s Problem}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {9:1--9:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.9},
URN = {urn:nbn:de:0030-drops-205653},
doi = {10.4230/LIPIcs.MFCS.2024.9},
annote = {Keywords: Quantum algorithms, Quantum walks, Computational Geometry}
}
Document
A New Characterization of FAC⁰ via Discrete Ordinary Differential Equations
Abstract
Implicit computational complexity is an active area of theoretical computer science, which aims at providing machine-independent characterizations of relevant complexity classes. One of the seminal works in this field appeared in 1965, when Cobham introduced a function algebra closed under bounded recursion on notation to capture FP. Later on, several complexity classes have been characterized using limited recursion schemas. In this context, a new approach was recently introduced, showing that ordinary differential equations (ODEs) offer a natural tool for algorithmic design and providing a characterization of FP by an ODE-schema. The overall goal of the present work is precisely that of generalizing this approach to parallel computation, obtaining an original ODE-characterization for the small circuit classes FAC⁰ and FTC⁰.
Cite as
Melissa Antonelli, Arnaud Durand, and Juha Kontinen. A New Characterization of FAC⁰ via Discrete Ordinary Differential Equations. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{antonelli_et_al:LIPIcs.MFCS.2024.10,
author = {Antonelli, Melissa and Durand, Arnaud and Kontinen, Juha},
title = {{A New Characterization of FAC⁰ via Discrete Ordinary Differential Equations}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {10:1--10:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.10},
URN = {urn:nbn:de:0030-drops-205664},
doi = {10.4230/LIPIcs.MFCS.2024.10},
annote = {Keywords: Implicit computational complexity, parallel computation, ordinary differential equations, circuit complexity}
}
Document
Switching Classes: Characterization and Computation
Abstract
In a graph, the switching operation reverses adjacencies between a subset of vertices and the others. For a hereditary graph class 𝒢, we are concerned with the maximum subclass and the minimum superclass of 𝒢 that are closed under switching. We characterize the maximum subclass for many important classes 𝒢, and prove that it is finite when 𝒢 is minor-closed and omits at least one graph. For several graph classes, we develop polynomial-time algorithms to recognize the minimum superclass. We also show that the recognition of the superclass is NP-hard for H-free graphs when H is a sufficiently long path or cycle, and it cannot be solved in subexponential time assuming the Exponential Time Hypothesis.
Cite as
Dhanyamol Antony, Yixin Cao, Sagartanu Pal, and R. B. Sandeep. Switching Classes: Characterization and Computation. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{antony_et_al:LIPIcs.MFCS.2024.11,
author = {Antony, Dhanyamol and Cao, Yixin and Pal, Sagartanu and Sandeep, R. B.},
title = {{Switching Classes: Characterization and Computation}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {11:1--11:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.11},
URN = {urn:nbn:de:0030-drops-205678},
doi = {10.4230/LIPIcs.MFCS.2024.11},
annote = {Keywords: Switching, Graph modification, Minor-closed graph class, Hereditary graph class}
}
Document
Generalizing Roberts' Characterization of Unit Interval Graphs
Abstract
For any natural number d, a graph G is a (disjoint) d-interval graph if it is the intersection graph of (disjoint) d-intervals, the union of d (disjoint) intervals on the real line. Two important subclasses of d-interval graphs are unit and balanced d-interval graphs (where every interval has unit length or all the intervals associated to a same vertex have the same length, respectively). A celebrated result by Roberts gives a simple characterization of unit interval graphs being exactly claw-free interval graphs. Here, we study the generalization of this characterization for d-interval graphs. In particular, we prove that for any d ⩾ 2, if G is a K_{1,2d+1}-free interval graph, then G is a unit d-interval graph. However, somehow surprisingly, under the same assumptions, G is not always a disjoint unit d-interval graph. This implies that the class of disjoint unit d-interval graphs is strictly included in the class of unit d-interval graphs. Finally, we study the relationships between the classes obtained under disjoint and non-disjoint d-intervals in the balanced case and show that the classes of disjoint balanced 2-intervals and balanced 2-intervals coincide, but this is no longer true for d > 2.
Cite as
Virginia Ardévol Martínez, Romeo Rizzi, Abdallah Saffidine, Florian Sikora, and Stéphane Vialette. Generalizing Roberts' Characterization of Unit Interval Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ardevolmartinez_et_al:LIPIcs.MFCS.2024.12,
author = {Ard\'{e}vol Mart{\'\i}nez, Virginia and Rizzi, Romeo and Saffidine, Abdallah and Sikora, Florian and Vialette, St\'{e}phane},
title = {{Generalizing Roberts' Characterization of Unit Interval Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {12:1--12:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.12},
URN = {urn:nbn:de:0030-drops-205687},
doi = {10.4230/LIPIcs.MFCS.2024.12},
annote = {Keywords: Interval graphs, Multiple Interval Graphs, Unit Interval Graphs, Characterization}
}
Document
A Direct Translation from LTL with Past to Deterministic Rabin Automata
Abstract
We present a translation from linear temporal logic with past to deterministic Rabin automata. The translation is direct in the sense that it does not rely on intermediate non-deterministic automata, and asymptotically optimal, resulting in Rabin automata of doubly exponential size. It is based on two main notions. One is that it is possible to encode the history contained in the prefix of a word, as relevant for the formula under consideration, by performing simple rewrites of the formula itself. As a consequence, a formula involving past operators can (through such rewrites, which involve alternating between weak and strong versions of past operators in the formula’s syntax tree) be correctly evaluated at an arbitrary point in the future without requiring backtracking through the word. The other is that this allows us to generalize to linear temporal logic with past the result that the language of a pure-future formula can be decomposed into a Boolean combination of simpler languages, for which deterministic automata with simple acceptance conditions are easily constructed.
Cite as
Shaun Azzopardi, David Lidell, and Nir Piterman. A Direct Translation from LTL with Past to Deterministic Rabin Automata. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{azzopardi_et_al:LIPIcs.MFCS.2024.13,
author = {Azzopardi, Shaun and Lidell, David and Piterman, Nir},
title = {{A Direct Translation from LTL with Past to Deterministic Rabin Automata}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.13},
URN = {urn:nbn:de:0030-drops-205694},
doi = {10.4230/LIPIcs.MFCS.2024.13},
annote = {Keywords: Linear temporal logic, \omega-automata, determinization}
}
Document
Logical Characterizations of Weighted Complexity Classes
Abstract
Fagin’s seminal result characterizing NP in terms of existential second-order logic started the fruitful field of descriptive complexity theory. In recent years, there has been much interest in the investigation of quantitative (weighted) models of computations. In this paper, we start the study of descriptive complexity based on weighted Turing machines over arbitrary semirings. We provide machine-independent characterizations (over ordered structures) of the weighted complexity classes NP[𝒮], FP[𝒮], FPLOG[𝒮], FPSPACE[𝒮], and FPSPACE_poly[𝒮] in terms of definability in suitable weighted logics for an arbitrary semiring 𝒮. In particular, we prove weighted versions of Fagin’s theorem (even for arbitrary structures, not necessarily ordered, provided that the semiring is idempotent and commutative), the Immerman-Vardi’s theorem (originally for 𝖯) and the Abiteboul-Vianu-Vardi’s theorem (originally for PSPACE). We also discuss a recent open problem proposed by Eiter and Kiesel.
Recently, the above mentioned weighted complexity classes have been investigated in connection to classical counting complexity classes. Furthermore, several classical counting complexity classes have been characterized in terms of particular weighted logics over the semiring ℕ of natural numbers. In this work, we cover several of these classes and obtain new results for others such as NPMV, ⊕𝖯, or the collection of real-valued languages realized by polynomial-time real-valued nondeterministic Turing machines. Furthermore, our results apply to classes based on many other important semirings, such as the max-plus and the min-plus semirings over the natural numbers which correspond to the classical classes MaxP[O(log n)] and MinP[O(log n)], respectively.
Cite as
Guillermo Badia, Manfred Droste, Carles Noguera, and Erik Paul. Logical Characterizations of Weighted Complexity Classes. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{badia_et_al:LIPIcs.MFCS.2024.14,
author = {Badia, Guillermo and Droste, Manfred and Noguera, Carles and Paul, Erik},
title = {{Logical Characterizations of Weighted Complexity Classes}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {14:1--14:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.14},
URN = {urn:nbn:de:0030-drops-205707},
doi = {10.4230/LIPIcs.MFCS.2024.14},
annote = {Keywords: Descriptive complexity, Weighted Turing machines, Weighted logics, Semirings}
}
Document
Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs
Abstract
The Subset Feedback Vertex Set problem (SFVS) is to delete k vertices from a given graph such that in the remaining graph, any vertex in a subset T of vertices (called a terminal set) is not in a cycle. The famous Feedback Vertex Set problem is the special case of SFVS with T being the whole set of vertices. In this paper, we study exact algorithms for SFVS in Split Graphs (SFVS-S) and SFVS in Chordal Graphs (SFVS-C). SFVS-S generalizes the minimum vertex cover problem and the prize-collecting version of the maximum independent set problem in hypergraphs (PCMIS), and SFVS-C further generalizes SFVS-S. Both SFVS-S and SFVS-C are implicit 3-Hitting Set problems. However, it is not easy to solve them faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS-C can be solved in 𝒪^*(2^k) time, slightly improving the best result 𝒪^*(2.0755^k) for 3-Hitting Set. In this paper, we break the "2^k-barrier" for SFVS-S and SFVS-C by introducing an 𝒪^*(1.8192^k)-time algorithm. This achievement also indicates that PCMIS can be solved in 𝒪^*(1.8192ⁿ) time, marking the first exact algorithm for PCMIS that outperforms the trivial 𝒪^*(2ⁿ) threshold. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.
Cite as
Tian Bai and Mingyu Xiao. Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bai_et_al:LIPIcs.MFCS.2024.15,
author = {Bai, Tian and Xiao, Mingyu},
title = {{Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {15:1--15:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.15},
URN = {urn:nbn:de:0030-drops-205711},
doi = {10.4230/LIPIcs.MFCS.2024.15},
annote = {Keywords: Subset Feedback Vertex Set, Prize-Collecting Maximum Independent Set, Parameterized Algorithms, Split Graphs, Chordal Graphs, Dulmage-Mendelsohn Decomposition}
}
Document
Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints
Abstract
Given an undirected graph G and a set A ⊆ V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G)⧵A. An A-path is called an (A,𝓁)-path if the length of the path is exactly 𝓁. In the (A, 𝓁)-Path Packing problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, 𝓁)-paths in G or not.
The problem is already known to be fixed-parmeter tractable when parameterized by k+𝓁 via color coding while it remains Para-NP-hard when parameterized by k (Hamiltonian Path) or 𝓁 (P₃-Partition) alone. Therefore, a logical direction to pursue this problem is to examine it in relation to structural parameters. Belmonte et al. initiated a study along these lines and proved that ALPP parameterized by pw+|A| is W[1]-hard where pw is the pathwidth of G. In this paper, we strengthen their result and prove that it is unlikely that ALPP is fixed-parameter tractable even with respect to a bigger parameter (|A|+dtp) where dtp denotes the distance between G and a path graph (distance to path). We use a randomized reduction to achieve the mentioned result. Toward this, we prove a lemma similar to the influential "isolation lemma": Given a set system (X,ℱ) if the elements of X are assigned a weight uniformly at random from a set of values fairly large, then each subset in ℱ will have a unique weight with high probability. We believe that this result will be useful beyond the scope of this paper.
ALPP being hard even for structural parameters like distance to path+|A| rules out the possibility of any FPT algorithms for many well-known other structural parameters, including FVS+|A| and treewidth+|A|. There is a straightforward FPT algorithm for ALPP parameterized by vc, the vertex cover number of the input graph. Following this, we consider the parameters CVD(cluster vertex deletion)+|A| and CVD+|𝓁| and show the problem to be FPT with respect to these parameters. Note that CVD is incomparable to the treewidth of a graph and has been in vogue recently.
Cite as
Susobhan Bandopadhyay, Aritra Banik, Diptapriyo Majumdar, and Abhishek Sahu. Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bandopadhyay_et_al:LIPIcs.MFCS.2024.16,
author = {Bandopadhyay, Susobhan and Banik, Aritra and Majumdar, Diptapriyo and Sahu, Abhishek},
title = {{Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {16:1--16:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.16},
URN = {urn:nbn:de:0030-drops-205725},
doi = {10.4230/LIPIcs.MFCS.2024.16},
annote = {Keywords: Parameterized complexity, (A,𝓁)-Path Packing, Kernelization, Randomized-Exponential Time Hypothesis, Graph Classes}
}
Document
On the Descriptive Complexity of Vertex Deletion Problems
Abstract
Vertex deletion problems for graphs are studied intensely in classical and parameterized complexity theory. They ask whether we can delete at most k vertices from an input graph such that the resulting graph has a certain property. Regarding k as the parameter, a dichotomy was recently shown based on the number of quantifier alternations of first-order formulas that describe the property. In this paper, we refine this classification by moving from quantifier alternations to individual quantifier patterns and from a dichotomy to a trichotomy, resulting in a complete classification of the complexity of vertex deletion problems based on their quantifier pattern. The more fine-grained approach uncovers new tractable fragments, which we show to not only lie in FPT, but even in parameterized constant-depth circuit complexity classes. On the other hand, we show that vertex deletion becomes intractable already for just one quantifier per alternation, that is, there is a formula of the form ∀ x∃ y∀ z (ψ), with ψ quantifier-free, for which the vertex deletion problem is W[1]-hard. The fine-grained analysis also allows us to uncover differences in the complexity landscape when we consider different kinds of graphs and more general structures: While basic graphs (undirected graphs without self-loops), undirected graphs, and directed graphs each have a different frontier of tractability, the frontier for arbitrary logical structures coincides with that of directed graphs.
Cite as
Max Bannach, Florian Chudigiewitsch, and Till Tantau. On the Descriptive Complexity of Vertex Deletion Problems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bannach_et_al:LIPIcs.MFCS.2024.17,
author = {Bannach, Max and Chudigiewitsch, Florian and Tantau, Till},
title = {{On the Descriptive Complexity of Vertex Deletion Problems}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {17:1--17:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.17},
URN = {urn:nbn:de:0030-drops-205733},
doi = {10.4230/LIPIcs.MFCS.2024.17},
annote = {Keywords: graph problems, fixed-parameter tractability, descriptive complexity, vertex deletion}
}
Document
Sparse Graphic Degree Sequences Have Planar Realizations
Abstract
A sequence d = (d_1,d_2, …, d_n) of positive integers is graphic if it is the degree sequence of some simple graph G, and planaric if it is the degree sequence of some simple planar graph G. It is known that if ∑ d ≤ 2n - 2, then d has a realization by a forest, hence it is trivially planaric. In this paper, we seek bounds on ∑ d that guarantee that if d is graphic then it is also planaric. We show that this holds true when ∑ d ≤ 4n-4-2ω₁, where ω₁ is the number of 1’s in d. Conversely, we show that there are graphic sequences with ∑ d = 4n-2ω₁ that are non-planaric. For the case ω₁ = 0, we show that d is planaric when ∑ d ≤ 4n-4. Conversely, we show that there is a graphic sequence with ∑ d = 4n-2 that is non-planaric. In fact, when ∑ d ≤ 4n-6-2ω₁, d can be realized by a graph with a 2-page book embedding.
Cite as
Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. Sparse Graphic Degree Sequences Have Planar Realizations. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{barnoy_et_al:LIPIcs.MFCS.2024.18,
author = {Bar-Noy, Amotz and B\"{o}hnlein, Toni and Peleg, David and Ran, Yingli and Rawitz, Dror},
title = {{Sparse Graphic Degree Sequences Have Planar Realizations}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {18:1--18:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.18},
URN = {urn:nbn:de:0030-drops-205745},
doi = {10.4230/LIPIcs.MFCS.2024.18},
annote = {Keywords: Degree Sequences, Graph Algorithms, Graph Realization, Planar Graphs}
}
Document
The Canadian Traveller Problem on Outerplanar Graphs
Abstract
We study the k-Canadian Traveller Problem, where a weighted graph G = (V,E,ω) with a source s ∈ V and a target t ∈ V are given. This problem also has a hidden input E_* ⊊ E of cardinality at most k representing blocked edges. The objective is to travel from s to t with the minimum distance. At the beginning of the walk, the blockages E_* are unknown: the traveller discovers that an edge is blocked when visiting one of its endpoints. Online algorithms, also called strategies, have been proposed for this problem and assessed with the competitive ratio, i.e., the ratio between the distance actually traversed by the traveller divided by the distance he would have traversed knowing the blockages in advance.
Even though the optimal competitive ratio is 2k+1 even on unit-weighted planar graphs of treewidth 2, we design a polynomial-time strategy achieving competitive ratio 9 on unit-weighted outerplanar graphs. This value 9 also stands as a lower bound for this family of graphs as we prove that, for any ε > 0, no strategy can achieve a competitive ratio 9-ε. Finally, we show that it is not possible to achieve a constant competitive ratio (independent of G and k) on weighted outerplanar graphs.
Cite as
Laurent Beaudou, Pierre Bergé, Vsevolod Chernyshev, Antoine Dailly, Yan Gerard, Aurélie Lagoutte, Vincent Limouzy, and Lucas Pastor. The Canadian Traveller Problem on Outerplanar Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{beaudou_et_al:LIPIcs.MFCS.2024.19,
author = {Beaudou, Laurent and Berg\'{e}, Pierre and Chernyshev, Vsevolod and Dailly, Antoine and Gerard, Yan and Lagoutte, Aur\'{e}lie and Limouzy, Vincent and Pastor, Lucas},
title = {{The Canadian Traveller Problem on Outerplanar Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {19:1--19:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.19},
URN = {urn:nbn:de:0030-drops-205750},
doi = {10.4230/LIPIcs.MFCS.2024.19},
annote = {Keywords: Canadian Traveller Problem, Online algorithms, Competitive analysis, Outerplanar graphs}
}
Document
Simple Qudit ZX and ZH Calculi, via Integrals
Abstract
The ZX calculus and ZH calculus use diagrams to denote and compute properties of quantum operations, using "rewrite rules" to transform between diagrams which denote the same operator through a functorial semantic map. Different semantic maps give rise to different rewrite systems, which may prove more convenient for different purposes. Using discrete measures, we describe semantic maps for ZX and ZH diagrams, well-suited to analyse unitary circuits and measurements on qudits of any fixed dimension D > 1 as a single "ZXH-calculus". We demonstrate rewrite rules for the "stabiliser fragment" of the ZX calculus and a "multicharacter fragment" of the ZH calculus.
Cite as
Niel de Beaudrap and Richard D. P. East. Simple Qudit ZX and ZH Calculi, via Integrals. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{debeaudrap_et_al:LIPIcs.MFCS.2024.20,
author = {de Beaudrap, Niel and East, Richard D. P.},
title = {{Simple Qudit ZX and ZH Calculi, via Integrals}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {20:1--20:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.20},
URN = {urn:nbn:de:0030-drops-205761},
doi = {10.4230/LIPIcs.MFCS.2024.20},
annote = {Keywords: ZX-calculus, ZH-calculus, qudits, string diagrams, discrete integrals}
}
Document
On Complexity of Confluence and Church-Rosser Proofs
Abstract
In this paper, we investigate confluence and the Church-Rosser property - two well-studied properties of rewriting and the λ-calculus - from the viewpoint of proof complexity. With respect to confluence, and focusing on orthogonal term rewrite systems, our main contribution is that the size, measured in number of symbols, of the smallest rewrite proof is polynomial in the size of the peak. For the Church-Rosser property we obtain exponential lower bounds for the size of the join in the size of the equality proof. Finally, we study the complexity of proving confluence in the context of the λ-calculus. Here, we establish an exponential (worst-case) lower bound of the size of the join in the size of the peak.
Cite as
Arnold Beckmann and Georg Moser. On Complexity of Confluence and Church-Rosser Proofs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{beckmann_et_al:LIPIcs.MFCS.2024.21,
author = {Beckmann, Arnold and Moser, Georg},
title = {{On Complexity of Confluence and Church-Rosser Proofs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {21:1--21:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.21},
URN = {urn:nbn:de:0030-drops-205774},
doi = {10.4230/LIPIcs.MFCS.2024.21},
annote = {Keywords: logic, bounded arithmetic, consistency, rewriting}
}
Document
Graph Search Trees and the Intermezzo Problem
Abstract
The last in-tree recognition problem asks whether a given spanning tree can be derived by connecting each vertex with its rightmost left neighbor of some search ordering. In this study, we demonstrate that the last-in-tree recognition problem for Generic Search is NP-complete. We utilize this finding to strengthen a complexity result from order theory. Given a partial order π and a set of triples, the NP-complete intermezzo problem asks for a linear extension of π where each first element of a triple is not between the other two. We show that this problem remains NP-complete even when the Hasse diagram of the partial order forms a tree of bounded height. In contrast, we give an XP-algorithm for the problem when parameterized by the width of the partial order. Furthermore, we show that - under the assumption of the Exponential Time Hypothesis - the running time of this algorithm is asymptotically optimal.
Cite as
Jesse Beisegel, Ekkehard Köhler, Fabienne Ratajczak, Robert Scheffler, and Martin Strehler. Graph Search Trees and the Intermezzo Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{beisegel_et_al:LIPIcs.MFCS.2024.22,
author = {Beisegel, Jesse and K\"{o}hler, Ekkehard and Ratajczak, Fabienne and Scheffler, Robert and Strehler, Martin},
title = {{Graph Search Trees and the Intermezzo Problem}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {22:1--22:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.22},
URN = {urn:nbn:de:0030-drops-205781},
doi = {10.4230/LIPIcs.MFCS.2024.22},
annote = {Keywords: graph search trees, intermezzo problem, algorithm, parameterized complexity}
}
Document
Minimizing Cost Register Automata over a Field
Abstract
Weighted automata (WA) are an extension of finite automata that define functions from words to values in a given semiring. An alternative deterministic model, called Cost Register Automata (CRA), was introduced by Alur et al. It enriches deterministic finite automata with a finite number of registers, which store values, updated at each transition using the operations of the semiring. It is known that CRA with register updates defined by linear maps have the same expressiveness as WA. Previous works have studied the register minimization problem: given a function computable by a WA and an integer k, is it possible to realize it using a CRA with at most k registers?
In this paper, we solve this problem for CRA over a field with linear register updates, using the notion of linear hull, an algebraic invariant of WA introduced recently by Bell and Smertnig. We then generalise the approach to solve a more challenging problem, that consists in minimizing simultaneously the number of states and that of registers. In addition, we also lift our results to the setting of CRA with affine updates. Last, while the linear hull was recently shown to be computable by Bell and Smertnig, no complexity bounds were given. To fill this gap, we provide two new algorithms to compute invariants of WA. This allows us to show that the register (resp. state-register) minimization problem can be solved in 2-ExpTime (resp. in NExpTime).
Cite as
Yahia Idriss Benalioua, Nathan Lhote, and Pierre-Alain Reynier. Minimizing Cost Register Automata over a Field. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{benalioua_et_al:LIPIcs.MFCS.2024.23,
author = {Benalioua, Yahia Idriss and Lhote, Nathan and Reynier, Pierre-Alain},
title = {{Minimizing Cost Register Automata over a Field}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {23:1--23:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.23},
URN = {urn:nbn:de:0030-drops-205798},
doi = {10.4230/LIPIcs.MFCS.2024.23},
annote = {Keywords: Weighted automata, Cost Register automata, Zariski topology}
}
Document
Breaking a Graph into Connected Components with Small Dominating Sets
Abstract
We study DOMINATED CLUSTER DELETION. Therein, we are given an undirected graph G = (V,E) and integers k and d and the task is to find a set of at most k vertices such that removing these vertices results in a graph in which each connected component has a dominating set of size at most d. We also consider the special case where d is a constant. We show an almost complete tetrachotomy in terms of para-NP-hardness, containment in XP, containment in FPT, and admitting a polynomial kernel with respect to parameterizations that are a combination of k,d,c, and Δ, where c and Δ are the degeneracy and the maximum degree of the input graph, respectively. As a main contribution, we show that the problem can be solved in f(k,d) ⋅ n^O(d) time, that is, the problem is FPT when parameterized by k when d is a constant. This answers an open problem asked in a recent Dagstuhl seminar (23331). For the special case d = 1, we provide an algorithm with running time 2^𝒪(klog k) nm. Furthermore, we show that even for d = 1, the problem does not admit a polynomial kernel with respect to k + c.
Cite as
Matthias Bentert, Michael R. Fellows, Petr A. Golovach, Frances A. Rosamond, and Saket Saurabh. Breaking a Graph into Connected Components with Small Dominating Sets. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bentert_et_al:LIPIcs.MFCS.2024.24,
author = {Bentert, Matthias and Fellows, Michael R. and Golovach, Petr A. and Rosamond, Frances A. and Saurabh, Saket},
title = {{Breaking a Graph into Connected Components with Small Dominating Sets}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {24:1--24:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.24},
URN = {urn:nbn:de:0030-drops-205801},
doi = {10.4230/LIPIcs.MFCS.2024.24},
annote = {Keywords: Parameterized Algorithms, Recursive Understanding, Polynomial Kernels, Degeneracy}
}
Document
Multiway Cuts with a Choice of Representatives
Abstract
In the Multiway Cut problem, we are given an undirected graph with nonnegative edge weights and a subset of k terminals, and the goal is to determine a set of edges of minimum total weight whose removal disconnects each terminal from the rest. The problem is APX-hard for k ≥ 3, and an extensive line of research has concentrated on closing the gap between the best upper and lower bounds for approximability and inapproximability, respectively.
In this paper, we study several generalizations of Multiway Cut where the terminals can be chosen as representatives from sets of candidates T₁,…,T_q. In this setting, one is allowed to choose these representatives so that the minimum-weight cut separating these sets via their representatives is as small as possible. We distinguish different cases depending on (A) whether the representative of a candidate set has to be separated from the other candidate sets completely or only from the representatives, and (B) whether there is a single representative for each candidate set or the choice of representative is independent for each pair of candidate sets.
For fixed q, we give approximation algorithms for each of these problems that match the best known approximation guarantee for Multiway Cut. Our technical contribution is a new extension of the CKR relaxation that preserves approximation guarantees. For general q, we show o(log q)-inapproximability for all cases where the choice of representatives may depend on the pair of candidate sets, as well as for the case where the goal is to separate a fixed node from a single representative from each candidate set. As a positive result, we give a 2-approximation algorithm for the case where we need to choose a single representative from each candidate set. This is a generalization of the (2-2/k)-approximation for k-Cut, and we can solve it by relating the tree case to optimization over a gammoid.
Cite as
Kristóf Bérczi, Tamás Király, and Daniel P. Szabo. Multiway Cuts with a Choice of Representatives. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{berczi_et_al:LIPIcs.MFCS.2024.25,
author = {B\'{e}rczi, Krist\'{o}f and Kir\'{a}ly, Tam\'{a}s and Szabo, Daniel P.},
title = {{Multiway Cuts with a Choice of Representatives}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {25:1--25:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.25},
URN = {urn:nbn:de:0030-drops-205813},
doi = {10.4230/LIPIcs.MFCS.2024.25},
annote = {Keywords: Approximation algorithms, Multiway cut, CKR relaxation, Steiner multicut}
}
Document
Capturing the Shape of a Point Set with a Line Segment
Abstract
Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group’s shape carries meaning as well. In this paper, we represent a group’s shape using a simple geometric object, a line segment. Specifically, given a radius r, we say a line segment is representative of a point set P of n points if it is within distance r of each point p ∈ P. We aim to find the shortest such line segment. This problem is equivalent to stabbing a set of circles of radius r using the shortest line segment. We describe an algorithm to find the shortest representative segment in O(n log h + h log³h) time, where h is the size of the convex hull of P. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in P move.
Cite as
Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, and Jules Wulms. Capturing the Shape of a Point Set with a Line Segment. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{vanbeusekom_et_al:LIPIcs.MFCS.2024.26,
author = {van Beusekom, Nathan and van Kreveld, Marc and van Mulken, Max and Roeloffzen, Marcel and Speckmann, Bettina and Wulms, Jules},
title = {{Capturing the Shape of a Point Set with a Line Segment}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {26:1--26:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.26},
URN = {urn:nbn:de:0030-drops-205820},
doi = {10.4230/LIPIcs.MFCS.2024.26},
annote = {Keywords: Shape descriptor, Stabbing, Rotating calipers}
}
Document
Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree
Abstract
We initiate an in-depth proof-complexity analysis of polynomial calculus (𝒬-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of 𝒬-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for 𝒬-PC, similar in spirit, yet different from the classic size-degree formula for propositional PC by Impagliazzo, Pudlák and Sgall (1999).
We use the circuit characterisation together with the size-degree relation to obtain various new lower bounds on proof size in 𝒬-PC. This leads to incomparability results for 𝒬-PC systems over different fields.
Cite as
Olaf Beyersdorff, Tim Hoffmann, Kaspar Kasche, and Luc Nicolas Spachmann. Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{beyersdorff_et_al:LIPIcs.MFCS.2024.27,
author = {Beyersdorff, Olaf and Hoffmann, Tim and Kasche, Kaspar and Spachmann, Luc Nicolas},
title = {{Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {27:1--27:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.27},
URN = {urn:nbn:de:0030-drops-205834},
doi = {10.4230/LIPIcs.MFCS.2024.27},
annote = {Keywords: proof complexity, QBF, polynomial calculus, circuits, lower bounds}
}
Document
Generalized Completion Problems with Forbidden Tournaments
Abstract
A recent result by Bodirsky and Guzmán-Pro gives a complexity dichotomy for the following class of computational problems, parametrized by a finite family F of finite tournaments: given an undirected graph, does there exist an orientation of the graph that avoids every tournament in F? One can see the edges of the input graphs as constraints imposing to find an orientation. In this paper, we consider a more general version of this problem where the constraints in the input are not necessarily about pairs of variables and impose local constraints on the global oriented graph to be found. Our main result is a complexity dichotomy for such problems, as well as a classification of such problems where the yes-instances have bounded treewidth duality. As a consequence, we obtain a streamlined proof of the result by Bodirsky and Guzmán-Pro using the theory of smooth approximations due to Mottet and Pinsker.
Cite as
Zeno Bitter and Antoine Mottet. Generalized Completion Problems with Forbidden Tournaments. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bitter_et_al:LIPIcs.MFCS.2024.28,
author = {Bitter, Zeno and Mottet, Antoine},
title = {{Generalized Completion Problems with Forbidden Tournaments}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {28:1--28:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.28},
URN = {urn:nbn:de:0030-drops-205844},
doi = {10.4230/LIPIcs.MFCS.2024.28},
annote = {Keywords: Tournaments, completion problems, constraint satisfaction problems, homogeneous structures, polymorphisms}
}
Document
Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra
Abstract
In the Equitable Connected Partition (ECP for short) problem, we are given a graph G = (V,E) together with an integer p ∈ ℕ, and our goal is to find a partition of V into p parts such that each part induces a connected sub-graph of G and the size of each two parts differs by at most 1. On the one hand, the problem is known to be NP-hard in general and W[1]-hard with respect to the path-width, the feedback-vertex set, and the number of parts p combined. On the other hand, fixed-parameter algorithms are known for parameters the vertex-integrity and the max leaf number.
In this work, we systematically study ECP with respect to various structural restrictions of the underlying graph and provide a clear dichotomy of its parameterised complexity. Specifically, we show that the problem is in FPT when parameterized by the modular-width and the distance to clique. Next, we prove W[1]-hardness with respect to the distance to cluster, the 4-path vertex cover number, the distance to disjoint paths, and the feedback-edge set, and NP-hardness for constant shrub-depth graphs. Our hardness results are complemented by matching algorithmic upper-bounds: we give an XP algorithm for parameterisation by the tree-width and the distance to cluster. We also give an improved FPT algorithm for parameterisation by the vertex integrity and the first explicit FPT algorithm for the 3-path vertex cover number. The main ingredient of these algorithms is a formulation of ECP as N-fold IP, which clearly indicates that such formulations may, in certain scenarios, significantly outperform existing algorithms based on the famous algorithm of Lenstra.
Cite as
Václav Blažej, Dušan Knop, Jan Pokorný, and Šimon Schierreich. Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{blazej_et_al:LIPIcs.MFCS.2024.29,
author = {Bla\v{z}ej, V\'{a}clav and Knop, Du\v{s}an and Pokorn\'{y}, Jan and Schierreich, \v{S}imon},
title = {{Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {29:1--29:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.29},
URN = {urn:nbn:de:0030-drops-205857},
doi = {10.4230/LIPIcs.MFCS.2024.29},
annote = {Keywords: Equitable Connected Partition, structural parameters, fixed-parameter tractability, N-fold integer programming, tree-width, shrub-depth, modular-width}
}
Document
When Lawvere Meets Peirce: An Equational Presentation of Boolean Hyperdoctrines
Abstract
Fo-bicategories are a categorification of Peirce’s calculus of relations. Notably, their laws provide a proof system for first-order logic that is both purely equational and complete. This paper illustrates a correspondence between fo-bicategories and Lawvere’s hyperdoctrines. To streamline our proof, we introduce peircean bicategories, which offer a more succinct characterization of fo-bicategories.
Cite as
Filippo Bonchi, Alessandro Di Giorgio, and Davide Trotta. When Lawvere Meets Peirce: An Equational Presentation of Boolean Hyperdoctrines. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bonchi_et_al:LIPIcs.MFCS.2024.30,
author = {Bonchi, Filippo and Di Giorgio, Alessandro and Trotta, Davide},
title = {{When Lawvere Meets Peirce: An Equational Presentation of Boolean Hyperdoctrines}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {30:1--30:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.30},
URN = {urn:nbn:de:0030-drops-205867},
doi = {10.4230/LIPIcs.MFCS.2024.30},
annote = {Keywords: relational algebra, hyperdoctrines, cartesian bicategories, string diagrams}
}
Document
Unveiling the Connection Between the Lyndon Factorization and the Canonical Inverse Lyndon Factorization via a Border Property
Abstract
The notion of Lyndon word and Lyndon factorization has shown to have unexpected applications in theory as well as in developing novel algorithms on words. A counterpart to these notions are those of inverse Lyndon word and inverse Lyndon factorization. Differently from the Lyndon words, the inverse Lyndon words may be bordered. The relationship between the two factorizations is related to the inverse lexicographic ordering, and has only been recently explored. More precisely, a main open question is how to get an inverse Lyndon factorization from a classical Lyndon factorization under the inverse lexicographic ordering, named CFL_in. In this paper we reveal a strong connection between these two factorizations where the border plays a relevant role. More precisely, we show two main results. We say that a factorization has the border property if a nonempty border of a factor cannot be a prefix of the next factor. First we show that there exists a unique inverse Lyndon factorization having the border property. Then we show that this unique factorization with the border property is the so-called canonical inverse Lyndon factorization, named ICFL. By showing that ICFL is obtained by compacting factors of the Lyndon factorization over the inverse lexicographic ordering, we provide a linear time algorithm for computing ICFL from CFL_in.
Cite as
Paola Bonizzoni, Clelia De Felice, Brian Riccardi, Rocco Zaccagnino, and Rosalba Zizza. Unveiling the Connection Between the Lyndon Factorization and the Canonical Inverse Lyndon Factorization via a Border Property. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bonizzoni_et_al:LIPIcs.MFCS.2024.31,
author = {Bonizzoni, Paola and De Felice, Clelia and Riccardi, Brian and Zaccagnino, Rocco and Zizza, Rosalba},
title = {{Unveiling the Connection Between the Lyndon Factorization and the Canonical Inverse Lyndon Factorization via a Border Property}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.31},
URN = {urn:nbn:de:0030-drops-205879},
doi = {10.4230/LIPIcs.MFCS.2024.31},
annote = {Keywords: Lyndon words, Lyndon factorization, Combinatorial algorithms on words}
}
Document
Symmetric-Difference (Degeneracy) and Signed Tree Models
Abstract
We introduce a dense counterpart of graph degeneracy, which extends the recently-proposed invariant symmetric difference. We say that a graph has sd-degeneracy (for symmetric-difference degeneracy) at most d if it admits an elimination order of its vertices where a vertex u can be removed whenever it has a d-twin, i.e., another vertex v such that at most d vertices outside {u,v} are neighbors of exactly one of u, v. The family of graph classes of bounded sd-degeneracy is a superset of that of graph classes of bounded degeneracy or of bounded flip-width, and more generally, of bounded symmetric difference. Unlike most graph parameters, sd-degeneracy is not hereditary: it may be strictly smaller on a graph than on some of its induced subgraphs. In particular, every n-vertex graph is an induced subgraph of some O(n²)-vertex graph of sd-degeneracy 1. In spite of this and the breadth of classes of bounded sd-degeneracy, we devise Õ(√n)-bit adjacency labeling schemes for them, which are optimal up to the hidden polylogarithmic factor. This is attained on some even more general classes, consisting of graphs G whose vertices bijectively map to the leaves of a tree T, where transversal edges and anti-edges added to T define the edge set of G. We call such graph representations signed tree models as they extend the so-called tree models (or twin-decompositions) developed in the context of twin-width, by adding transversal anti-edges.
While computing the degeneracy of a graph takes linear time, we show that determining its symmetric difference is para-co-NP-complete. This may seem surprising as symmetric difference can serve as a short-sighted first approximation of twin-width, whose computation is para-NP-complete. Indeed, we show that deciding if the symmetric difference of an input graph is at most 8 is co-NP-complete. We also show that deciding if the sd-degeneracy is at most 6 is NP-complete, contrasting with the symmetric difference.
Cite as
Édouard Bonnet, Julien Duron, John Sylvester, and Viktor Zamaraev. Symmetric-Difference (Degeneracy) and Signed Tree Models. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bonnet_et_al:LIPIcs.MFCS.2024.32,
author = {Bonnet, \'{E}douard and Duron, Julien and Sylvester, John and Zamaraev, Viktor},
title = {{Symmetric-Difference (Degeneracy) and Signed Tree Models}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {32:1--32:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.32},
URN = {urn:nbn:de:0030-drops-205886},
doi = {10.4230/LIPIcs.MFCS.2024.32},
annote = {Keywords: symmetric difference, degeneracy, adjacency labeling schemes, NP-hardness}
}
Document
First-Fit Coloring of Forests in Random Arrival Model
Abstract
We consider a graph coloring algorithm that processes vertices in order taken uniformly at random and assigns colors to them using First-Fit strategy. We show that this algorithm uses, in expectation, at most (1+o(1))⋅ln n / ln ln n different colors to color any forest with n vertices. We also construct a family of forests that shows that this bound is best possible.
Cite as
Bartłomiej Bosek, Grzegorz Gutowski, Michał Lasoń, and Jakub Przybyło. First-Fit Coloring of Forests in Random Arrival Model. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 33:1-33:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bosek_et_al:LIPIcs.MFCS.2024.33,
author = {Bosek, Bart{\l}omiej and Gutowski, Grzegorz and Laso\'{n}, Micha{\l} and Przyby{\l}o, Jakub},
title = {{First-Fit Coloring of Forests in Random Arrival Model}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {33:1--33:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.33},
URN = {urn:nbn:de:0030-drops-205892},
doi = {10.4230/LIPIcs.MFCS.2024.33},
annote = {Keywords: First-Fit, Online Algorithms, Graph Coloring, Random Arrival Model}
}
Document
On the Number of Quantifiers Needed to Define Boolean Functions
Abstract
The number of quantifiers needed to express first-order (FO) properties is captured by two-player combinatorial games called multi-structural games. We analyze these games on binary strings with an ordering relation, using a technique we call parallel play, which significantly reduces the number of quantifiers needed in many cases. Ordered structures such as strings have historically been notoriously difficult to analyze in the context of these and similar games. Nevertheless, in this paper, we provide essentially tight bounds on the number of quantifiers needed to characterize different-sized subsets of strings. The results immediately give bounds on the number of quantifiers necessary to define several different classes of Boolean functions. One of our results is analogous to Lupanov’s upper bounds on circuit size and formula size in propositional logic: we show that every Boolean function on n-bit inputs can be defined by a FO sentence having (1+ε)n/log(n) + O(1) quantifiers, and that this is essentially tight. We reduce this number to (1 + ε)log(n) + O(1) when the Boolean function in question is sparse.
Cite as
Marco Carmosino, Ronald Fagin, Neil Immerman, Phokion G. Kolaitis, Jonathan Lenchner, and Rik Sengupta. On the Number of Quantifiers Needed to Define Boolean Functions. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{carmosino_et_al:LIPIcs.MFCS.2024.34,
author = {Carmosino, Marco and Fagin, Ronald and Immerman, Neil and Kolaitis, Phokion G. and Lenchner, Jonathan and Sengupta, Rik},
title = {{On the Number of Quantifiers Needed to Define Boolean Functions}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {34:1--34:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.34},
URN = {urn:nbn:de:0030-drops-205907},
doi = {10.4230/LIPIcs.MFCS.2024.34},
annote = {Keywords: logic, combinatorial games, Boolean functions, quantifier number}
}
Document
The Complexity of Simplifying ω-Automata Through the Alternating Cycle Decomposition
Abstract
In 2021, Casares, Colcombet and Fijalkow introduced the Alternating Cycle Decomposition (ACD), a structure used to define optimal transformations of Muller into parity automata and to obtain theoretical results about the possibility of relabelling automata with different acceptance conditions. In this work, we study the complexity of computing the ACD and its DAG-version, proving that this can be done in polynomial time for suitable representations of the acceptance condition of the Muller automaton. As corollaries, we obtain that we can decide typeness of Muller automata in polynomial time, as well as the parity index of the languages they recognise.
Furthermore, we show that we can minimise in polynomial time the number of colours (resp. Rabin pairs) defining a Muller (resp. Rabin) acceptance condition, but that these problems become NP-complete when taking into account the structure of an automaton using such a condition.
Cite as
Antonio Casares and Corto Mascle. The Complexity of Simplifying ω-Automata Through the Alternating Cycle Decomposition. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{casares_et_al:LIPIcs.MFCS.2024.35,
author = {Casares, Antonio and Mascle, Corto},
title = {{The Complexity of Simplifying \omega-Automata Through the Alternating Cycle Decomposition}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {35:1--35:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.35},
URN = {urn:nbn:de:0030-drops-205916},
doi = {10.4230/LIPIcs.MFCS.2024.35},
annote = {Keywords: Omega-regular languages, Muller automata, Zielonka tree}
}
Document
Distance to Transitivity: New Parameters for Taming Reachability in Temporal Graphs
Abstract
A temporal graph is a graph whose edges only appear at certain points in time. Reachability in these graphs is defined in terms of paths that traverse the edges in chronological order (temporal paths). This form of reachability is neither symmetric nor transitive, the latter having important consequences on the computational complexity of even basic questions, such as computing temporal connected components. In this paper, we introduce several parameters that capture how far a temporal graph 𝒢 is from being transitive, namely, vertex-deletion distance to transitivity and arc-modification distance to transitivity, both being applied to the reachability graph of 𝒢. We illustrate the impact of these parameters on the temporal connected component problem, obtaining several tractability results in terms of fixed-parameter tractability and polynomial kernels. Significantly, these results are obtained without restrictions of the underlying graph, the snapshots, or the lifetime of the input graph. As such, our results isolate the impact of non-transitivity and confirm the key role that it plays in the hardness of temporal graph problems.
Cite as
Arnaud Casteigts, Nils Morawietz, and Petra Wolf. Distance to Transitivity: New Parameters for Taming Reachability in Temporal Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{casteigts_et_al:LIPIcs.MFCS.2024.36,
author = {Casteigts, Arnaud and Morawietz, Nils and Wolf, Petra},
title = {{Distance to Transitivity: New Parameters for Taming Reachability in Temporal Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {36:1--36:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.36},
URN = {urn:nbn:de:0030-drops-205923},
doi = {10.4230/LIPIcs.MFCS.2024.36},
annote = {Keywords: Temporal graphs, Parameterized complexity, Reachability, Transitivity}
}
Document
Quasi-Isometric Reductions Between Infinite Strings
Abstract
This paper studies the recursion-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We investigate several notions of quasi-isometric reductions between recursive infinite strings and prove various results on the equivalence classes of such reductions. The main result is the construction of two infinite recursive strings α and β such that α is strictly quasi-isometrically reducible to β, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka.
Cite as
Karen Frilya Celine, Ziyuan Gao, Sanjay Jain, Ryan Lou, Frank Stephan, and Guohua Wu. Quasi-Isometric Reductions Between Infinite Strings. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{celine_et_al:LIPIcs.MFCS.2024.37,
author = {Celine, Karen Frilya and Gao, Ziyuan and Jain, Sanjay and Lou, Ryan and Stephan, Frank and Wu, Guohua},
title = {{Quasi-Isometric Reductions Between Infinite Strings}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {37:1--37:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.37},
URN = {urn:nbn:de:0030-drops-205931},
doi = {10.4230/LIPIcs.MFCS.2024.37},
annote = {Keywords: Quasi-isometry, recursion theory, infinite strings}
}
Document
Algorithms and Complexity for Path Covers of Temporal DAGs
Abstract
A path cover of a digraph is a collection of paths collectively containing its vertex set. A path cover with minimum cardinality for a directed acyclic graph can be found in polynomial time [Fulkerson, AMS'56; Cáceres et al., SODA'22]. Moreover, Dilworth’s celebrated theorem on chain coverings of partially ordered sets equivalently states that the minimum size of a path cover of a DAG is equal to the maximum size of a set of mutually unreachable vertices. In this paper, we examine how far these classic results can be extended to a dynamic setting.
A temporal digraph has an arc set that changes over discrete time-steps; if the underlying digraph is acyclic, then it is a temporal DAG. A temporal path is a directed path in the underlying digraph, such that the time-steps of arcs are strictly increasing along the path. Two temporal paths are temporally disjoint if they do not occupy any vertex at the same time. A temporal path cover is a collection 𝒞 of temporal paths that covers all vertices, and 𝒞 is temporally disjoint if all its temporal paths are pairwise temporally disjoint. We study the computational complexities of the problems of finding a minimum-size temporal (disjoint) path cover (denoted as Temporal Path Cover and Temporally Disjoint Path Cover).
On the negative side, we show that both Temporal Path Cover and Temporally Disjoint Path Cover are NP-hard even when the underlying DAG is planar, bipartite, subcubic, and there are only two arc-disjoint time-steps. Moreover, Temporally Disjoint Path Cover remains NP-hard even on temporal oriented trees. We also observe that natural temporal analogues of Dilworth’s theorem on these classes of temporal DAGs do not hold.
In contrast, we show that Temporal Path Cover is polynomial-time solvable on temporal oriented trees by a reduction to Clique Cover for (static undirected) weakly chordal graphs (a subclass of perfect graphs for which Clique Cover admits an efficient algorithm). This highlights an interesting algorithmic difference between the two problems. Although it is NP-hard on temporal oriented trees, Temporally Disjoint Path Cover becomes polynomial-time solvable on temporal oriented lines and temporal rooted directed trees.
Motivated by the hardness result on trees, we show that, in contrast, Temporal Path Cover admits an XP time algorithm with respect to parameter t_max + tw, where t_max is the maximum time-step and tw is the treewidth of the underlying static undirected graph; moreover, Temporally Disjoint Path Cover admits an FPT algorithm with respect to the same parameterization.
Cite as
Dibyayan Chakraborty, Antoine Dailly, Florent Foucaud, and Ralf Klasing. Algorithms and Complexity for Path Covers of Temporal DAGs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2024.38,
author = {Chakraborty, Dibyayan and Dailly, Antoine and Foucaud, Florent and Klasing, Ralf},
title = {{Algorithms and Complexity for Path Covers of Temporal DAGs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {38:1--38:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.38},
URN = {urn:nbn:de:0030-drops-205940},
doi = {10.4230/LIPIcs.MFCS.2024.38},
annote = {Keywords: Temporal Graphs, Dilworth’s Theorem, DAGs, Path Cover, Temporally Disjoint Paths, Algorithms, Oriented Trees, Treewidth}
}
Document
Covering and Partitioning of Split, Chain and Cographs with Isometric Paths
Abstract
Given a graph G, an isometric path cover of a graph is a set of isometric paths that collectively contain all vertices of G. An isometric path cover 𝒞 of a graph G is also an isometric path partition if no vertex lies in two paths in 𝒞. Given a graph G, and an integer k, the objective of Isometric Path Cover (resp. Isometric Path Partition) is to decide whether G has an isometric path cover (resp. partition) of cardinality k.
In this paper, we show that Isometric Path Partition is NP-complete even on split graphs, i.e. graphs whose vertex set can be partitioned into a clique and an independent set. In contrast, we show that both Isometric Path Cover and Isometric Path Partition admit polynomial time algorithms on cographs (graphs with no induced P₄) and chain graphs (bipartite graphs with no induced 2K₂).
Cite as
Dibyayan Chakraborty, Haiko Müller, Sebastian Ordyniak, Fahad Panolan, and Mateusz Rychlicki. Covering and Partitioning of Split, Chain and Cographs with Isometric Paths. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2024.39,
author = {Chakraborty, Dibyayan and M\"{u}ller, Haiko and Ordyniak, Sebastian and Panolan, Fahad and Rychlicki, Mateusz},
title = {{Covering and Partitioning of Split, Chain and Cographs with Isometric Paths}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {39:1--39:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.39},
URN = {urn:nbn:de:0030-drops-205959},
doi = {10.4230/LIPIcs.MFCS.2024.39},
annote = {Keywords: Isometric path partition (cover), chordal graphs, chain graphs, split graphs}
}
Document
On Fourier Analysis of Sparse Boolean Functions over Certain Abelian Groups
Abstract
Given an Abelian group 𝒢, a Boolean-valued function f: 𝒢 → {-1,+1}, is said to be s-sparse, if it has at most s-many non-zero Fourier coefficients over the domain 𝒢. In a seminal paper, Gopalan et al. [Gopalan et al., 2011] proved "Granularity" for Fourier coefficients of Boolean valued functions over ℤ₂ⁿ, that have found many diverse applications in theoretical computer science and combinatorics. They also studied structural results for Boolean functions over ℤ₂ⁿ which are approximately Fourier-sparse. In this work, we obtain structural results for approximately Fourier-sparse Boolean valued functions over Abelian groups 𝒢 of the form, 𝒢: = ℤ_{p_1}^{n_1} × ⋯ × ℤ_{p_t}^{n_t}, for distinct primes p_i. We also obtain a lower bound of the form 1/(m²s)^⌈φ(m)/2⌉, on the absolute value of the smallest non-zero Fourier coefficient of an s-sparse function, where m = p_1 ⋯ p_t, and φ(m) = (p_1-1) ⋯ (p_t-1). We carefully apply probabilistic techniques from [Gopalan et al., 2011], to obtain our structural results, and use some non-trivial results from algebraic number theory to get the lower bound.
We construct a family of at most s-sparse Boolean functions over ℤ_pⁿ, where p > 2, for arbitrarily large enough s, where the minimum non-zero Fourier coefficient is o(1/s). The "Granularity" result of Gopalan et al. implies that the absolute values of non-zero Fourier coefficients of any s-sparse Boolean valued function over ℤ₂ⁿ are Ω(1/s). So, our result shows that one cannot expect such a lower bound for general Abelian groups.
Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient sparsity testing algorithm for Boolean function, which tests whether the given function is s-sparse, or ε-far from any sparse Boolean function, and it requires poly((ms)^φ(m),1/ε)-many queries. Further, we generalize the notion of degree of a Boolean function over an Abelian group 𝒢. We use it to prove an Ω(√s) lower bound on the query complexity of any adaptive sparsity testing algorithm.
Cite as
Sourav Chakraborty, Swarnalipa Datta, Pranjal Dutta, Arijit Ghosh, and Swagato Sanyal. On Fourier Analysis of Sparse Boolean Functions over Certain Abelian Groups. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2024.40,
author = {Chakraborty, Sourav and Datta, Swarnalipa and Dutta, Pranjal and Ghosh, Arijit and Sanyal, Swagato},
title = {{On Fourier Analysis of Sparse Boolean Functions over Certain Abelian Groups}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {40:1--40:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.40},
URN = {urn:nbn:de:0030-drops-205963},
doi = {10.4230/LIPIcs.MFCS.2024.40},
annote = {Keywords: Fourier coefficients, sparse, Abelian, granularity}
}
Document
Krenn-Gu Conjecture for Sparse Graphs
Abstract
Greenberger–Horne–Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. Krenn, Gu and Zeilinger discovered a correspondence between a large class of quantum optical experiments which produce GHZ states and edge-weighted edge-coloured multi-graphs with some special properties called the GHZ graphs. On such GHZ graphs, a graph parameter called dimension can be defined, which is the same as the dimension of the GHZ state produced by the corresponding experiment. Krenn and Gu conjectured that the dimension of any GHZ graph with more than 4 vertices is at most 2. An affirmative resolution of the Krenn-Gu conjecture has implications for quantum resource theory. Moreover, this would save huge computational resources used for finding experiments which lead to higher dimensional GHZ states. On the other hand, the construction of a GHZ graph on a large number of vertices with a high dimension would lead to breakthrough results.
In this paper, we study the existence of GHZ graphs from the perspective of the Krenn-Gu conjecture and show that the conjecture is true for graphs of vertex connectivity at most 2 and for cubic graphs. We also show that the minimal counterexample to the conjecture should be 4-connected. Such information could be of great help in the search for GHZ graphs using existing tools like PyTheus. While the impact of the work is in quantum physics, the techniques in this paper are purely combinatorial, and no background in quantum physics is required to understand them.
Cite as
L. Sunil Chandran, Rishikesh Gajjala, and Abraham M. Illickan. Krenn-Gu Conjecture for Sparse Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chandran_et_al:LIPIcs.MFCS.2024.41,
author = {Chandran, L. Sunil and Gajjala, Rishikesh and Illickan, Abraham M.},
title = {{Krenn-Gu Conjecture for Sparse Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {41:1--41:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.41},
URN = {urn:nbn:de:0030-drops-205978},
doi = {10.4230/LIPIcs.MFCS.2024.41},
annote = {Keywords: Graph colourings, Perfect matchings, Quantum Physics}
}
Document
Applications of Littlestone Dimension to Query Learning and to Compression
Abstract
In this paper we give several applications of Littlestone dimension. The first is to the model of [Angluin and Dohrn, 2017], where we extend their results for learning by equivalence queries with random counterexamples. Second, we extend that model to infinite concept classes with an additional source of randomness. Third, we give improved results on the relationship of Littlestone dimension to classes with extended d-compression schemes, proving the analog of a conjecture of [Floyd and Warmuth, 1995] for Littlestone dimension.
Cite as
Hunter Chase, James Freitag, and Lev Reyzin. Applications of Littlestone Dimension to Query Learning and to Compression. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 42:1-42:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chase_et_al:LIPIcs.MFCS.2024.42,
author = {Chase, Hunter and Freitag, James and Reyzin, Lev},
title = {{Applications of Littlestone Dimension to Query Learning and to Compression}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {42:1--42:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.42},
URN = {urn:nbn:de:0030-drops-205988},
doi = {10.4230/LIPIcs.MFCS.2024.42},
annote = {Keywords: compression scheme, query learning, random queries, Littlestone dimension}
}
Document
The Even-Path Problem in Directed Single-Crossing-Minor-Free Graphs
Abstract
Finding a simple path of even length between two designated vertices in a directed graph is a fundamental NP-complete problem [Andrea S. LaPaugh and Christos H. Papadimitriou, 1984] known as the EP problem. Nedev [Zhivko Prodanov Nedev, 1999] proved in 1999, that for directed planar graphs, the problem can be solved in polynomial time. More than two decades since then, we make the first progress in extending the tractable classes of graphs for this problem. We give a polynomial time algorithm to solve the EP problem for classes of H-minor-free directed graphs, where H is a single-crossing graph.
We make two new technical contributions along the way, that might be of independent interest. The first, and perhaps our main, contribution is the construction of small, planar, parity-mimicking networks. These are graphs that mimic parities of all possible paths between a designated set of terminals of the original graph.
Finding vertex disjoint paths between given source-destination pairs of vertices is another fundamental problem, known to be NP-complete in directed graphs [Steven Fortune et al., 1980], though known to be tractable in planar directed graphs [Alexander Schrijver, 1994]. We encounter a natural variant of this problem, that of finding disjoint paths between given pairs of vertices, but with constraints on parity of the total length of paths. The other significant contribution of our paper is to give a polynomial time algorithm for the 3-disjoint paths with total parity problem, in directed planar graphs with some restrictions (and also in directed graphs of bounded treewidth).
Cite as
Archit Chauhan, Samir Datta, Chetan Gupta, and Vimal Raj Sharma. The Even-Path Problem in Directed Single-Crossing-Minor-Free Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chauhan_et_al:LIPIcs.MFCS.2024.43,
author = {Chauhan, Archit and Datta, Samir and Gupta, Chetan and Sharma, Vimal Raj},
title = {{The Even-Path Problem in Directed Single-Crossing-Minor-Free Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {43:1--43:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.43},
URN = {urn:nbn:de:0030-drops-205992},
doi = {10.4230/LIPIcs.MFCS.2024.43},
annote = {Keywords: Graph Algorithms, EvenPath, Polynomial-time Algorithms, Reachability}
}
Document
The Freeness Problem for Automaton Semigroups
Abstract
We present a new technique to encode Post’s Correspondence Problem into automaton semigroups and monoids. The encoding allows us to precisely control whether there exists a relation in the generated semigroup/monoid and thus show that the freeness problems for automaton semigroups and for automaton monoids (listed as open problems by Grigorchuk, Nekrashevych and Sushchanskĭi) are undecidable. The construction seems to be quite versatile and we obtain the undecidability of further problems: Is a given automaton semigroup (monoid) (left) cancellative? Is it equidivisible (which - together with the existence of a (proper) length function - characterizes free semigroups and monoids)? Does a given map extend into a homomorphism between given automaton semigroups? Finally, our construction can be adapted to show that it is undecidable whether a given automaton generates a free monoid whose basis is given by the states (but where we allow one state to act as the identity). In the semigroup case, we show a weaker version of this.
Cite as
Daniele D'Angeli, Emanuele Rodaro, and Jan Philipp Wächter. The Freeness Problem for Automaton Semigroups. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dangeli_et_al:LIPIcs.MFCS.2024.44,
author = {D'Angeli, Daniele and Rodaro, Emanuele and W\"{a}chter, Jan Philipp},
title = {{The Freeness Problem for Automaton Semigroups}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {44:1--44:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.44},
URN = {urn:nbn:de:0030-drops-206002},
doi = {10.4230/LIPIcs.MFCS.2024.44},
annote = {Keywords: Automaton Monoid, Automaton Semigroup, Freeness Problem, Free Presentation}
}
Document
Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs
Abstract
Flow sparsification is a classic graph compression technique which, given a capacitated graph G on k terminals, aims to construct another capacitated graph H, called a flow sparsifier, that preserves, either exactly or approximately, every multicommodity flow between terminals (ideally, with size as a small function of k). Cut sparsifiers are a restricted variant of flow sparsifiers which are only required to preserve maximum flows between bipartitions of the terminal set. It is known that exact cut sparsifiers require 2^Ω(k) many vertices [Krauthgamer and Rika, SODA 2013], with the hard instances being quasi-bipartite graphs, where there are no edges between non-terminals. On the other hand, it has been shown recently that exact (or even (1+ε)-approximate) flow sparsifiers on networks with just 6 terminals require unbounded size [Krauthgamer and Mosenzon, SODA 2023, Chen and Tan, SODA 2024].
In this paper, we construct exact flow sparsifiers of size 3^k³ and exact cut sparsifiers of size 2^k² for quasi-bipartite graphs. In particular, the flow sparsifiers are contraction-based, that is, they are obtained from the input graph by (vertex) contraction operations. Our main contribution is a new technique to construct sparsifiers that exploits connections to polyhedral geometry, and that can be generalized to graphs with a small separator that separates the graph into small components. We also give an improved reduction theorem for graphs of bounded treewidth [Andoni et al., SODA 2011], implying a flow sparsifier of size O(k⋅w) and quality O((log w)/log log w), where w is the treewidth.
Cite as
Syamantak Das, Nikhil Kumar, and Daniel Vaz. Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{das_et_al:LIPIcs.MFCS.2024.45,
author = {Das, Syamantak and Kumar, Nikhil and Vaz, Daniel},
title = {{Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {45:1--45:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.45},
URN = {urn:nbn:de:0030-drops-206018},
doi = {10.4230/LIPIcs.MFCS.2024.45},
annote = {Keywords: Graph Sparsification, Cut Sparsifiers, Flow Sparsifiers, Quasi-bipartite Graphs, Bounded Treewidth}
}
Document
Query Maintenance Under Batch Changes with Small-Depth Circuits
Abstract
Which dynamic queries can be maintained efficiently? For constant-size changes, it is known that constant-depth circuits or, equivalently, first-order updates suffice for maintaining many important queries, among them reachability, tree isomorphism, and the word problem for context-free languages. In other words, these queries are in the dynamic complexity class DynFO. We show that most of the existing results for constant-size changes can be recovered for batch changes of polylogarithmic size if one allows circuits of depth 𝒪(log log n) or, equivalently, first-order updates that are iterated 𝒪(log log n) times.
Cite as
Samir Datta, Asif Khan, Anish Mukherjee, Felix Tschirbs, Nils Vortmeier, and Thomas Zeume. Query Maintenance Under Batch Changes with Small-Depth Circuits. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{datta_et_al:LIPIcs.MFCS.2024.46,
author = {Datta, Samir and Khan, Asif and Mukherjee, Anish and Tschirbs, Felix and Vortmeier, Nils and Zeume, Thomas},
title = {{Query Maintenance Under Batch Changes with Small-Depth Circuits}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {46:1--46:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.46},
URN = {urn:nbn:de:0030-drops-206027},
doi = {10.4230/LIPIcs.MFCS.2024.46},
annote = {Keywords: Dynamic complexity theory, parallel computation, dynamic algorithms}
}
Document
Preservation Theorems on Sparse Classes Revisited
Abstract
We revisit the work studying homomorphism preservation for first-order logic in sparse classes of structures initiated in [Atserias et al., JACM 2006] and [Dawar, JCSS 2010]. These established that first-order logic has the homomorphism preservation property in any sparse class that is monotone and addable. It turns out that the assumption of addability is not strong enough for the proofs given. We demonstrate this by constructing classes of graphs of bounded treewidth which are monotone and addable but fail to have homomorphism preservation. We also show that homomorphism preservation fails on the class of planar graphs. On the other hand, the proofs of homomorphism preservation can be recovered by replacing addability by a stronger condition of amalgamation over bottlenecks. This is analogous to a similar condition formulated for extension preservation in [Atserias et al., SiCOMP 2008].
Cite as
Anuj Dawar and Ioannis Eleftheriadis. Preservation Theorems on Sparse Classes Revisited. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dawar_et_al:LIPIcs.MFCS.2024.47,
author = {Dawar, Anuj and Eleftheriadis, Ioannis},
title = {{Preservation Theorems on Sparse Classes Revisited}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {47:1--47:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.47},
URN = {urn:nbn:de:0030-drops-206036},
doi = {10.4230/LIPIcs.MFCS.2024.47},
annote = {Keywords: Homomorphism preservation, sparsity, finite model theory, planar graphs}
}
Document
Local Certification of Geometric Graph Classes
Abstract
The goal of local certification is to locally convince the vertices of a graph G that G satisfies a given property. A prover assigns short certificates to the vertices of the graph, then the vertices are allowed to check their certificates and the certificates of their neighbors, and based only on this local view and their own unique identifier, they must decide whether G satisfies the given property. If the graph indeed satisfies the property, all vertices must accept the instance, and otherwise at least one vertex must reject the instance (for any possible assignment of certificates). The goal is to minimize the size of the certificates.
In this paper we study the local certification of geometric and topological graph classes. While it is known that in n-vertex graphs, planarity can be certified locally with certificates of size O(log n), we show that several closely related graph classes require certificates of size Ω(n). This includes penny graphs, unit-distance graphs, (induced) subgraphs of the square grid, 1-planar graphs, and unit-square graphs. These bounds are tight up to a constant factor and give the first known examples of hereditary (and even monotone) graph classes for which the certificates must have linear size. For unit-disk graphs we obtain a lower bound of Ω(n^{1-δ}) for any δ > 0 on the size of the certificates, and an upper bound of O(n log n). The lower bounds are obtained by proving rigidity properties of the considered graphs, which might be of independent interest.
Cite as
Oscar Defrain, Louis Esperet, Aurélie Lagoutte, Pat Morin, and Jean-Florent Raymond. Local Certification of Geometric Graph Classes. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{defrain_et_al:LIPIcs.MFCS.2024.48,
author = {Defrain, Oscar and Esperet, Louis and Lagoutte, Aur\'{e}lie and Morin, Pat and Raymond, Jean-Florent},
title = {{Local Certification of Geometric Graph Classes}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {48:1--48:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.48},
URN = {urn:nbn:de:0030-drops-206042},
doi = {10.4230/LIPIcs.MFCS.2024.48},
annote = {Keywords: Local certification, proof labeling schemes, geometric intersection graphs}
}
Document
Efficient Computation in Congested Anonymous Dynamic Networks
Abstract
An anonymous dynamic network is a network of indistinguishable processes whose communication links may appear or disappear unpredictably over time. Previous research has shown that deterministically computing an arbitrary function of a multiset of input values given to these processes takes only a linear number of communication rounds (Di Luna-Viglietta, FOCS 2022).
However, fast algorithms for anonymous dynamic networks rely on the construction and transmission of large data structures called history trees, whose size is polynomial in the number of processes. This approach is unfeasible if the network is congested, and only messages of logarithmic size can be sent through its links. Observe that sending a large message piece by piece over several rounds is not in itself a solution, due to the anonymity of the processes combined with the dynamic nature of the network. Moreover, it is known that certain basic tasks such as all-to-all token dissemination (by means of single-token forwarding) require Ω(n²/log n) rounds in congested networks (Dutta et al., SODA 2013).
In this work, we develop a series of practical and efficient techniques that make it possible to use history trees in congested anonymous dynamic networks. Among other applications, we show how to compute arbitrary functions in such networks in O(n³) communication rounds, greatly improving upon previous state-of-the-art algorithms for congested networks.
Cite as
Giuseppe A. Di Luna and Giovanni Viglietta. Efficient Computation in Congested Anonymous Dynamic Networks. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 49:1-49:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{diluna_et_al:LIPIcs.MFCS.2024.49,
author = {Di Luna, Giuseppe A. and Viglietta, Giovanni},
title = {{Efficient Computation in Congested Anonymous Dynamic Networks}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {49:1--49:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.49},
URN = {urn:nbn:de:0030-drops-206056},
doi = {10.4230/LIPIcs.MFCS.2024.49},
annote = {Keywords: anonymous dynamic network, congested network, history tree}
}
Document
An Oracle with no UP-Complete Sets, but NP = PSPACE
Abstract
We construct an oracle relative to which NP = PSPACE, but UP has no many-one complete sets. This combines the properties of an oracle by Hartmanis and Hemachandra [J. Hartmanis and L. A. Hemachandra, 1988] and one by Ogiwara and Hemachandra [Ogiwara and Hemachandra, 1991].
The oracle provides new separations of classical conjectures on optimal proof systems and complete sets in promise classes. This answers several questions by Pudlák [P. Pudlák, 2017], e.g., the implications UP ⟹ CON^𝖭 and SAT ⟹ TFNP are false relative to our oracle.
Moreover, the oracle demonstrates that, in principle, it is possible that TFNP-complete problems exist, while at the same time SAT has no p-optimal proof systems.
Cite as
David Dingel, Fabian Egidy, and Christian Glaßer. An Oracle with no UP-Complete Sets, but NP = PSPACE. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 50:1-50:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dingel_et_al:LIPIcs.MFCS.2024.50,
author = {Dingel, David and Egidy, Fabian and Gla{\ss}er, Christian},
title = {{An Oracle with no UP-Complete Sets, but NP = PSPACE}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {50:1--50:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.50},
URN = {urn:nbn:de:0030-drops-206063},
doi = {10.4230/LIPIcs.MFCS.2024.50},
annote = {Keywords: Computational Complexity, Promise Classes, Complete Sets, Oracle Construction}
}
Document
Half-Space Separation in Monophonic Convexity
Abstract
We study half-space separation in the convexity of chordless paths of a graph, i.e., monophonic convexity. In this problem, one is given a graph and two (disjoint) subsets of vertices and asks whether these two sets can be separated by complementary convex sets, called half-spaces. While it is known this problem is NP-complete for geodesic convexity - the convexity of shortest paths - we show that it can be solved in polynomial time for monophonic convexity.
Cite as
Mohammed Elaroussi, Lhouari Nourine, and Simon Vilmin. Half-Space Separation in Monophonic Convexity. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{elaroussi_et_al:LIPIcs.MFCS.2024.51,
author = {Elaroussi, Mohammed and Nourine, Lhouari and Vilmin, Simon},
title = {{Half-Space Separation in Monophonic Convexity}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {51:1--51:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.51},
URN = {urn:nbn:de:0030-drops-206070},
doi = {10.4230/LIPIcs.MFCS.2024.51},
annote = {Keywords: chordless paths, monophonic convexity, separation, half-space}
}
Document
Structural Parameters for Dense Temporal Graphs
Abstract
Temporal graphs provide a useful model for many real-world networks. Unfortunately, the majority of algorithmic problems we might consider on such graphs are intractable. There has been recent progress in defining structural parameters which describe tractable cases by simultaneously restricting the underlying structure and the times at which edges appear in the graph. These all rely on the temporal graph being sparse in some sense. We introduce temporal analogues of three increasingly restrictive static graph parameters - cliquewidth, modular-width and neighbourhood diversity - which take small values for highly structured temporal graphs, even if a large number of edges are active at each timestep. The computational problems solvable efficiently when the temporal cliquewidth of the input graph is bounded form a subset of those solvable efficiently when the temporal modular-width is bounded, which is in turn a subset of problems efficiently solvable when the temporal neighbourhood diversity is bounded. By considering specific temporal graph problems, we demonstrate that (up to standard complexity theoretic assumptions) these inclusions are strict.
Cite as
Jessica Enright, Samuel D. Hand, Laura Larios-Jones, and Kitty Meeks. Structural Parameters for Dense Temporal Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{enright_et_al:LIPIcs.MFCS.2024.52,
author = {Enright, Jessica and Hand, Samuel D. and Larios-Jones, Laura and Meeks, Kitty},
title = {{Structural Parameters for Dense Temporal Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {52:1--52:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.52},
URN = {urn:nbn:de:0030-drops-206082},
doi = {10.4230/LIPIcs.MFCS.2024.52},
annote = {Keywords: Graph algorithms, Parameterized Algorithms, Temporal Graphs}
}
Document
A Robust Measure on FDFAs Following Duo-Normalized Acceptance
Abstract
Families of DFAs (FDFAs) are a computational model recognizing ω-regular languages. They were introduced in the quest of finding a Myhill-Nerode theorem for ω-regular languages and obtaining learning algorithms. FDFAs have been shown to have good qualities in terms of the resources required for computing Boolean operations on them (complementation, union, and intersection) and answering decision problems (emptiness and equivalence); all can be done in non-deterministic logarithmic space. In this paper we study FDFAs with a new type of acceptance condition, duo-normalization, that generalizes the traditional normalization acceptance type. We show that duo-normalized FDFAs are advantageous to normalized FDFAs in terms of succinctness as they can be exponentially smaller. Fortunately this added succinctness doesn't come at the cost of increasing the complexity of Boolean operations and decision problems - they can still be preformed in NLOGSPACE.
An important measure of the complexity of an ω-regular language is its position in the Wagner hierarchy (aka the Rabin Index). It is based on the inclusion measure of Muller automata, and for the common ω-automata there exist algorithms computing their position. We develop a similarly robust measure for duo-normalized (and normalized) FDFAs, which we term the diameter measure. We show that the diameter measure corresponds one-to-one to the position in the Wagner hierarchy. We show that computing it for duo-normalized FDFAs is PSPACE-complete, while it can be done in NLOGSPACE for traditional FDFAs.
Cite as
Dana Fisman, Emmanuel Goldberg, and Oded Zimerman. A Robust Measure on FDFAs Following Duo-Normalized Acceptance. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 53:1-53:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{fisman_et_al:LIPIcs.MFCS.2024.53,
author = {Fisman, Dana and Goldberg, Emmanuel and Zimerman, Oded},
title = {{A Robust Measure on FDFAs Following Duo-Normalized Acceptance}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {53:1--53:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.53},
URN = {urn:nbn:de:0030-drops-206093},
doi = {10.4230/LIPIcs.MFCS.2024.53},
annote = {Keywords: Omega-Regular Languages, Families of DFAs, Complexity Measure, Wagner Hierarchy, Rabin Index}
}
Document
Romeo and Juliet Is EXPTIME-Complete
Abstract
Romeo and Juliet is a two player Rendezvous game played on graphs where one player controls two agents, Romeo (ℛ) and Juliet (𝒥) who aim to meet at a vertex against k adversaries, called dividers, controlled by the other player. The optimization in this game lies at deciding the minimum number of dividers sufficient to restrict ℛ and 𝒥 from meeting in a graph, called the dynamic separation number. We establish that Romeo and Juliet is EXPTIME-complete, settling a conjecture of Fomin, Golovach, and Thilikos [Inf. and Comp., 2023] positively. We also consider the game for directed graphs and establish that although the game is EXPTIME-complete for general directed graphs, it is PSPACE-complete and co-W[2]-hard for directed acyclic graphs.
Cite as
Harmender Gahlawat, Jan Matyáš Křišťan, and Tomáš Valla. Romeo and Juliet Is EXPTIME-Complete. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{gahlawat_et_al:LIPIcs.MFCS.2024.54,
author = {Gahlawat, Harmender and K\v{r}i\v{s}\v{t}an, Jan Maty\'{a}\v{s} and Valla, Tom\'{a}\v{s}},
title = {{Romeo and Juliet Is EXPTIME-Complete}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {54:1--54:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.54},
URN = {urn:nbn:de:0030-drops-206106},
doi = {10.4230/LIPIcs.MFCS.2024.54},
annote = {Keywords: Rendezvous Games on graphs, EXPTIME-completeness, Dynamic Separators}
}
Document
Minimal Obstructions to C₅-Coloring in Hereditary Graph Classes
Abstract
For graphs G and H, an H-coloring of G is an edge-preserving mapping from V(G) to V(H). Note that if H is the triangle, then H-colorings are equivalent to 3-colorings. In this paper we are interested in the case that H is the five-vertex cycle C₅.
A minimal obstruction to C₅-coloring is a graph that does not have a C₅-coloring, but every proper induced subgraph thereof has a C₅-coloring. In this paper we are interested in minimal obstructions to C₅-coloring in F-free graphs, i.e., graphs that exclude some fixed graph F as an induced subgraph. Let P_t denote the path on t vertices, and let S_{a,b,c} denote the graph obtained from paths P_{a+1},P_{b+1},P_{c+1} by identifying one of their endvertices.
We show that there is only a finite number of minimal obstructions to C₅-coloring among F-free graphs, where F ∈ {P₈, S_{2,2,1}, S_{3,1,1}} and explicitly determine all such obstructions. This extends the results of Kamiński and Pstrucha [Discr. Appl. Math. 261, 2019] who proved that there is only a finite number of P₇-free minimal obstructions to C₅-coloring, and of Dębski et al. [ISAAC 2022 Proc.] who showed that the triangle is the unique S_{2,1,1}-free minimal obstruction to C₅-coloring.
We complement our results with a construction of an infinite family of minimal obstructions to C₅-coloring, which are simultaneously P_{13}-free and S_{2,2,2}-free. We also discuss infinite families of F-free minimal obstructions to H-coloring for other graphs H.
Cite as
Jan Goedgebeur, Jorik Jooken, Karolina Okrasa, Paweł Rzążewski, and Oliver Schaudt. Minimal Obstructions to C₅-Coloring in Hereditary Graph Classes. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{goedgebeur_et_al:LIPIcs.MFCS.2024.55,
author = {Goedgebeur, Jan and Jooken, Jorik and Okrasa, Karolina and Rz\k{a}\.{z}ewski, Pawe{\l} and Schaudt, Oliver},
title = {{Minimal Obstructions to C₅-Coloring in Hereditary Graph Classes}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {55:1--55:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.55},
URN = {urn:nbn:de:0030-drops-206110},
doi = {10.4230/LIPIcs.MFCS.2024.55},
annote = {Keywords: graph homomorphism, critical graphs, hereditary graph classes}
}
Document
Specification and Automatic Verification of Computational Reductions
Abstract
We are interested in the following validation problem for computational reductions: for algorithmic problems P and P^⋆, is a given candidate reduction indeed a reduction from P to P^⋆? Unsurprisingly, this problem is undecidable even for very restricted classes of reductions. This leads to the question: Is there a natural, expressive class of reductions for which the validation problem can be attacked algorithmically? We answer this question positively by introducing an easy-to-use graphical specification mechanism for computational reductions, called cookbook reductions. We show that cookbook reductions are sufficiently expressive to cover many classical graph reductions and expressive enough so that SAT remains NP-complete (in the presence of a linear order). Surprisingly, the validation problem is decidable for natural and expressive subclasses of cookbook reductions.
Cite as
Julien Grange, Fabian Vehlken, Nils Vortmeier, and Thomas Zeume. Specification and Automatic Verification of Computational Reductions. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{grange_et_al:LIPIcs.MFCS.2024.56,
author = {Grange, Julien and Vehlken, Fabian and Vortmeier, Nils and Zeume, Thomas},
title = {{Specification and Automatic Verification of Computational Reductions}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {56:1--56:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.56},
URN = {urn:nbn:de:0030-drops-206122},
doi = {10.4230/LIPIcs.MFCS.2024.56},
annote = {Keywords: Computational reductions, automatic verification, decidability}
}
Document
Higher-Order Constrained Dependency Pairs for (Universal) Computability
Abstract
Dependency pairs constitute a series of very effective techniques for the termination analysis of term rewriting systems. In this paper, we adapt the static dependency pair framework to logically constrained simply-typed term rewriting systems (LCSTRSs), a higher-order formalism with logical constraints built in. We also propose the concept of universal computability, which enables a form of open-world termination analysis through the use of static dependency pairs.
Cite as
Liye Guo, Kasper Hagens, Cynthia Kop, and Deivid Vale. Higher-Order Constrained Dependency Pairs for (Universal) Computability. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{guo_et_al:LIPIcs.MFCS.2024.57,
author = {Guo, Liye and Hagens, Kasper and Kop, Cynthia and Vale, Deivid},
title = {{Higher-Order Constrained Dependency Pairs for (Universal) Computability}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {57:1--57:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.57},
URN = {urn:nbn:de:0030-drops-206137},
doi = {10.4230/LIPIcs.MFCS.2024.57},
annote = {Keywords: Higher-order term rewriting, constrained rewriting, dependency pairs}
}
Document
Parameterized Vertex Integrity Revisited
Abstract
Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are very structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of structural parameterizations. We present a number of new results, which also answer some recently posed open questions from the literature. Specifically, we show that unweighted vertex integrity is W[1]-hard parameterized by treedepth; we show that the problem remains W[1]-hard if we parameterize by feedback edge set size (via a reduction from a Bin Packing variant which may be of independent interest); and complementing this we show that the problem is FPT by max-leaf number. Furthermore, for weighted vertex integrity, we show that the problem admits a single-exponential FPT algorithm parameterized by vertex cover or by modular width, the latter result improving upon a previous algorithm which required weights to be polynomially bounded.
Cite as
Tesshu Hanaka, Michael Lampis, Manolis Vasilakis, and Kanae Yoshiwatari. Parameterized Vertex Integrity Revisited. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hanaka_et_al:LIPIcs.MFCS.2024.58,
author = {Hanaka, Tesshu and Lampis, Michael and Vasilakis, Manolis and Yoshiwatari, Kanae},
title = {{Parameterized Vertex Integrity Revisited}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {58:1--58:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.58},
URN = {urn:nbn:de:0030-drops-206141},
doi = {10.4230/LIPIcs.MFCS.2024.58},
annote = {Keywords: Parameterized Complexity, Treedepth, Vertex Integrity}
}
Document
The Complexity of (P₃, H)-Arrowing and Beyond
Abstract
Often regarded as the study of how order emerges from randomness, Ramsey theory has played an important role in mathematics and computer science, giving rise to applications in numerous domains such as logic, parallel processing, and number theory. The core of graph Ramsey theory is arrowing: For fixed graphs F and H, the (F,H)-Arrowing problem asks whether a given graph, G, has a red/blue coloring of the edges of G such that there are no red copies of F and no blue copies of H. For some cases, the problem has been shown to be coNP-complete, or solvable in polynomial time. However, a more systematic approach is needed to categorize the complexity of all cases.
We focus on (P₃,H)-Arrowing as F = P₃ is the simplest meaningful case for which the complexity question remains open, and the hardness for this case likely extends to general (F,H)-Arrowing for nontrivial F. In this pursuit, we also gain insight into the complexity of a class of matching removal problems, since (P₃,H)-Arrowing is equivalent to H-free Matching Removal. We show that (P₃,H)-Arrowing is coNP-complete for all 2-connected H except when H = K₃, in which case the problem is in P. We introduce a new graph invariant to help us carefully combine graphs when constructing the gadgets for our reductions. Moreover, we show how (P₃,H)-Arrowing hardness results can be extended to other (F,H)-Arrowing problems. This allows for more intuitive and palatable hardness proofs instead of ad-hoc constructions of SAT gadgets, bringing us closer to categorizing the complexity of all (F,H)-Arrowing problems.
Cite as
Zohair Raza Hassan. The Complexity of (P₃, H)-Arrowing and Beyond. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 59:1-59:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hassan:LIPIcs.MFCS.2024.59,
author = {Hassan, Zohair Raza},
title = {{The Complexity of (P₃, H)-Arrowing and Beyond}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {59:1--59:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.59},
URN = {urn:nbn:de:0030-drops-206153},
doi = {10.4230/LIPIcs.MFCS.2024.59},
annote = {Keywords: Graph arrowing, Ramsey theory, Complexity}
}
Document
On the Complexity of Community-Aware Network Sparsification
Abstract
In the NP-hard Π-Network Sparsification problem, we are given an edge-weighted graph G, a collection 𝒞 of c subsets of V(G), called communities, and two numbers 𝓁 and b, and the question is whether there exists a spanning subgraph G' of G with at most 𝓁 edges of total weight at most b such that G'[C] fulfills Π for each community C ∈ 𝒞. We study the fine-grained and parameterized complexity of two special cases of this problem: Connectivity NWS where Π is the connectivity property and Stars NWS, where Π is the property of having a spanning star.
First, we provide a tight 2^Ω(n²+c)-time running time lower bound based on the ETH for both problems, where n is the number of vertices in G even if all communities have size at most 4, G is a clique, and every edge has unit weight. For the connectivity property, the unit weight case with G being a clique is the well-studied problem of computing a hypergraph support with a minimum number of edges. We then study the complexity of both problems parameterized by the feedback edge number t of the solution graph G'. For Stars NWS, we present an XP-algorithm for t answering an open question by Korach and Stern [Discret. Appl. Math. '08] who asked for the existence of polynomial-time algorithms for t = 0. In contrast, we show for Connectivity NWS that known polynomial-time algorithms for t = 0 [Korach and Stern, Math. Program. '03; Klemz et al., SWAT '14] cannot be extended to larger values of t by showing NP-hardness for t = 1.
Cite as
Emanuel Herrendorf, Christian Komusiewicz, Nils Morawietz, and Frank Sommer. On the Complexity of Community-Aware Network Sparsification. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 60:1-60:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{herrendorf_et_al:LIPIcs.MFCS.2024.60,
author = {Herrendorf, Emanuel and Komusiewicz, Christian and Morawietz, Nils and Sommer, Frank},
title = {{On the Complexity of Community-Aware Network Sparsification}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {60:1--60:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.60},
URN = {urn:nbn:de:0030-drops-206169},
doi = {10.4230/LIPIcs.MFCS.2024.60},
annote = {Keywords: parameterized complexity, hypergraph support, above guarantee parameterization, exponential-time-hypothesis}
}
Document
ℋ-Clique-Width and a Hereditary Analogue of Product Structure
Abstract
We introduce a novel generalization of the notion of clique-width which aims to bridge the gap between classical hereditary width measures and the recently introduced graph product structure theory. Bounding the new H-clique-width, in the special case of H being the class of paths, is equivalent to admitting a hereditary (i.e., induced) product structure of a path times a graph of bounded clique-width. Furthermore, every graph admitting the usual (non-induced) product structure of a path times a graph of bounded tree-width, has bounded H-clique-width and, as a consequence, it admits the usual product structure in an induced way. We prove further basic properties of H-clique-width in general.
Cite as
Petr Hliněný and Jan Jedelský. ℋ-Clique-Width and a Hereditary Analogue of Product Structure. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hlineny_et_al:LIPIcs.MFCS.2024.61,
author = {Hlin\v{e}n\'{y}, Petr and Jedelsk\'{y}, Jan},
title = {{ℋ-Clique-Width and a Hereditary Analogue of Product Structure}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {61:1--61:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.61},
URN = {urn:nbn:de:0030-drops-206176},
doi = {10.4230/LIPIcs.MFCS.2024.61},
annote = {Keywords: product structure, hereditary class, clique-width, twin-width}
}
Document
Randomness Versus Superspeedability
Abstract
Speedable numbers are real numbers which are algorithmically approximable from below and whose approximations can be accelerated nonuniformly. We begin this article by answering a question of Barmpalias by separating a strict subclass that we will refer to as superspeedable from the speedable numbers; for elements of this subclass, acceleration is possible uniformly and to an even higher degree. This new type of benign left-approximation of numbers then integrates itself into a hierarchy of other such notions studied in a growing body of recent work. We add a new perspective to this study by juxtaposing this hierachy with the well-studied hierachy of algorithmic randomness notions.
Cite as
Rupert Hölzl, Philip Janicki, Wolfgang Merkle, and Frank Stephan. Randomness Versus Superspeedability. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{holzl_et_al:LIPIcs.MFCS.2024.62,
author = {H\"{o}lzl, Rupert and Janicki, Philip and Merkle, Wolfgang and Stephan, Frank},
title = {{Randomness Versus Superspeedability}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {62:1--62:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.62},
URN = {urn:nbn:de:0030-drops-206187},
doi = {10.4230/LIPIcs.MFCS.2024.62},
annote = {Keywords: superspeedable numbers, speedable numbers, regainingly approximable numbers, regular numbers, left-computable numbers}
}
Document
Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs
Abstract
A classical problem in computational geometry and graph algorithms is: given a dynamic set 𝒮 of geometric shapes in the plane, efficiently maintain the connectivity of the intersection graph of 𝒮. Previous papers studied the setting where, before the updates, the data structure receives some parameter P. Then, updates could insert and delete disks as long as at all times the disks have a diameter that lies in a fixed range [1/P, 1]. As a consequence of that prerequisite, the aspect ratio ψ (i.e. the ratio between the largest and smallest diameter) of the disks would at all times satisfy ψ ≤ P. The state-of-the-art for storing disks in a dynamic connectivity data structure is a data structure that uses O(Pn) space and that has amortized O(P log⁴ n) expected amortized update time. Connectivity queries between disks are supported in O(log n / log log n) time.
In the dynamic setting, one wishes for a more flexible data structure in which disks of any diameter may arrive and leave, independent of their diameter, changing the aspect ratio freely. Ideally, the aspect ratio should merely be part of the analysis. We restrict our attention to axis-aligned squares, and study fully-dynamic square intersection graph connectivity. Our result is fully-adaptive to the aspect ratio, spending time proportional to the current aspect ratio ψ, as opposed to some previously given maximum P. Our focus on squares allows us to simplify and streamline the connectivity pipeline from previous work. When n is the number of squares and ψ is the aspect ratio after insertion (or before deletion), our data structure answers connectivity queries in O(log n / log log n) time. We can update connectivity information in O(ψ log⁴ n + log⁶ n) amortized time. We also improve space usage from O(P ⋅ n log n) to O(n log³ n log ψ) - while generalizing to a fully-adaptive aspect ratio - which yields a space usage that is near-linear in n for any polynomially bounded ψ.
Cite as
Ivor van der Hoog, André Nusser, Eva Rotenberg, and Frank Staals. Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 63:1-63:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{vanderhoog_et_al:LIPIcs.MFCS.2024.63,
author = {van der Hoog, Ivor and Nusser, Andr\'{e} and Rotenberg, Eva and Staals, Frank},
title = {{Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {63:1--63:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.63},
URN = {urn:nbn:de:0030-drops-206197},
doi = {10.4230/LIPIcs.MFCS.2024.63},
annote = {Keywords: Computational geometry, planar geometry, data structures, geometric intersection graphs, fully-dynamic algorithms}
}
Document
Pebble Games and Algebraic Proof Systems
Abstract
Analyzing refutations of the well known pebbling formulas Peb(G) we prove some new strong connections between pebble games and algebraic proof system, showing that there is a parallelism between the reversible, black and black-white pebbling games on one side, and the three algebraic proof systems Nullstellensatz, Monomial Calculus and Polynomial Calculus on the other side. In particular we prove that for any DAG G with a single sink, if there is a Monomial Calculus refutation for Peb(G) having simultaneously degree s and size t then there is a black pebbling strategy on G with space s and time t+s. Also if there is a black pebbling strategy for G with space s and time t it is possible to extract from it a MC refutation for Peb(G) having simultaneously degree s and size ts. These results are analogous to those proven in [Susanna F. de Rezende et al., 2021] for the case of reversible pebbling and Nullstellensatz. Using them we prove degree separations between NS, MC and PC, as well as strong degree-size tradeoffs for MC.
We also notice that for any directed acyclic graph G the space needed in a pebbling strategy on G, for the three versions of the game, reversible, black and black-white, exactly matches the variable space complexity of a refutation of the corresponding pebbling formula Peb(G) in each of the algebraic proof systems NS,MC and PC. Using known pebbling bounds on graphs, this connection implies separations between the corresponding variable space measures.
Cite as
Lisa-Marie Jaser and Jacobo Torán. Pebble Games and Algebraic Proof Systems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{jaser_et_al:LIPIcs.MFCS.2024.64,
author = {Jaser, Lisa-Marie and Tor\'{a}n, Jacobo},
title = {{Pebble Games and Algebraic Proof Systems}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {64:1--64:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.64},
URN = {urn:nbn:de:0030-drops-206200},
doi = {10.4230/LIPIcs.MFCS.2024.64},
annote = {Keywords: Proof Complexity, Algebraic Proof Systems, Pebble Games}
}
Document
Punctual Presentability in Certain Classes of Algebraic Structures
Abstract
Punctual structure theory is a rapidly emerging subfield of computable structure theory which aims at understanding the primitive recursive content of algebraic structures. A structure with domain ℕ is punctual if its relations and functions are (uniformly) primitive recursive. One of the fundamental problems of this area is to understand which computable members of a given class of structures admit a punctual presentation. We investigate such a problem for a number of familiar classes of algebraic structures, paying special attention to the case of trees, presented both in a relational and functional signature.
Cite as
Dariusz Kalociński, Luca San Mauro, and Michał Wrocławski. Punctual Presentability in Certain Classes of Algebraic Structures. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 65:1-65:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kalocinski_et_al:LIPIcs.MFCS.2024.65,
author = {Kaloci\'{n}ski, Dariusz and San Mauro, Luca and Wroc{\l}awski, Micha{\l}},
title = {{Punctual Presentability in Certain Classes of Algebraic Structures}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {65:1--65:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.65},
URN = {urn:nbn:de:0030-drops-206212},
doi = {10.4230/LIPIcs.MFCS.2024.65},
annote = {Keywords: fully primitive recursive structures, punctual presentability, trees, injection structures}
}
Document
Twin-Width of Graphs on Surfaces
Abstract
Twin-width is a width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. Hliněný and Jedelský [ICALP'23] showed that every planar graph has twin-width at most 8. We prove that the twin-width of every graph embeddable in a surface of Euler genus g is at most 18√{47g} + O(1), which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus g that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size max{6, 32g-37}.
Cite as
Daniel Kráľ, Kristýna Pekárková, and Kenny Štorgel. Twin-Width of Graphs on Surfaces. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kral_et_al:LIPIcs.MFCS.2024.66,
author = {Kr\'{a}\v{l}, Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na and \v{S}torgel, Kenny},
title = {{Twin-Width of Graphs on Surfaces}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {66:1--66:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.66},
URN = {urn:nbn:de:0030-drops-206226},
doi = {10.4230/LIPIcs.MFCS.2024.66},
annote = {Keywords: twin-width, graphs on surfaces, fixed parameter tractability}
}
Document
Agafonov’s Theorem for Probabilistic Selectors
Abstract
A normal sequence over {0,1} is an infinite sequence for which every word of length k appears with frequency 2^{-k}. Agafonov’s eponymous theorem states that selection by a finite state selector preserves normality, i.e. if α is a normal sequence and A is a finite state selector, then the subsequence A(α) is either finite or a normal sequence.
In this work, we address the following question: does this result hold when considering probabilistic selectors? We provide a partial positive answer, in the case where the probabilities involved are rational. More formally, we prove that given a normal sequence α and a rational probabilistic selector P, the selected subsequence P(α) will be a normal sequence with probability 1.
Cite as
Ulysse Léchine, Thomas Seiller, and Jakob Grue Simonsen. Agafonov’s Theorem for Probabilistic Selectors. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{lechine_et_al:LIPIcs.MFCS.2024.67,
author = {L\'{e}chine, Ulysse and Seiller, Thomas and Simonsen, Jakob Grue},
title = {{Agafonov’s Theorem for Probabilistic Selectors}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {67:1--67:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.67},
URN = {urn:nbn:de:0030-drops-206238},
doi = {10.4230/LIPIcs.MFCS.2024.67},
annote = {Keywords: Normal sequences, probabilistic automata, Agafonov’s theorem}
}
Document
Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems
Abstract
Given a set P of n points and a set S of m disks in the plane, the disk hitting set problem asks for a smallest subset of P such that every disk of S contains at least one point in the subset. The problem is NP-hard. This paper considers a line-constrained version in which all disks have their centers on a line. We present an O(mlog²n+(n+m)log(n+m)) time algorithm for the problem. This improves the previous result of O(m²log m+(n+m)log(n+m)) time for the weighted case of the problem where every point of P has a weight and the objective is to minimize the total weight of the hitting set. Our algorithm also solves a more general line-separable problem with a single intersection property: The points of P and the disk centers are separated by a line 𝓁 and the boundary of every two disks intersect at most once on the side of 𝓁 containing P.
Cite as
Gang Liu and Haitao Wang. Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{liu_et_al:LIPIcs.MFCS.2024.68,
author = {Liu, Gang and Wang, Haitao},
title = {{Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {68:1--68:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.68},
URN = {urn:nbn:de:0030-drops-206240},
doi = {10.4230/LIPIcs.MFCS.2024.68},
annote = {Keywords: hitting set, line-constrained, line-separable, unit disks, half-planes, coverage}
}
Document
Scheduling with Locality by Routing
Abstract
This work examines a strongly NP-hard routing problem on trees, in which multiple servers need to serve a given set of requests (on vertices), where the routes of the servers start from a common source and end at their respective terminals. Each server can travel free of cost on its source-to-terminal path but has to pay for travel on other edges. The objective is to minimize the maximum cost over all servers. As the servers may pay different costs for traveling through a common edge, balancing the loads of the servers can be difficult. We propose a polynomial-time 4-approximation algorithm that applies the parametric pruning framework but consists of two phases. The first phase of the algorithm partitions the requests into packets, and the second phase of the algorithm assigns the packets to the servers. Unlike the standard parametric pruning techniques, the challenge of our algorithm design and analysis is to harmoniously relate the quality of the partition in the first phase, the balances of the servers' loads in the second phase, and the hypothetical optimal values of the framework. For the problem in general graphs, we show that there is no algorithm better than 2-approximate unless P = NP. The problem is a generalization of unrelated machine scheduling and other classic scheduling problems. It also models scheduling problems where the job processing times depend on the machine serving the job and the other jobs served by that machine. This modeling provides a framework that physicalizes scheduling problems through the graph’s point of view.
Cite as
Alison Hsiang-Hsuan Liu and Fu-Hong Liu. Scheduling with Locality by Routing. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 69:1-69:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{liu_et_al:LIPIcs.MFCS.2024.69,
author = {Liu, Alison Hsiang-Hsuan and Liu, Fu-Hong},
title = {{Scheduling with Locality by Routing}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {69:1--69:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.69},
URN = {urn:nbn:de:0030-drops-206250},
doi = {10.4230/LIPIcs.MFCS.2024.69},
annote = {Keywords: Makespan minimization, Approximation algorithms, Routing problems, Parametric pruning framework}
}
Document
On Line-Separable Weighted Unit-Disk Coverage and Related Problems
Abstract
Given a set P of n points and a set S of n weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of P. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of P by a line 𝓁. We present an O(n^{3/2}log² n) time algorithm for the problem. This improves the previously best work of O(n²log n) time. Our result leads to an algorithm of O(n^{7/2}log² n) time for the halfplane coverage problem (i.e., using n weighted halfplanes to cover n points), an improvement over the previous O(n⁴log n) time solution. If all halfplanes are lower ones, our algorithm runs in O(n^{3/2}log² n) time, while the previous best algorithm takes O(n²log n) time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities.
Cite as
Gang Liu and Haitao Wang. On Line-Separable Weighted Unit-Disk Coverage and Related Problems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{liu_et_al:LIPIcs.MFCS.2024.70,
author = {Liu, Gang and Wang, Haitao},
title = {{On Line-Separable Weighted Unit-Disk Coverage and Related Problems}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {70:1--70:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.70},
URN = {urn:nbn:de:0030-drops-206265},
doi = {10.4230/LIPIcs.MFCS.2024.70},
annote = {Keywords: Line-separable, unit disks, halfplanes, geometric coverage, geometric hitting set}
}
Document
Streaming in Graph Products
Abstract
We investigate the streaming space complexity of word problems for groups. Using so-called distinguishers, we prove a transfer theorem for graph products of groups. Moreover, we use distinguishers to obtain a logspace streaming algorithm for the membership problem in a finitely generated subgroup of a free group.
Cite as
Markus Lohrey and Julio Xochitemol. Streaming in Graph Products. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 71:1-71:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2024.71,
author = {Lohrey, Markus and Xochitemol, Julio},
title = {{Streaming in Graph Products}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {71:1--71:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.71},
URN = {urn:nbn:de:0030-drops-206271},
doi = {10.4230/LIPIcs.MFCS.2024.71},
annote = {Keywords: word problems for groups, streaming algorithms, graph products}
}
Document
Algorithmic Dimensions via Learning Functions
Abstract
We characterize the algorithmic dimensions (i.e., the lower and upper asymptotic densities of information) of infinite binary sequences in terms of the inability of learning functions having an algorithmic constraint to detect patterns in them. Our pattern detection criterion is a quantitative extension of the criterion that Zaffora Blando used to characterize the algorithmically random (i.e., Martin-Löf random) sequences. Our proof uses Lutz’s and Mayordomo’s respective characterizations of algorithmic dimension in terms of gales and Kolmogorov complexity.
Cite as
Jack H. Lutz and Andrei N. Migunov. Algorithmic Dimensions via Learning Functions. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 72:1-72:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{lutz_et_al:LIPIcs.MFCS.2024.72,
author = {Lutz, Jack H. and Migunov, Andrei N.},
title = {{Algorithmic Dimensions via Learning Functions}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {72:1--72:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.72},
URN = {urn:nbn:de:0030-drops-206282},
doi = {10.4230/LIPIcs.MFCS.2024.72},
annote = {Keywords: algorithmic dimensions, learning functions, randomness}
}
Document
On the Complexity of the Conditional Independence Implication Problem with Bounded Cardinalities
Abstract
We show that the conditional independence (CI) implication problem with bounded cardinalities, which asks whether a given CI implication holds for all discrete random variables with given cardinalities, is co-NEXPTIME-hard. The problem remains co-NEXPTIME-hard if all variables are binary. The reduction goes from a variant of the tiling problem and is based on a prior construction used by Cheuk Ting Li to show the undecidability of a related problem where the cardinality of some variables remains unbounded. The CI implication problem with bounded cardinalities is known to be in EXPSPACE, as its negation can be stated as an existential first-order logic formula over the reals of size exponential with regard to the size of the input.
Cite as
Michał Makowski. On the Complexity of the Conditional Independence Implication Problem with Bounded Cardinalities. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{makowski:LIPIcs.MFCS.2024.73,
author = {Makowski, Micha{\l}},
title = {{On the Complexity of the Conditional Independence Implication Problem with Bounded Cardinalities}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {73:1--73:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.73},
URN = {urn:nbn:de:0030-drops-206291},
doi = {10.4230/LIPIcs.MFCS.2024.73},
annote = {Keywords: Conditional independence implication, exponential time, tiling problem}
}
Document
Synthesis of Robust Optimal Real-Time Systems
Abstract
Weighted Timed Games (WTGs for short) are widely used to describe real-time controller synthesis problems, but they rely on an unrealistic perfect measure of time elapse. In order to produce strategies tolerant to timing imprecisions, we consider a notion of robustness, expressed as a parametric semantics, first introduced for timed automata. WTGs are two-player zero-sum games played in a weighted timed automaton in which one of the players, that we call Min, wants to reach a target location while minimising the cumulated weight. The opponent player, in addition to controlling some of the locations, can perturb delays chosen by Min. The robust value problem asks, given some threshold, whether there exists a positive perturbation and a strategy for Min ensuring to reach the target, with an accumulated weight below the threshold, whatever the opponent does.
We provide in this article the first decidability result for this robust value problem. More precisely, we show that we can compute the robust value function, in a parametric way, for the class of divergent WTGs (this class has been introduced previously to obtain decidability of the (classical) value problem in WTGs without bounding the number of clocks). To this end, we show that the robust value is the fixpoint of some operators, as is classically done for value iteration algorithms. We then combine in a very careful way two representations: piecewise affine functions introduced in [Alur et al., 2004] to analyse WTGs, and shrunk Difference Bound Matrices (shrunk DBMs for short) considered in [Sankur et al., 2011] to analyse robustness in timed automata. The crux of our result consists in showing that using this representation, the operator of value iteration can be computed for infinitesimally small perturbations. Last, we also study qualitative decision problems and close an open problem on robust reachability, showing it is EXPTIME-complete for general WTGs.
Cite as
Benjamin Monmege, Julie Parreaux, and Pierre-Alain Reynier. Synthesis of Robust Optimal Real-Time Systems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 74:1-74:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{monmege_et_al:LIPIcs.MFCS.2024.74,
author = {Monmege, Benjamin and Parreaux, Julie and Reynier, Pierre-Alain},
title = {{Synthesis of Robust Optimal Real-Time Systems}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {74:1--74:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.74},
URN = {urn:nbn:de:0030-drops-206304},
doi = {10.4230/LIPIcs.MFCS.2024.74},
annote = {Keywords: Weighted timed games, Algorithmic game theory, Robustness}
}
Document
Edit and Alphabet-Ordering Sensitivity of Lex-Parse
Abstract
We investigate the compression sensitivity [Akagi et al., 2023] of lex-parse [Navarro et al., 2021] for two operations: (1) single character edit and (2) modification of the alphabet ordering, and give tight upper and lower bounds for both operations (i.e., we show Θ(log n) bounds for strings of length n). For both lower bounds, we use the family of Fibonacci words. For the bounds on edit operations, our analysis makes heavy use of properties of the Lyndon factorization of Fibonacci words to characterize the structure of lex-parse.
Cite as
Yuto Nakashima, Dominik Köppl, Mitsuru Funakoshi, Shunsuke Inenaga, and Hideo Bannai. Edit and Alphabet-Ordering Sensitivity of Lex-Parse. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 75:1-75:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{nakashima_et_al:LIPIcs.MFCS.2024.75,
author = {Nakashima, Yuto and K\"{o}ppl, Dominik and Funakoshi, Mitsuru and Inenaga, Shunsuke and Bannai, Hideo},
title = {{Edit and Alphabet-Ordering Sensitivity of Lex-Parse}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {75:1--75:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.75},
URN = {urn:nbn:de:0030-drops-206314},
doi = {10.4230/LIPIcs.MFCS.2024.75},
annote = {Keywords: Compression sensitivity, Lex-parse, Fibonacci words}
}
Document
Point-To-Set Principle and Constructive Dimension Faithfulness
Abstract
Hausdorff Φ-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce a constructive analogue of Φ-dimension using the notion of constructive Φ-s-supergales. We prove a Point-to-Set Principle for Φ-dimension, through which we get Point-to-Set Principles for Hausdorff dimension, continued-fraction dimension and dimension of Cantor coverings as special cases. We also provide a Kolmogorov complexity characterization of constructive Φ-dimension.
A class of covering sets Φ is said to be "faithful" to Hausdorff dimension if the Φ-dimension and Hausdorff dimension coincide for every set. Similarly, Φ is said to be "faithful" to constructive dimension if the constructive Φ-dimension and constructive dimension coincide for every set. Using the Point-to-Set Principle for Cantor coverings and a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of "faithfulness" of Cantor coverings at the Hausdorff and constructive levels are equivalent.
We adapt the result by Albeverio, Ivanenko, Lebid, and Torbin [Albeverio et al., 2020] to derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion.
Cite as
Satyadev Nandakumar, Subin Pulari, and Akhil S. Point-To-Set Principle and Constructive Dimension Faithfulness. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 76:1-76:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2024.76,
author = {Nandakumar, Satyadev and Pulari, Subin and S, Akhil},
title = {{Point-To-Set Principle and Constructive Dimension Faithfulness}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {76:1--76:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.76},
URN = {urn:nbn:de:0030-drops-206321},
doi = {10.4230/LIPIcs.MFCS.2024.76},
annote = {Keywords: Kolmogorov complexity, Constructive dimension, Faithfulness, Point to set principle, Continued fraction dimension, Cantor series expansion}
}
Document
Toward Grünbaum’s Conjecture for 4-Connected Graphs
Abstract
Given a spanning tree T of a 3-connected planar graph G, the co-tree of T is the spanning tree of the dual graph G^* given by the duals of the edges that are not in T. Grünbaum conjectured in 1970 that there is such a spanning tree T such that T and its co-tree both have maximum degree at most 3.
In 2014, Biedl proved that there is a spanning tree T such that T and its co-tree have maximum degree at most 5. Using structural insights into Schnyder woods, Schmidt and the author recently improved this bound on the maximum degree to 4. In this paper, we prove that in a 4-connected planar graph there exists a spanning tree T of maximum degree at most 3 such its co-tree has maximum degree at most 4. This almost solves Grünbaum’s conjecture for 4-connected graphs.
Cite as
Christian Ortlieb. Toward Grünbaum’s Conjecture for 4-Connected Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 77:1-77:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ortlieb:LIPIcs.MFCS.2024.77,
author = {Ortlieb, Christian},
title = {{Toward Gr\"{u}nbaum’s Conjecture for 4-Connected Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {77:1--77:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.77},
URN = {urn:nbn:de:0030-drops-206339},
doi = {10.4230/LIPIcs.MFCS.2024.77},
annote = {Keywords: 4-connected planar graph, spanning tree, maximum degree, Schnyder wood, Gr\"{u}nbaum}
}
Document
C_{2k+1}-Coloring of Bounded-Diameter Graphs
Abstract
For a fixed graph H, in the graph homomorphism problem, denoted by Hom(H), we are given a graph G and we have to determine whether there exists an edge-preserving mapping φ: V(G) → V(H). Note that Hom(C₃), where C₃ is the cycle of length 3, is equivalent to 3-Coloring. The question of whether 3-Coloring is polynomial-time solvable on diameter-2 graphs is a well-known open problem. In this paper we study the Hom(C_{2k+1}) problem on bounded-diameter graphs for k ≥ 2, so we consider all other odd cycles than C₃. We prove that for k ≥ 2, the Hom(C_{2k+1}) problem is polynomial-time solvable on diameter-(k+1) graphs - note that such a result for k = 1 would be precisely a polynomial-time algorithm for 3-Coloring of diameter-2 graphs. Furthermore, we give subexponential-time algorithms for diameter-(k+2) and -(k+3) graphs.
We complement these results with a lower bound for diameter-(2k+2) graphs - in this class of graphs the Hom(C_{2k+1}) problem is NP-hard and cannot be solved in subexponential-time, unless the ETH fails.
Finally, we consider another direction of generalizing 3-Coloring on diameter-2 graphs. We consider other target graphs H than odd cycles but we restrict ourselves to diameter 2. We show that if H is triangle-free, then Hom(H) is polynomial-time solvable on diameter-2 graphs.
Cite as
Marta Piecyk. C_{2k+1}-Coloring of Bounded-Diameter Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 78:1-78:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{piecyk:LIPIcs.MFCS.2024.78,
author = {Piecyk, Marta},
title = {{C\underline\{2k+1\}-Coloring of Bounded-Diameter Graphs}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {78:1--78:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.78},
URN = {urn:nbn:de:0030-drops-206348},
doi = {10.4230/LIPIcs.MFCS.2024.78},
annote = {Keywords: graph homomorphism, odd cycles, diameter}
}
Document
Demonic Variance and a Non-Determinism Score for Markov Decision Processes
Abstract
This paper studies the influence of probabilism and non-determinism on some quantitative aspect X of the execution of a system modeled as a Markov decision process (MDP). To this end, the novel notion of demonic variance is introduced: For a random variable X in an MDP ℳ, it is defined as 1/2 times the maximal expected squared distance of the values of X in two independent execution of ℳ in which also the non-deterministic choices are resolved independently by two distinct schedulers.
It is shown that the demonic variance is between 1 and 2 times as large as the maximal variance of X in ℳ that can be achieved by a single scheduler. This allows defining a non-determinism score for ℳ and X measuring how strongly the difference of X in two executions of ℳ can be influenced by the non-deterministic choices. Properties of MDPs ℳ with extremal values of the non-determinism score are established. Further, the algorithmic problems of computing the maximal variance and the demonic variance are investigated for two random variables, namely weighted reachability and accumulated rewards. In the process, also the structure of schedulers maximizing the variance and of scheduler pairs realizing the demonic variance is analyzed.
Cite as
Jakob Piribauer. Demonic Variance and a Non-Determinism Score for Markov Decision Processes. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{piribauer:LIPIcs.MFCS.2024.79,
author = {Piribauer, Jakob},
title = {{Demonic Variance and a Non-Determinism Score for Markov Decision Processes}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {79:1--79:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.79},
URN = {urn:nbn:de:0030-drops-206358},
doi = {10.4230/LIPIcs.MFCS.2024.79},
annote = {Keywords: Markov decision processes, variance, non-determinism, probabilism}
}
Document
Computational Model for Parsing Expression Grammars
Abstract
We present a computational model for Parsing Expression Grammars (PEGs). The predecessor of PEGs top-down parsing languages (TDPLs) were discovered by A. Birman and J. Ullman in the 1960-s, B. Ford showed in 2004 that both formalisms recognize the same class named Parsing Expression Languages (PELs). A. Birman and J. Ullman established such important properties like TDPLs generate any DCFL and some non-context-free languages like a^n b^n c^n, a linear-time parsing algorithm was constructed as well. But since this parsing algorithm was impractical in the 60-s TDPLs were abandoned and then upgraded by B. Ford to PEGs, so the parsing algorithm was improved (from the practical point of view) as well. Now PEGs are actively used in compilers (eg., Python replaced LL(1)-parser with a PEG one) so as for text processing as well. In this paper, we present a computational model for PEG, obtain structural properties of PELs, namely proof that PELs contain Boolean closure of regular closure of DCFLs and PELs are closed over left concatenation with regular closure of DCFLs. We present an extension of the PELs class based on the extension of our computational model. Our model is an upgrade of deterministic pushdown automata (DPDA) such that during the pop of a symbol it is allowed to return the head to the position of the push of the symbol. We provide a linear-time simulation algorithm for the 2-way version of this model, which is similar to the famous S. Cook linear-time simulation algorithm of 2-way DPDA.
Cite as
Alexander Rubtsov and Nikita Chudinov. Computational Model for Parsing Expression Grammars. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 80:1-80:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{rubtsov_et_al:LIPIcs.MFCS.2024.80,
author = {Rubtsov, Alexander and Chudinov, Nikita},
title = {{Computational Model for Parsing Expression Grammars}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {80:1--80:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.80},
URN = {urn:nbn:de:0030-drops-206362},
doi = {10.4230/LIPIcs.MFCS.2024.80},
annote = {Keywords: PEG, formal languages, pushdown automata, two-way pushdown automata}
}
Document
Monoids of Upper Triangular Matrices over the Boolean Semiring
Abstract
Given a finite set 𝒜 of square matrices and a square matrix B, all of the same dimension, the membership problem asks if B belongs to the monoid ℳ(𝒜) generated by 𝒜. The rank one problem asks if there is a matrix of rank one in ℳ(𝒜). We study the membership and the rank one problems in the case where all matrices are upper triangular matrices over the Boolean semiring. We characterize the computational complexity of these problems, and identify their PSPACE-complete and NP-complete special cases.
We then consider, for a set 𝒜 of matrices from the same class, the problem of finding in ℳ(𝒜) a matrix of minimum rank with no zero rows. We show that the minimum rank of such matrix can be computed in linear time.We also characterize the space complexity of this problem depending on the size of 𝒜, and apply all these results to the ergodicity problem asking if ℳ(𝒜) contains a matrix with a column consisting of all ones. Finally, we show that our results give better upper bounds for the case where each row of every matrix in 𝒜 contains at most one non-zero entry than for the general case.
Cite as
Andrew Ryzhikov and Petra Wolf. Monoids of Upper Triangular Matrices over the Boolean Semiring. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 81:1-81:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ryzhikov_et_al:LIPIcs.MFCS.2024.81,
author = {Ryzhikov, Andrew and Wolf, Petra},
title = {{Monoids of Upper Triangular Matrices over the Boolean Semiring}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {81:1--81:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.81},
URN = {urn:nbn:de:0030-drops-206377},
doi = {10.4230/LIPIcs.MFCS.2024.81},
annote = {Keywords: matrix monoids, membership, rank, ergodicity, partially ordered automata}
}
Document
An Algorithmic Meta Theorem for Homomorphism Indistinguishability
Abstract
Two graphs G and H are homomorphism indistinguishable over a family of graphs ℱ if for all graphs F ∈ ℱ the number of homomorphisms from F to G is equal to the number of homomorphism from F to H. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, cospectrality, and logical equivalences can be characterised as homomorphism indistinguishability relations over various graph classes.
The wealth of such results motivates a more fundamental study of homomorphism indistinguishability. From a computational perspective, the central object of interest is the decision problem HomInd(ℱ) which asks to determine whether two input graphs G and H are homomorphism indistinguishable over a fixed graph class ℱ. The problem HomInd(ℱ) is known to be decidable only for few graph classes ℱ. Due to a conjecture by Roberson (2022) and results by Seppelt (MFCS 2023), homomorphism indistinguishability relations over minor-closed graph classes are of special interest. We show that HomInd(ℱ) admits a randomised polynomial-time algorithm for every minor-closed graph class ℱ of bounded treewidth.
This result extends to a version of HomInd where the graph class ℱ is specified by a sentence in counting monadic second-order logic and a bound k on the treewidth, which are given as input. For fixed k, this problem is randomised fixed-parameter tractable. If k is part of the input, then it is coNP- and coW[1]-hard. Addressing a problem posed by Berkholz (2012), we show coNP-hardness by establishing that deciding indistinguishability under the k-dimensional Weisfeiler-Leman algorithm is coNP-hard when k is part of the input.
Cite as
Tim Seppelt. An Algorithmic Meta Theorem for Homomorphism Indistinguishability. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 82:1-82:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{seppelt:LIPIcs.MFCS.2024.82,
author = {Seppelt, Tim},
title = {{An Algorithmic Meta Theorem for Homomorphism Indistinguishability}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {82:1--82:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.82},
URN = {urn:nbn:de:0030-drops-206387},
doi = {10.4230/LIPIcs.MFCS.2024.82},
annote = {Keywords: homomorphism indistinguishability, graph homomorphism, graph minor, recognisability, randomised algorithm, Courcelle’s Theorem}
}
Document
Leakage-Resilient Hardness Equivalence to Logspace Derandomization
Abstract
Efficient derandomization has long been a goal in complexity theory, and a major recent result by Yanyi Liu and Rafael Pass identifies a new class of hardness assumption under which it is possible to perform time-bounded derandomization efficiently: that of "leakage-resilient hardness." They identify a specific form of this assumption which is equivalent to prP = prBPP.
In this paper, we pursue an equivalence to derandomization of prBP⋅L (logspace promise problems with two-way randomness) through techniques analogous to Liu and Pass.
We are able to obtain an equivalence between a similar "leakage-resilient hardness" assumption and a slightly stronger statement than derandomization of prBP⋅L, that of finding "non-no" instances of "promise search problems."
Cite as
Yakov Shalunov. Leakage-Resilient Hardness Equivalence to Logspace Derandomization. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 83:1-83:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{shalunov:LIPIcs.MFCS.2024.83,
author = {Shalunov, Yakov},
title = {{Leakage-Resilient Hardness Equivalence to Logspace Derandomization}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {83:1--83:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.83},
URN = {urn:nbn:de:0030-drops-206395},
doi = {10.4230/LIPIcs.MFCS.2024.83},
annote = {Keywords: Derandomization, logspace computation, leakage-resilient hardness, psuedorandom generators}
}
Document
Faster Approximation Schemes for (Constrained) k-Means with Outliers
Abstract
Given a set of n points in ℝ^d and two positive integers k and m, the Euclidean k-means with outliers problem aims to remove at most m points, referred to as outliers, and minimize the k-means cost function for the remaining points. Developing algorithms for this problem remains an active area of research due to its prevalence in applications involving noisy data. In this paper, we give a (1+ε)-approximation algorithm that runs in n²d((k+m)ε^{-1})^O(kε^{-1}) time for the problem. When combined with a coreset construction method, the running time of the algorithm can be improved to be linear in n. For the case where k is a constant, this represents the first polynomial-time approximation scheme for the problem: Existing algorithms with the same approximation guarantee run in polynomial time only when both k and m are constants. Furthermore, our approach generalizes to variants of k-means with outliers incorporating additional constraints on instances, such as those related to capacities and fairness.
Cite as
Zhen Zhang, Junyu Huang, and Qilong Feng. Faster Approximation Schemes for (Constrained) k-Means with Outliers. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{zhang_et_al:LIPIcs.MFCS.2024.84,
author = {Zhang, Zhen and Huang, Junyu and Feng, Qilong},
title = {{Faster Approximation Schemes for (Constrained) k-Means with Outliers}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {84:1--84:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.84},
URN = {urn:nbn:de:0030-drops-206408},
doi = {10.4230/LIPIcs.MFCS.2024.84},
annote = {Keywords: Approximation algorithms, clustering}
}
Document
Approximate Suffix-Prefix Dictionary Queries
Abstract
In the all-pairs suffix-prefix (APSP) problem [Gusfield et al., Inf. Process. Lett. 1992], we are given a dictionary R of r strings, S₁,…,S_r, of total length n, and we are asked to find the length SPL_{i,j} of the longest string that is both a suffix of S_i and a prefix of S_j, for all i,j ∈ [1..r]. APSP is a classic problem in string algorithms with applications in bioinformatics, especially in sequence assembly. Since r = |R| is typically very large in real-world applications, considering all r² pairs of strings explicitly is prohibitive. This is when the data structure variant of APSP makes sense; in the same spirit as distance oracles computing shortest paths between any two vertices given online.
We show how to quickly locate k-approximate matches (under the Hamming or the edit distance) in R using a version of the k-errata tree [Cole et al., STOC 2004] that we introduce. Let SPL^k_{i,j} be the length of the longest suffix of S_i that is at distance at most k from a prefix of S_j. In particular, for any k = 𝒪(1), we show an 𝒪(nlog^k n)-sized data structure to support the following queries:
- One-to-One^k(i,j): output SPL^k_{i,j} in 𝒪(log^k nlog log n) time.
- Report^k(i,d): output all j ∈ [1..r], such that SPL^k_{i,j} ≥ d, in 𝒪(log^{k}n(log n/log log n+output)) time, where output denotes the size of the output.
In fact, our algorithms work for any value of k not just for k = 𝒪(1), but the formulas bounding the complexities get much more complicated for larger values of k.
Cite as
Wiktor Zuba, Grigorios Loukides, Solon P. Pissis, and Sharma V. Thankachan. Approximate Suffix-Prefix Dictionary Queries. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 85:1-85:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{zuba_et_al:LIPIcs.MFCS.2024.85,
author = {Zuba, Wiktor and Loukides, Grigorios and Pissis, Solon P. and Thankachan, Sharma V.},
title = {{Approximate Suffix-Prefix Dictionary Queries}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {85:1--85:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.85},
URN = {urn:nbn:de:0030-drops-206416},
doi = {10.4230/LIPIcs.MFCS.2024.85},
annote = {Keywords: all-pairs suffix-prefix, suffix-prefix queries, suffix tree, k-errata tree}
}