Volume
LIPIcs, Volume 170
45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS)
Event
MFCS 2020, August 24-28, 2020, Prague, Czech RepublicEditors
Publication Details
- published at: 2020-08-18
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-159-7
- DBLP: db/conf/mfcs/mfcs2020
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Document
Complete Volume
LIPIcs, Volume 170, MFCS 2020, Complete Volume
Abstract
LIPIcs, Volume 170, MFCS 2020, Complete Volume
Cite as
45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 1-1216, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@Proceedings{esparza_et_al:LIPIcs.MFCS.2020,
title = {{LIPIcs, Volume 170, MFCS 2020, Complete Volume}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {1--1216},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020},
URN = {urn:nbn:de:0030-drops-126703},
doi = {10.4230/LIPIcs.MFCS.2020},
annote = {Keywords: LIPIcs, Volume 170, MFCS 2020, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{esparza_et_al:LIPIcs.MFCS.2020.0,
author = {Esparza, Javier and Kr\'{a}l', Daniel},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {0:i--0:xviii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.0},
URN = {urn:nbn:de:0030-drops-126714},
doi = {10.4230/LIPIcs.MFCS.2020.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Concurrent Games with Arbitrarily Many Players (Invited Talk)
Abstract
Traditional concurrent games on graphs involve a fixed number of players, who take decisions simultaneously, determining the next state of the game. With Anirban Majumdar and Patricia Bouyer, we introduced a parameterized variant of concurrent games on graphs, where the parameter is precisely the number of players. Parameterized concurrent games are described by finite graphs, in which the transitions bear finite-word languages to describe the possible move combinations that lead from one vertex to another.
We report on results on two problems for such concurrent games with arbitrary many players. To start with, we studied the problem of determining whether the first player, say Eve, has a strategy to ensure a reachability objective against any strategy profile of her opponents as a coalition. In particular Eve’s strategy should be independent of the number of opponents she actually has. We establish the precise complexities of the problem for reachability objectives. Second, we considered a synthesis problem, where one aims at designing a strategy for each of the (arbitrarily many) players so as to achieve a common objective. For safety objectives, we show that this kind of distributed synthesis problem is decidable.
Cite as
Nathalie Bertrand. Concurrent Games with Arbitrarily Many Players (Invited Talk). In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 1:1-1:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bertrand:LIPIcs.MFCS.2020.1,
author = {Bertrand, Nathalie},
title = {{Concurrent Games with Arbitrarily Many Players}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {1:1--1:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.1},
URN = {urn:nbn:de:0030-drops-126724},
doi = {10.4230/LIPIcs.MFCS.2020.1},
annote = {Keywords: concurrent games, parameterized verification}
}
Document
Invited Talk
Some Open Problems in Computational Geometry (Invited Talk)
Abstract
In this paper we shall encounter three open problems in Computational Geometry that are, in my opinion, interesting for a general audience interested in algorithms.
Cite as
Sergio Cabello. Some Open Problems in Computational Geometry (Invited Talk). In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 2:1-2:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cabello:LIPIcs.MFCS.2020.2,
author = {Cabello, Sergio},
title = {{Some Open Problems in Computational Geometry}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {2:1--2:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.2},
URN = {urn:nbn:de:0030-drops-126734},
doi = {10.4230/LIPIcs.MFCS.2020.2},
annote = {Keywords: barrier resilience, maximum matching, geometric graphs, fixed-parameter tractability, stochastic computational geometry}
}
Document
Invited Talk
List-Decodability of Structured Ensembles of Codes (Invited Talk)
Abstract
What combinatorial properties are satisfied by a random subspace over a finite field? For example, is it likely that not too many points lie in any Hamming ball? What about any cube? In this talk, I will discuss the answer to these questions, along with a more general characterization of the properties that are likely to be satisfied by a random subspace. The motivation for this characterization comes from error correcting codes. I will discuss how to use this characterization to make progress on the questions of list-decoding and list-recovery for random linear codes, and also to establish the list-decodability of random Low Density Parity-Check (LDPC) codes.
This talk is based on the works [Mosheiff et al., 2019] and [Guruswami et al., 2020], which are joint works with Venkatesan Guruswami, Ray Li, Jonathan Mosheiff, Nicolas Resch, Noga Ron-Zewi, and Shashwat Silas.
Cite as
Mary Wootters. List-Decodability of Structured Ensembles of Codes (Invited Talk). In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 3:1-3:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{wootters:LIPIcs.MFCS.2020.3,
author = {Wootters, Mary},
title = {{List-Decodability of Structured Ensembles of Codes}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {3:1--3:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.3},
URN = {urn:nbn:de:0030-drops-126742},
doi = {10.4230/LIPIcs.MFCS.2020.3},
annote = {Keywords: Error Correcting Codes, List-Decoding}
}
Document
Isomorphism Problem for S_d-Graphs
Abstract
An H-graph is the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph H, introduced by Biró, Hujter and Tuza (1992). We focus on S_d-graphs as a special case generalizing interval graphs. A graph G is an S_d-graph iff it is the intersection graph of connected subgraphs of a subdivision of a star S_d with d rays.
We give an FPT algorithm to solve the isomorphism problem for S_d-graphs with the parameter d. This solves an open problem of Chaplick, Töpfer, Voborník and Zeman (2016). In the course of our proof, we also show that the isomorphism problem of S_d-graphs is computationally at least as hard as the isomorphism problem of posets of bounded width.
Cite as
Deniz Ağaoğlu and Petr Hliněný. Isomorphism Problem for S_d-Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{agaoglu_et_al:LIPIcs.MFCS.2020.4,
author = {A\u{g}ao\u{g}lu, Deniz and Hlin\v{e}n\'{y}, Petr},
title = {{Isomorphism Problem for S\underlined-Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {4:1--4:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.4},
URN = {urn:nbn:de:0030-drops-126754},
doi = {10.4230/LIPIcs.MFCS.2020.4},
annote = {Keywords: intersection graph, isomorphism testing, interval graph, H-graph}
}
Document
A Polynomial Kernel for 3-Leaf Power Deletion
Abstract
For a non-negative integer 𝓁, a graph G is an 𝓁-leaf power of a tree T if V(G) is equal to the set of leaves of T, and distinct vertices v and w of G are adjacent if and only if the distance between v and w in T is at most 𝓁. Given a graph G, 3-Leaf Power Deletion asks whether there is a set S ⊆ V(G) of size at most k such that G\S is a 3-leaf power of some treeT. We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance (G,k) to output an equivalent instance (G',k') such that k'≤ k and G' has at most O(k^14) vertices.
Cite as
Jungho Ahn, Eduard Eiben, O-joung Kwon, and Sang-il Oum. A Polynomial Kernel for 3-Leaf Power Deletion. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ahn_et_al:LIPIcs.MFCS.2020.5,
author = {Ahn, Jungho and Eiben, Eduard and Kwon, O-joung and Oum, Sang-il},
title = {{A Polynomial Kernel for 3-Leaf Power Deletion}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {5:1--5:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.5},
URN = {urn:nbn:de:0030-drops-126763},
doi = {10.4230/LIPIcs.MFCS.2020.5},
annote = {Keywords: 𝓁-leaf power, parameterized algorithms, kernelization}
}
Document
Complexity of Computing the Anti-Ramsey Numbers for Paths
Abstract
The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar(G,H) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color.
Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar(G,P_k), where P_k is a path of length k. First, we observe that when k is close to n, the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant.
We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar(G,P_k) for every integer k > 2 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating ar(G,P₃) to a factor of n^{-1/2 - ε} is hard already in 3-partite graphs, unless P = NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k.
Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. Along this line, we first introduce the notion of color connected coloring, and, employing this structural property, we obtain a linear time algorithm to compute ar(G,P_k), for every integer k, when the host graph, G, is a tree.
Cite as
Saeed Akhoondian Amiri, Alexandru Popa, Mohammad Roghani, Golnoosh Shahkarami, Reza Soltani, and Hossein Vahidi. Complexity of Computing the Anti-Ramsey Numbers for Paths. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{akhoondianamiri_et_al:LIPIcs.MFCS.2020.6,
author = {Akhoondian Amiri, Saeed and Popa, Alexandru and Roghani, Mohammad and Shahkarami, Golnoosh and Soltani, Reza and Vahidi, Hossein},
title = {{Complexity of Computing the Anti-Ramsey Numbers for Paths}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {6:1--6:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.6},
URN = {urn:nbn:de:0030-drops-126781},
doi = {10.4230/LIPIcs.MFCS.2020.6},
annote = {Keywords: Coloring, Anti-Ramsey, Approximation, NP-hard, Algorithm, ETH}
}
Document
Best Fit Bin Packing with Random Order Revisited
Abstract
Best Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon’s result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon.
We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case.
In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.
Cite as
Susanne Albers, Arindam Khan, and Leon Ladewig. Best Fit Bin Packing with Random Order Revisited. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{albers_et_al:LIPIcs.MFCS.2020.7,
author = {Albers, Susanne and Khan, Arindam and Ladewig, Leon},
title = {{Best Fit Bin Packing with Random Order Revisited}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {7:1--7:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.7},
URN = {urn:nbn:de:0030-drops-127300},
doi = {10.4230/LIPIcs.MFCS.2020.7},
annote = {Keywords: Online bin packing, random arrival order, probabilistic analysis}
}
Document
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
Abstract
We study the quantum query complexity of two problems.
First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyck_{k,n} problem. We prove a lower bound of Ω(c^k √n), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative example of star-free languages for which a surprising Õ(√n) query quantum algorithm was recently constructed by Aaronson et al. [Scott Aaronson et al., 2018]. Their proof does not give rise to a general algorithm. When k is not a constant, Dyck_{k,n} is not context-free. We give an algorithm with O(√n(log n)^{0.5k}) quantum queries for Dyck_{k,n} for all k. This is better than the trival upper bound n for k = o({log(n)}/{log log n}).
Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of Ω(n^{1.5-ε}) for the directed 2D grid and Ω(n^{2-ε}) for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.
Cite as
Andris Ambainis, Kaspars Balodis, Jānis Iraids, Kamil Khadiev, Vladislavs Kļevickis, Krišjānis Prūsis, Yixin Shen, Juris Smotrovs, and Jevgēnijs Vihrovs. Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ambainis_et_al:LIPIcs.MFCS.2020.8,
author = {Ambainis, Andris and Balodis, Kaspars and Iraids, J\={a}nis and Khadiev, Kamil and K\c{l}evickis, Vladislavs and Pr\={u}sis, Kri\v{s}j\={a}nis and Shen, Yixin and Smotrovs, Juris and Vihrovs, Jevg\={e}nijs},
title = {{Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {8:1--8:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.8},
URN = {urn:nbn:de:0030-drops-126774},
doi = {10.4230/LIPIcs.MFCS.2020.8},
annote = {Keywords: Quantum query complexity, Quantum algorithms, Dyck language, Grid path}
}
Document
Dynamic Time Warping-Based Proximity Problems
Abstract
Dynamic Time Warping (DTW) is a well-known similarity measure for curves, i.e., sequences of points, and especially for time series. We study several proximity problems for curves, where dynamic time warping is the underlying similarity measure. More precisely, we focus on the variants of these problems, in which, whenever we refer to the dynamic time warping distance between two curves, one of them is a line segment (i.e., a sequence of length two). These variants already reveal some of the difficulties that occur when dealing with the more general ones.
Specifically, we study the following three problems: (i) distance oracle: given a curve C in ℝ^d, preprocess it to accommodate distance computations between query segments and C, (ii) segment center: given a set 𝒞 of curves in ℝ^d, find a segment s that minimizes the maximum distance between s and a curve in 𝒞, and (iii) segment nearest neighbor: given 𝒞, construct a data structure for segment nearest neighbor queries, i.e., return the curve in 𝒞 which is closest to a query segment s. We present solutions to these problems in any constant dimension d ≥ 1, using L_∞ for inter-point distances. We also consider the approximation version of the first problem, using L₁ for inter-point distances. That is, given a length-m curve C in ℝ^d, we construct a data structure of size O(m log m) that allows one to compute a 2-approximation of the distance between a query segment s and C in O(log³ m) time.
Finally, we describe an interesting experimental study that we performed, which is related to the first problem above.
Cite as
Boris Aronov, Matthew J. Katz, and Elad Sulami. Dynamic Time Warping-Based Proximity Problems. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{aronov_et_al:LIPIcs.MFCS.2020.9,
author = {Aronov, Boris and Katz, Matthew J. and Sulami, Elad},
title = {{Dynamic Time Warping-Based Proximity Problems}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {9:1--9:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.9},
URN = {urn:nbn:de:0030-drops-126794},
doi = {10.4230/LIPIcs.MFCS.2020.9},
annote = {Keywords: dynamic time warping, distance oracle, clustering, nearest-neighbor search}
}
Document
A Special Case of Rational Identity Testing and the Brešar-Klep Theorem
Abstract
We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Brešar-Klep theorem [Brešar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q<x_1,x_2,...,x_n> over rationals.
1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q<(X)>. Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem.
2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f.
3) The Brešar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d.
Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].
Cite as
V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. A Special Case of Rational Identity Testing and the Brešar-Klep Theorem. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{arvind_et_al:LIPIcs.MFCS.2020.10,
author = {Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha},
title = {{A Special Case of Rational Identity Testing and the Bre\v{s}ar-Klep Theorem}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {10:1--10:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.10},
URN = {urn:nbn:de:0030-drops-126807},
doi = {10.4230/LIPIcs.MFCS.2020.10},
annote = {Keywords: Rational identity testing, ABP with inverses, Bre\v{s}ar-Klep Theorem, Invertible image, Amitsur’s theorem}
}
Document
Solving Packing Problems with Few Small Items Using Rainbow Matchings
Abstract
An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number k of small items.
Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by k. The algorithms are randomized with one-sided error and run in time 4^k⋅ k!⋅ n^{O(1)}. To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. We also present a deterministic fixed-parameter algorithm for Bin Covering with run time O((k!)² ⋅ k ⋅ 2^k ⋅ n log(n)).
Cite as
Max Bannach, Sebastian Berndt, Marten Maack, Matthias Mnich, Alexandra Lassota, Malin Rau, and Malte Skambath. Solving Packing Problems with Few Small Items Using Rainbow Matchings. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bannach_et_al:LIPIcs.MFCS.2020.11,
author = {Bannach, Max and Berndt, Sebastian and Maack, Marten and Mnich, Matthias and Lassota, Alexandra and Rau, Malin and Skambath, Malte},
title = {{Solving Packing Problems with Few Small Items Using Rainbow Matchings}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {11:1--11:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.11},
URN = {urn:nbn:de:0030-drops-126816},
doi = {10.4230/LIPIcs.MFCS.2020.11},
annote = {Keywords: Bin Packing, Knapsack, matching, fixed-parameter tractable}
}
Document
Decidability in Group Shifts and Group Cellular Automata
Abstract
Many undecidable questions concerning cellular automata are known to be decidable when the cellular automaton has a suitable algebraic structure. Typical situations include linear cellular automata where the states come from a finite field or a finite commutative ring, and so-called additive cellular automata in the case the states come from a finite commutative group and the cellular automaton is a group homomorphism. In this paper we generalize the setup and consider so-called group cellular automata whose state set is any (possibly non-commutative) finite group and the cellular automaton is a group homomorphism. The configuration space may be any subshift that is a subgroup of the full shift and still many properties are decidable in any dimension of the cellular space. Decidable properties include injectivity, surjectivity, equicontinuity, sensitivity and nilpotency. Non-transitivity is semi-decidable. It also turns out that the the trace shift and the limit set can be effectively constructed, that injectivity always implies surjectivity, and that jointly periodic points are dense in the limit set. Our decidability proofs are based on developing algorithms to manipulate arbitrary group shifts, and viewing the set of space-time diagrams of group cellular automata as multidimensional group shifts.
Cite as
Pierre Béaur and Jarkko Kari. Decidability in Group Shifts and Group Cellular Automata. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 12:1-12:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{beaur_et_al:LIPIcs.MFCS.2020.12,
author = {B\'{e}aur, Pierre and Kari, Jarkko},
title = {{Decidability in Group Shifts and Group Cellular Automata}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {12:1--12:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.12},
URN = {urn:nbn:de:0030-drops-127206},
doi = {10.4230/LIPIcs.MFCS.2020.12},
annote = {Keywords: group cellular automata, group shifts, symbolic dynamics, decidability}
}
Document
Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain
Abstract
The Ideal Membership Problem (IMP) asks if an input polynomial f ∈ 𝔽[x₁,… ,x_n] with coefficients from a field 𝔽 belongs to an input ideal I ⊆ 𝔽[x₁,… ,x_n]. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial f has degree at most d = O(1) (we call this problem IMP_d). Our main interest is in understanding when the inherent combinatorial structure of the ideals makes the IMP_d "hard" (NP-hard) or "easy" (polynomial time) to solve.
Such a dichotomy result between "hard" and "easy" IMPs was recently achieved for Constraint Satisfaction Problems over finite domains [Andrei A. Bulatov, 2017; Dmitriy Zhuk, 2017] (this is equivalent to IMP₀) and IMP_d for the Boolean domain [Mastrolilli, 2019], both based on the classification of the IMP through functions called polymorphisms. For the latter result, each polymorphism determined the complexity of the computation of a suitable Gröbner basis.
In this paper we consider a 3-element domain and a majority polymorphism (constraints under this polymorphism are a generalisation of the 2-SAT problem). By using properties of the majority polymorphism and assuming graded lexicographic ordering of monomials, we show that the reduced Gröbner basis of ideals whose varieties are closed under the majority polymorphism can be computed in polynomial time. This proves polynomial time solvability of the IMP_d for these constrained problems. We conjecture that this result can be extended to a general finite domain of size k = O(1). This is a first step towards the long term and challenging goal of generalizing the dichotomy results of solvability of the IMP_d for a finite domain.
Cite as
Arpitha P. Bharathi and Monaldo Mastrolilli. Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bharathi_et_al:LIPIcs.MFCS.2020.13,
author = {Bharathi, Arpitha P. and Mastrolilli, Monaldo},
title = {{Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {13:1--13:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.13},
URN = {urn:nbn:de:0030-drops-126829},
doi = {10.4230/LIPIcs.MFCS.2020.13},
annote = {Keywords: Polynomial ideal membership, Polymorphisms, Gr\"{o}bner basis theory, Constraint satisfaction problems}
}
Document
Layered Fan-Planar Graph Drawings
Abstract
In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. In this paper, we study fan-planar drawings that use h (horizontal) layers and are proper, i.e., edges connect adjacent layers. We show that if the embedding of the graph is fixed, then testing the existence of such drawings is fixed-parameter tractable in h, via a reduction to a similar result for planar graphs by Dujmović et al. If the embedding is not fixed, then we give partial results for h = 2: It was already known how to test the existence of fan-planar proper 2-layer drawings for 2-connected graphs, and we show here how to test this for trees. Along the way, we exhibit other interesting results for graphs with a fan-planar proper h-layer drawing; in particular we bound their pathwidth and show that they have a bar-1-visibility representation.
Cite as
Therese Biedl, Steven Chaplick, Michael Kaufmann, Fabrizio Montecchiani, Martin Nöllenburg, and Chrysanthi Raftopoulou. Layered Fan-Planar Graph Drawings. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{biedl_et_al:LIPIcs.MFCS.2020.14,
author = {Biedl, Therese and Chaplick, Steven and Kaufmann, Michael and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Raftopoulou, Chrysanthi},
title = {{Layered Fan-Planar Graph Drawings}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {14:1--14:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.14},
URN = {urn:nbn:de:0030-drops-126835},
doi = {10.4230/LIPIcs.MFCS.2020.14},
annote = {Keywords: Graph Drawing, Parameterized Complexity, Beyond Planar Graphs}
}
Document
Topological Influence and Locality in Swap Schelling Games
Abstract
Residential segregation is a wide-spread phenomenon that can be observed in almost every major city. In these urban areas residents with different racial or socioeconomic background tend to form homogeneous clusters. Schelling’s famous agent-based model for residential segregation explains how such clusters can form even if all agents are tolerant, i.e., if they agree to live in mixed neighborhoods. For segregation to occur, all it needs is a slight bias towards agents preferring similar neighbors. Very recently, Schelling’s model has been investigated from a game-theoretic point of view with selfish agents that strategically select their residential location. In these games, agents can improve on their current location by performing a location swap with another agent who is willing to swap.
We significantly deepen these investigations by studying the influence of the underlying topology modeling the residential area on the existence of equilibria, the Price of Anarchy and on the dynamic properties of the resulting strategic multi-agent system. Moreover, as a new conceptual contribution, we also consider the influence of locality, i.e., if the location swaps are restricted to swaps of neighboring agents. We give improved almost tight bounds on the Price of Anarchy for arbitrary underlying graphs and we present (almost) tight bounds for regular graphs, paths and cycles. Moreover, we give almost tight bounds for grids, which are commonly used in empirical studies. For grids we also show that locality has a severe impact on the game dynamics.
Cite as
Davide Bilò, Vittorio Bilò, Pascal Lenzner, and Louise Molitor. Topological Influence and Locality in Swap Schelling Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bilo_et_al:LIPIcs.MFCS.2020.15,
author = {Bil\`{o}, Davide and Bil\`{o}, Vittorio and Lenzner, Pascal and Molitor, Louise},
title = {{Topological Influence and Locality in Swap Schelling Games}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {15:1--15:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.15},
URN = {urn:nbn:de:0030-drops-126841},
doi = {10.4230/LIPIcs.MFCS.2020.15},
annote = {Keywords: Residential Segregation, Schelling’s Segregation Model, Non-cooperative Games, Price of Anarchy, Game Dynamics}
}
Document
Approximation in (Poly-) Logarithmic Space
Abstract
We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for d–Hitting Set that runs in time n^{O(d² + (d / ε))}, uses O(d² + (d / ε) log n) bits of space, and achieves an approximation ratio of O((d / ε) n^ε) for any positive ε ≤ 1 and any constant d ∈ ℕ. In particular, this yields a factor-O(d log n) approximation algorithm which uses O(log² n) bits of space. As a corollary, we obtain similar bounds on space and approximation ratio for Vertex Cover and several graph deletion problems. For graphs with maximum degree Δ, one can do better. We give a factor-2 approximation algorithm for Vertex Cover which runs in time n^{O(Δ)} and uses O(Δ log n) bits of space.
For Independent Set on graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses O(log n) bits of space. We also devise a factor-O(d²) approximation algorithm for Dominating Set on d-degenerate graphs which runs in time n^{O(log n)} and uses O(log² n) bits of space. For d-regular graphs, we observe that a known randomized algorithm which achieves an approximation ratio of O(log d) can be derandomized to run in polynomial time and use O(log n) bits of space.
Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.
Cite as
Arindam Biswas, Venkatesh Raman, and Saket Saurabh. Approximation in (Poly-) Logarithmic Space. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{biswas_et_al:LIPIcs.MFCS.2020.16,
author = {Biswas, Arindam and Raman, Venkatesh and Saurabh, Saket},
title = {{Approximation in (Poly-) Logarithmic Space}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {16:1--16:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.16},
URN = {urn:nbn:de:0030-drops-126852},
doi = {10.4230/LIPIcs.MFCS.2020.16},
annote = {Keywords: approximation, logspace, logarithmic, log, space, small, limited, memory, ROM, read-only}
}
Document
Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras
Abstract
Determining the exponent of matrix multiplication ω is one of the central open problems in algebraic complexity theory. All approaches to design fast matrix multiplication algorithms follow the following general pattern: We start with one "efficient" tensor T of fixed size and then we use a way to get a large matrix multiplication out of a large tensor power of T. In the recent years, several so-called barrier results have been established. A barrier result shows a lower bound on the best upper bound for the exponent of matrix multiplication that can be obtained by a certain restriction starting with a certain tensor.
We prove the following barrier over C: Starting with a tensor of minimal border rank satisfying a certain genericity condition, except for the diagonal tensor, it is impossible to prove ω = 2 using arbitrary restrictions. This is astonishing since the tensors of minimal border rank look like the most natural candidates for designing fast matrix multiplication algorithms. We prove this by showing that all of these tensors are irreversible, using a structural characterisation of these tensors. To obtain our result, we relate irreversibility to asymptotic slice rank and instability of tensors and prove that the instability of block tensors can often be decided by looking only on the sizes of nonzero blocks.
Cite as
Markus Bläser and Vladimir Lysikov. Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{blaser_et_al:LIPIcs.MFCS.2020.17,
author = {Bl\"{a}ser, Markus and Lysikov, Vladimir},
title = {{Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {17:1--17:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.17},
URN = {urn:nbn:de:0030-drops-126869},
doi = {10.4230/LIPIcs.MFCS.2020.17},
annote = {Keywords: Tensors, Slice rank, Barriers, Matrix multiplication, GIT stability}
}
Document
Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics
Abstract
We consider the problem of computing a Steiner tree of minimum cost under a k-hop constraint which requires the depth of the tree to be at most k. Our main result is an exact algorithm for metrics induced by graphs of bounded treewidth that runs in time n^O(k). For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if k is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the k-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension and bounded doubling dimension.
Cite as
Martin Böhm, Ruben Hoeksma, Nicole Megow, Lukas Nölke, and Bertrand Simon. Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bohm_et_al:LIPIcs.MFCS.2020.18,
author = {B\"{o}hm, Martin and Hoeksma, Ruben and Megow, Nicole and N\"{o}lke, Lukas and Simon, Bertrand},
title = {{Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {18:1--18:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.18},
URN = {urn:nbn:de:0030-drops-126870},
doi = {10.4230/LIPIcs.MFCS.2020.18},
annote = {Keywords: k-hop Steiner tree, dynamic programming, bounded treewidth}
}
Document
Some Remarks on Deciding Equivalence for Graph-To-Graph Transducers
Abstract
We study the following decision problem: given two mso transductions that input and output graphs of bounded treewidth, decide if they are equivalent, i.e. isomorphic inputs give isomorphic outputs. We do not know how to decide it, but we propose an approach that uses automata manipulating elements of a ring extended with division. The approach works for a variant of the problem, where isomorphism on output graphs is replaced by a relaxation of isomorphism.
Cite as
Mikołaj Bojańczyk and Janusz Schmude. Some Remarks on Deciding Equivalence for Graph-To-Graph Transducers. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bojanczyk_et_al:LIPIcs.MFCS.2020.19,
author = {Boja\'{n}czyk, Miko{\l}aj and Schmude, Janusz},
title = {{Some Remarks on Deciding Equivalence for Graph-To-Graph Transducers}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {19:1--19:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.19},
URN = {urn:nbn:de:0030-drops-127529},
doi = {10.4230/LIPIcs.MFCS.2020.19},
annote = {Keywords: equivalence for mso transductions, bounded treewidth, Hilbert basis theorem}
}
Document
List Homomorphism Problems for Signed Graphs
Abstract
We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph (G,σ), equipped with lists L(v) ⊆ V(H), v ∈ V(G), of allowed images, to a fixed target signed graph (H,π). The complexity of the similar homomorphism problem without lists (corresponding to all lists being L(v) = V(H)) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. Both versions (with lists or without lists) can be formulated as constraint satisfaction problems, and hence enjoy the algebraic dichotomy classification recently verified by Bulatov and Zhuk. By contrast, we seek a combinatorial classification for the list version, akin to the combinatorial classification for the version without lists completed by Brewster and Siggers. We illustrate the possible complications by classifying the complexity of the list homomorphism problem when H is a (reflexive or irreflexive) signed tree. It turns out that the problems are polynomial-time solvable for certain caterpillar-like trees, and are NP-complete otherwise. The tools we develop will be useful for classifications of other classes of signed graphs, and we mention some follow-up research of this kind; those classifications are surprisingly complex.
Cite as
Jan Bok, Richard Brewster, Tomás Feder, Pavol Hell, and Nikola Jedličková. List Homomorphism Problems for Signed Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bok_et_al:LIPIcs.MFCS.2020.20,
author = {Bok, Jan and Brewster, Richard and Feder, Tom\'{a}s and Hell, Pavol and Jedli\v{c}kov\'{a}, Nikola},
title = {{List Homomorphism Problems for Signed Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {20:1--20:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.20},
URN = {urn:nbn:de:0030-drops-126886},
doi = {10.4230/LIPIcs.MFCS.2020.20},
annote = {Keywords: complexity, dichotomy, graph homomorphism, signed graph}
}
Document
On a Temporal Logic of Prefixes and Infixes
Abstract
A classic result by Stockmeyer [Stockmeyer, 1974] gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of the chop operator under the homogeneity assumption [Halpern et al., 1983]. In this paper, we study the complexity of the satisfiability problem for a suitable weakening of the chop interval temporal logic, that can be equivalently viewed as a fragment of Halpern and Shoham interval logic featuring the operators B, for "begins", corresponding to the prefix relation on pairs of intervals, and D, for "during", corresponding to the infix relation. The homogeneous models of the considered logic naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations.
Cite as
Laura Bozzelli, Angelo Montanari, Adriano Peron, and Pietro Sala. On a Temporal Logic of Prefixes and Infixes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bozzelli_et_al:LIPIcs.MFCS.2020.21,
author = {Bozzelli, Laura and Montanari, Angelo and Peron, Adriano and Sala, Pietro},
title = {{On a Temporal Logic of Prefixes and Infixes}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {21:1--21:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.21},
URN = {urn:nbn:de:0030-drops-126898},
doi = {10.4230/LIPIcs.MFCS.2020.21},
annote = {Keywords: Interval Temporal Logic, Star-Free Regular Languages, Satisfiability, Complexity}
}
Document
Simplified Game of Life: Algorithms and Complexity
Abstract
Game of Life is a simple and elegant model to study dynamical system over networks. The model consists of a graph where every vertex has one of two types, namely, dead or alive. A configuration is a mapping of the vertices to the types. An update rule describes how the type of a vertex is updated given the types of its neighbors. In every round, all vertices are updated synchronously, which leads to a configuration update. While in general, Game of Life allows a broad range of update rules, we focus on two simple families of update rules, namely, underpopulation and overpopulation, that model several interesting dynamics studied in the literature. In both settings, a dead vertex requires at least a desired number of live neighbors to become alive. For underpopulation (resp., overpopulation), a live vertex requires at least (resp. at most) a desired number of live neighbors to remain alive. We study the basic computation problems, e.g., configuration reachability, for these two families of rules. For underpopulation rules, we show that these problems can be solved in polynomial time, whereas for overpopulation rules they are PSPACE-complete.
Cite as
Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, and Jakub Svoboda. Simplified Game of Life: Algorithms and Complexity. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2020.22,
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Jecker, Isma\"{e}l and Svoboda, Jakub},
title = {{Simplified Game of Life: Algorithms and Complexity}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {22:1--22:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.22},
URN = {urn:nbn:de:0030-drops-126903},
doi = {10.4230/LIPIcs.MFCS.2020.22},
annote = {Keywords: game of life, cellular automata, computational complexity, dynamical systems}
}
Document
Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming
Abstract
Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with m constraint matrices, each of dimension n and rank r, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time O(m⋅poly(log n,r,1/ε)) given access to a sampling-based low-overhead data structure for the constraint matrices, where ε is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest:
- Weighted sampling: assuming sampling access to each individual constraint matrix A₁,…,A_τ, we propose a procedure that gives a good approximation of A = A₁+⋯+A_τ.
- Symmetric approximation: we propose a sampling procedure that gives the spectral decomposition of a low-rank Hermitian matrix A. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.
Cite as
Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chia_et_al:LIPIcs.MFCS.2020.23,
author = {Chia, Nai-Hui and Li, Tongyang and Lin, Han-Hsuan and Wang, Chunhao},
title = {{Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {23:1--23:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.23},
URN = {urn:nbn:de:0030-drops-126919},
doi = {10.4230/LIPIcs.MFCS.2020.23},
annote = {Keywords: Spectral decomposition, Semi-definite programming, Quantum-inspired algorithm, Sublinear algorithm}
}
Document
PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations
Abstract
We introduce the PBS-calculus to represent and reason on quantum computations involving coherent control of quantum operations. Coherent control, and in particular indefinite causal order, is known to enable multiple computational and communication advantages over classically ordered models like quantum circuits. The PBS-calculus is inspired by quantum optics, in particular the polarising beam splitter (PBS for short). We formalise the syntax and the semantics of the PBS-diagrams, and we equip the language with an equational theory, which is proved to be sound and complete: two diagrams are representing the same quantum evolution if and only if one can be transformed into the other using the rules of the PBS-calculus. Moreover, we show that the equational theory is minimal. Finally, we consider applications like the implementation of controlled permutations and the unrolling of loops.
Cite as
Alexandre Clément and Simon Perdrix. PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{clement_et_al:LIPIcs.MFCS.2020.24,
author = {Cl\'{e}ment, Alexandre and Perdrix, Simon},
title = {{PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {24:1--24:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.24},
URN = {urn:nbn:de:0030-drops-126921},
doi = {10.4230/LIPIcs.MFCS.2020.24},
annote = {Keywords: Quantum Computing, Diagrammatic Language, Completeness, Quantum Control, Polarising Beam Splitter, Categorical Quantum Mechanics, Quantum Switch}
}
Document
Enumeration of s-d Separators in DAGs with Application to Reliability Analysis in Temporal Graphs
Abstract
Temporal graphs are graphs in which arcs have temporal labels, specifying at which time they can be traversed. Motivated by recent results concerning the reliability analysis of a temporal graph through the enumeration of minimal cutsets in the corresponding line graph, in this paper we attack the problem of enumerating minimal s-d separators in s-d directed acyclic graphs (in short, s-d DAGs), also known as 2-terminal DAGs or s-t digraphs. Our main result is an algorithm for enumerating all the minimal s-d separators in a DAG with O(nm) delay, where n and m are respectively the number of nodes and arcs, and the delay is the time between the output of two consecutive solutions. To this aim, we give a characterization of the minimal s-d separators in a DAG through vertex cuts of an expanded version of the DAG itself. As a consequence of our main result, we provide an algorithm for enumerating all the minimal s-d cutsets in a temporal graph with delay O(m³), where m is the number of temporal arcs.
Cite as
Alessio Conte, Pierluigi Crescenzi, Andrea Marino, and Giulia Punzi. Enumeration of s-d Separators in DAGs with Application to Reliability Analysis in Temporal Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{conte_et_al:LIPIcs.MFCS.2020.25,
author = {Conte, Alessio and Crescenzi, Pierluigi and Marino, Andrea and Punzi, Giulia},
title = {{Enumeration of s-d Separators in DAGs with Application to Reliability Analysis in Temporal Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {25:1--25:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.25},
URN = {urn:nbn:de:0030-drops-126932},
doi = {10.4230/LIPIcs.MFCS.2020.25},
annote = {Keywords: minimal cutset, temporal graph, minimal separator, directed acyclic graph}
}
Document
Span Programs and Quantum Time Complexity
Abstract
Span programs are an important model of quantum computation due to their correspondence with quantum query and space complexity. While the query complexity of quantum algorithms obtained from span programs is well-understood, it is not generally clear how to implement certain query-independent operations in a time-efficient manner. In this work, we prove an analogous connection for quantum time complexity. In particular, we show how to convert a sufficiently-structured quantum algorithm for f with time complexity T into a span program for f such that it compiles back into a quantum algorithm for f with time complexity 𝒪̃(T). This shows that for span programs derived from algorithms with a time-efficient implementation, we can preserve the time efficiency when implementing the span program, which means that span programs capture time, query and space complexities and are a complete model of quantum algorithms.
One practical advantage of being able to convert quantum algorithms to span programs in a way that preserves time complexity is that span programs compose very nicely. We demonstrate this by improving Ambainis’s variable-time quantum search result using our construction through a span program composition for the OR function.
Cite as
Arjan Cornelissen, Stacey Jeffery, Maris Ozols, and Alvaro Piedrafita. Span Programs and Quantum Time Complexity. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cornelissen_et_al:LIPIcs.MFCS.2020.26,
author = {Cornelissen, Arjan and Jeffery, Stacey and Ozols, Maris and Piedrafita, Alvaro},
title = {{Span Programs and Quantum Time Complexity}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {26:1--26:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.26},
URN = {urn:nbn:de:0030-drops-126947},
doi = {10.4230/LIPIcs.MFCS.2020.26},
annote = {Keywords: quantum query algorithms, span programs, variable-time quantum search}
}
Document
Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle
Abstract
In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph G and a Hamiltonian cycle C₀ of G, how can we compute a second Hamiltonian cycle C₁ ≠ C₀ of G? Cedric Smith and William Tutte proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is a deterministic algorithm which computes the second Hamiltonian cycle in O(n⋅2^0.299862744n) = O(1.23103ⁿ) time and in linear space, thus improving the state of the art running time of O*(2^0.3n) = O(1.2312ⁿ) for solving this problem (among deterministic algorithms running in polynomial space). Whenever the input graph G does not contain any induced cycle C₆ on 6 vertices, the running time becomes O(n⋅ 2^0.2971925n) = O(1.22876ⁿ). Our algorithm is based on a fundamental structural property of Thomason’s lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a (not necessarily cubic) Hamiltonian graph G with a given Hamiltonian cycle C₀ (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least n - 4α (√n+2α)+8, where α = (Δ-2)/(δ-2) and δ,Δ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art.
Cite as
Argyrios Deligkas, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{deligkas_et_al:LIPIcs.MFCS.2020.27,
author = {Deligkas, Argyrios and Mertzios, George B. and Spirakis, Paul G. and Zamaraev, Viktor},
title = {{Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {27:1--27:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.27},
URN = {urn:nbn:de:0030-drops-126953},
doi = {10.4230/LIPIcs.MFCS.2020.27},
annote = {Keywords: Hamiltonian cycle, cubic graph, exact algorithm, approximation algorithm}
}
Document
Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes
Abstract
We consider the bit-probe complexity of the set membership problem: represent an n-element subset S of an m-element universe as a succinct bit vector so that membership queries of the form "Is x ∈ S" can be answered using at most t probes into the bit vector. Let s(m,n,t) (resp. s_N(m,n,t)) denote the minimum number of bits of storage needed when the probes are adaptive (resp. non-adaptive). Lewenstein, Munro, Nicholson, and Raman (ESA 2014) obtain fully-explicit schemes that show that
s(m,n,t) = 𝒪((2^t-1)m^{1/(t - min{2⌊log n⌋, n-3/2})}) for n ≥ 2,t ≥ ⌊log n⌋+1 .
In this work, we improve this bound when the probes are allowed to be superlinear in n, i.e., when t ≥ Ω(nlog n), n ≥ 2, we design fully-explicit schemes that show that
s(m,n,t) = 𝒪((2^t-1)m^{1/(t-{n-1}/{2^{t/(2(n-1))}})}),
asymptotically (in the exponent of m) close to the non-explicit upper bound on s(m,n,t) derived by Radhakrishan, Shah, and Shannigrahi (ESA 2010), for constant n.
In the non-adaptive setting, it was shown by Garg and Radhakrishnan (STACS 2017) that for a large constant n₀, for n ≥ n₀, s_N(m,n,3) ≥ √{mn}. We improve this result by showing that the same lower bound holds even for storing sets of size 2, i.e., s_N(m,2,3) ≥ Ω(√m).
Cite as
Palash Dey, Jaikumar Radhakrishnan, and Santhoshini Velusamy. Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 28:1-28:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{dey_et_al:LIPIcs.MFCS.2020.28,
author = {Dey, Palash and Radhakrishnan, Jaikumar and Velusamy, Santhoshini},
title = {{Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {28:1--28:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.28},
URN = {urn:nbn:de:0030-drops-126965},
doi = {10.4230/LIPIcs.MFCS.2020.28},
annote = {Keywords: Set membership, Bit-probe model, Fully-explicit data structures, Adaptive data structures, Error-correcting codes}
}
Document
Register Transducers Are Marble Transducers
Abstract
Deterministic two-way transducers define the class of regular functions from words to words. Alur and Cerný introduced an equivalent model of transducers with registers called copyless streaming string transducers. In this paper, we drop the "copyless" restriction on these machines and show that they are equivalent to two-way transducers enhanced with the ability to drop marks, named "marbles", on the input. We relate the maximal number of marbles used with the amount of register copies performed by the streaming string transducer. Finally, we show that the class membership problems associated with these models are decidable. Our results can be interpreted in terms of program optimization for simple recursive and iterative programs.
Cite as
Gaëtan Douéneau-Tabot, Emmanuel Filiot, and Paul Gastin. Register Transducers Are Marble Transducers. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{doueneautabot_et_al:LIPIcs.MFCS.2020.29,
author = {Dou\'{e}neau-Tabot, Ga\"{e}tan and Filiot, Emmanuel and Gastin, Paul},
title = {{Register Transducers Are Marble Transducers}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {29:1--29:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.29},
URN = {urn:nbn:de:0030-drops-126979},
doi = {10.4230/LIPIcs.MFCS.2020.29},
annote = {Keywords: streaming string transducer, two-way transducer, marbles, pebbles}
}
Document
Randomization in Non-Uniform Finite Automata
Abstract
The non-uniform version of Turing machines with an extra advice input tape that depends on the length of the input but not the input itself is a well-studied model in complexity theory. We investigate the same notion of non-uniformity in weaker models, namely one-way finite automata. In particular, we are interested in the power of two-sided bounded-error randomization, and how it compares to determinism and non-determinism. We show that for unlimited advice, randomization is strictly stronger than determinism, and strictly weaker than non-determinism. However, when the advice is restricted to polynomial length, the landscape changes: the expressive power of determinism and randomization does not change, but the power of non-determinism is reduced to the extent that it becomes incomparable with randomization.
Cite as
Pavol Ďuriš, Rastislav Královič, Richard Královič, Dana Pardubská, Martin Pašen, and Peter Rossmanith. Randomization in Non-Uniform Finite Automata. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{duris_et_al:LIPIcs.MFCS.2020.30,
author = {\v{D}uri\v{s}, Pavol and Kr\'{a}lovi\v{c}, Rastislav and Kr\'{a}lovi\v{c}, Richard and Pardubsk\'{a}, Dana and Pa\v{s}en, Martin and Rossmanith, Peter},
title = {{Randomization in Non-Uniform Finite Automata}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {30:1--30:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.30},
URN = {urn:nbn:de:0030-drops-126987},
doi = {10.4230/LIPIcs.MFCS.2020.30},
annote = {Keywords: finite automata, non-uniform computation, randomization}
}
Document
Extending Nearly Complete 1-Planar Drawings in Polynomial Time
Abstract
The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph G, a connected subgraph H of G and a drawing H of H, the extension problem asks whether H can be extended into a drawing of G while maintaining some desired property of the drawing (e.g., planarity).
In their breakthrough result, Angelini et al. [ACM TALG 2015] showed that the extension problem is polynomial-time solvable when the aim is to preserve planarity. Very recently we considered this problem for partial 1-planar drawings [ICALP 2020], which are drawings in the plane that allow each edge to have at most one crossing. The most important question identified and left open in that work is whether the problem can be solved in polynomial time when H can be obtained from G by deleting a bounded number of vertices and edges. In this work, we answer this question positively by providing a constructive polynomial-time decision algorithm.
Cite as
Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending Nearly Complete 1-Planar Drawings in Polynomial Time. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{eiben_et_al:LIPIcs.MFCS.2020.31,
author = {Eiben, Eduard and Ganian, Robert and Hamm, Thekla and Klute, Fabian and N\"{o}llenburg, Martin},
title = {{Extending Nearly Complete 1-Planar Drawings in Polynomial Time}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {31:1--31:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.31},
URN = {urn:nbn:de:0030-drops-126998},
doi = {10.4230/LIPIcs.MFCS.2020.31},
annote = {Keywords: Extension problems, 1-planarity}
}
Document
The Mergegram of a Dendrogram and Its Stability
Abstract
This paper extends the key concept of persistence within Topological Data Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case the distance-based persistence is determined by a single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stable-under-noise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay.
Cite as
Yury Elkin and Vitaliy Kurlin. The Mergegram of a Dendrogram and Its Stability. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{elkin_et_al:LIPIcs.MFCS.2020.32,
author = {Elkin, Yury and Kurlin, Vitaliy},
title = {{The Mergegram of a Dendrogram and Its Stability}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {32:1--32:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.32},
URN = {urn:nbn:de:0030-drops-127281},
doi = {10.4230/LIPIcs.MFCS.2020.32},
annote = {Keywords: clustering dendrogram, topological data analysis, persistence, stability}
}
Document
Synchronizing Deterministic Push-Down Automata Can Be Really Hard
Abstract
The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata.
There are several interpretations of what synchronizability should mean for deterministic push-down automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference.
Cite as
Henning Fernau, Petra Wolf, and Tomoyuki Yamakami. Synchronizing Deterministic Push-Down Automata Can Be Really Hard. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{fernau_et_al:LIPIcs.MFCS.2020.33,
author = {Fernau, Henning and Wolf, Petra and Yamakami, Tomoyuki},
title = {{Synchronizing Deterministic Push-Down Automata Can Be Really Hard}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {33:1--33:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.33},
URN = {urn:nbn:de:0030-drops-127008},
doi = {10.4230/LIPIcs.MFCS.2020.33},
annote = {Keywords: Synchronizing automaton, Reset sequence, Real-time deterministic push-down automaton, Finite-turn push-down automaton, Computability, Computational complexity}
}
Document
Value Iteration Using Universal Graphs and the Complexity of Mean Payoff Games
Abstract
We study the computational complexity of solving mean payoff games. This class of games can be seen as an extension of parity games, and they have similar complexity status: in both cases solving them is in NP ∩ coNP and not known to be in P. In a breakthrough result Calude, Jain, Khoussainov, Li, and Stephan constructed in 2017 a quasipolynomial time algorithm for solving parity games, which was quickly followed by a few other algorithms with the same complexity. Our objective is to investigate how these techniques can be extended to mean payoff games.
The starting point is the combinatorial notion of universal trees: all quasipolynomial time algorithms for parity games have been shown to exploit universal trees. Universal graphs extend universal trees to arbitrary (positionally determined) objectives. We show that they yield a family of value iteration algorithms for solving mean payoff games which includes the value iteration algorithm due to Brim, Chaloupka, Doyen, Gentilini, and Raskin.
The contribution of this paper is to prove tight bounds on the complexity of algorithms for mean payoff games using universal graphs. We consider two parameters: the largest weight N in absolute value and the number k of weights. The dependence in N in the existing value iteration algorithm is linear, we show that this can be improved to N^{1 - 1/n} and obtain a matching lower bound. However, we show that we cannot break the linear dependence in the exponent in the number k of weights implying that universal graphs do not yield a quasipolynomial time algorithm for solving mean payoff games.
Cite as
Nathanaël Fijalkow, Paweł Gawrychowski, and Pierre Ohlmann. Value Iteration Using Universal Graphs and the Complexity of Mean Payoff Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{fijalkow_et_al:LIPIcs.MFCS.2020.34,
author = {Fijalkow, Nathana\"{e}l and Gawrychowski, Pawe{\l} and Ohlmann, Pierre},
title = {{Value Iteration Using Universal Graphs and the Complexity of Mean Payoff Games}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {34:1--34:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.34},
URN = {urn:nbn:de:0030-drops-127011},
doi = {10.4230/LIPIcs.MFCS.2020.34},
annote = {Keywords: Mean payoff games, Universal graphs, Value iteration}
}
Document
Building Large k-Cores from Sparse Graphs
Abstract
A popular model to measure network stability is the k-core, that is the maximal induced subgraph in which every vertex has degree at least k. For example, k-cores are commonly used to model the unraveling phenomena in social networks. In this model, users having less than k connections within the network leave it, so the remaining users form exactly the k-core. In this paper we study the question of whether it is possible to make the network more robust by spending only a limited amount of resources on new connections. A mathematical model for the k-core construction problem is the following Edge k-Core optimization problem. We are given a graph G and integers k, b and p. The task is to ensure that the k-core of G has at least p vertices by adding at most b edges.
The previous studies on Edge k-Core demonstrate that the problem is computationally challenging. In particular, it is NP-hard when k = 3, W[1]-hard when parameterized by k+b+p (Chitnis and Talmon, 2018), and APX-hard (Zhou et al, 2019). Nevertheless, we show that there are efficient algorithms with provable guarantee when the k-core has to be constructed from a sparse graph with some additional structural properties. Our results are
- When the input graph is a forest, Edge k-Core is solvable in polynomial time;
- Edge k-Core is fixed-parameter tractable (FPT) when parameterized by the minimum size of a vertex cover in the input graph. On the other hand, with such parameterization, the problem does not admit a polynomial kernel subject to a widely-believed assumption from complexity theory;
- Edge k-Core is FPT parameterized by the treewidth of the graph plus k. This improves upon a result of Chitnis and Talmon by not requiring b to be small. Each of our algorithms is built upon a new graph-theoretical result interesting in its own.
Cite as
Fedor V. Fomin, Danil Sagunov, and Kirill Simonov. Building Large k-Cores from Sparse Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{fomin_et_al:LIPIcs.MFCS.2020.35,
author = {Fomin, Fedor V. and Sagunov, Danil and Simonov, Kirill},
title = {{Building Large k-Cores from Sparse Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {35:1--35:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.35},
URN = {urn:nbn:de:0030-drops-127026},
doi = {10.4230/LIPIcs.MFCS.2020.35},
annote = {Keywords: parameterized complexity, k-core, vertex cover, treewidth}
}
Document
The Complexity of Approximating the Complex-Valued Potts Model
Abstract
We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane.
Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q ≥ 2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations.
Cite as
Andreas Galanis, Leslie Ann Goldberg, and Andrés Herrera-Poyatos. The Complexity of Approximating the Complex-Valued Potts Model. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{galanis_et_al:LIPIcs.MFCS.2020.36,
author = {Galanis, Andreas and Goldberg, Leslie Ann and Herrera-Poyatos, Andr\'{e}s},
title = {{The Complexity of Approximating the Complex-Valued Potts Model}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {36:1--36:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.36},
URN = {urn:nbn:de:0030-drops-127038},
doi = {10.4230/LIPIcs.MFCS.2020.36},
annote = {Keywords: approximate counting, Potts model, Tutte polynomial, partition function, complex numbers}
}
Document
Fast Algorithms for General Spin Systems on Bipartite Expanders
Abstract
A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs.
In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to "bicliques" of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ(n²) time, where n is the size of the graph.
Cite as
Andreas Galanis, Leslie Ann Goldberg, and James Stewart. Fast Algorithms for General Spin Systems on Bipartite Expanders. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{galanis_et_al:LIPIcs.MFCS.2020.37,
author = {Galanis, Andreas and Goldberg, Leslie Ann and Stewart, James},
title = {{Fast Algorithms for General Spin Systems on Bipartite Expanders}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {37:1--37:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.37},
URN = {urn:nbn:de:0030-drops-127049},
doi = {10.4230/LIPIcs.MFCS.2020.37},
annote = {Keywords: bipartite expanders, approximate counting, spin systems}
}
Document
Linear High-Order Deterministic Tree Transducers with Regular Look-Ahead
Abstract
We introduce the notion of high-order deterministic top-down tree transducers (HODT) whose outputs correspond to single-typed lambda-calculus formulas. These transducers are natural generalizations of known models of top-tree transducers such as: Deterministic Top-Down Tree Transducers, Macro Tree Transducers, Streaming Tree Transducers... We focus on the linear restriction of high order tree transducers with look-ahead (HODTR_lin), and prove this corresponds to tree to tree functional transformations defined by Monadic Second Order (MSO) logic. We give a specialized procedure for the composition of those transducers that uses a flow analysis based on coherence spaces and allows us to preserve the linearity of transducers. This procedure has a better complexity than classical algorithms for composition of other equivalent tree transducers, but raises the order of transducers. However, we also indicate that the order of a HODTR_lin can always be bounded by 3, and give a procedure that reduces the order of a HODTR_lin to 3. As those resulting HODTR_lin can then be transformed into other equivalent models, this gives an important insight on composition algorithm for other classes of transducers. Finally, we prove that those results partially translate to the case of almost linear HODTR: the class corresponds to the class of tree transformations performed by MSO with unfolding (not closed by composition), and provide a mechanism to reduce the order to 3 in this case.
Cite as
Paul D. Gallot, Aurélien Lemay, and Sylvain Salvati. Linear High-Order Deterministic Tree Transducers with Regular Look-Ahead. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gallot_et_al:LIPIcs.MFCS.2020.38,
author = {Gallot, Paul D. and Lemay, Aur\'{e}lien and Salvati, Sylvain},
title = {{Linear High-Order Deterministic Tree Transducers with Regular Look-Ahead}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {38:1--38:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.38},
URN = {urn:nbn:de:0030-drops-127050},
doi = {10.4230/LIPIcs.MFCS.2020.38},
annote = {Keywords: Transducers, \lambda-calculus, Trees}
}
Document
Graph Clustering in All Parameter Regimes
Abstract
Resolution parameters in graph clustering control the size and structure of clusters formed by solving a parametric objective function. Typically there is more than one meaningful way to cluster a graph, and solving the same objective function for different resolution parameters produces clusterings at different levels of granularity, each of which can be meaningful depending on the application. In this paper, we address the task of efficiently solving a parameterized graph clustering objective for all values of a resolution parameter. Specifically, we consider a new analysis-friendly objective we call LambdaPrime, involving a parameter λ ∈ (0,1). LambdaPrime is an adaptation of LambdaCC, a significant family of instances of the Correlation Clustering (minimization) problem. Indeed, LambdaPrime and LambdaCC are closely related to other parameterized clustering problems, such as parametric generalizations of modularity. They capture a number of specific clustering problems as special cases, including sparsest cut and cluster deletion. While previous work provides approximation results for a single value of the resolution parameter, we seek a set of approximately optimal clusterings for all values of λ in polynomial time.
More specifically, we show that when a graph has m edges and n nodes, there exists a set of at most m clusterings such that, for every λ ∈ (0,1), the family contains an optimal solution to the LambdaPrime objective. This bound is tight on star graphs. We obtain a family of O(log n) clusterings by solving the parametric linear programming (LP) relaxation of LambdaPrime at O(log n) λ values, and rounding each LP solution using existing approximation algorithms. We prove that this is asymptotically tight: for a certain class of ring graphs, for all values of λ, Ω(log n) feasible solutions are required to provide a constant-factor approximation for the LambdaPrime LP relaxation. To minimize the size of the clustering family, we further propose an algorithm that yields a family of solutions of a size no more than twice of the minimum LP-approximating family.
Cite as
Junhao Gan, David F. Gleich, Nate Veldt, Anthony Wirth, and Xin Zhang. Graph Clustering in All Parameter Regimes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gan_et_al:LIPIcs.MFCS.2020.39,
author = {Gan, Junhao and Gleich, David F. and Veldt, Nate and Wirth, Anthony and Zhang, Xin},
title = {{Graph Clustering in All Parameter Regimes}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {39:1--39:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.39},
URN = {urn:nbn:de:0030-drops-127065},
doi = {10.4230/LIPIcs.MFCS.2020.39},
annote = {Keywords: Graph Clustering, Algorithms, Parametric Linear Programming}
}
Document
A Quasiorder-Based Perspective on Residual Automata
Abstract
In this work, we define a framework of automata constructions based on quasiorders over words to provide new insights on the class of residual automata. We present a new residualization operation and a generalized double-reversal method for building the canonical residual automaton for a given language. Finally, we use our framework to offer a quasiorder-based perspective on NL^*, an online learning algorithm for residual automata. We conclude that quasiorders are fundamental to residual automata as congruences are to deterministic automata.
Cite as
Pierre Ganty, Elena Gutiérrez, and Pedro Valero. A Quasiorder-Based Perspective on Residual Automata. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ganty_et_al:LIPIcs.MFCS.2020.40,
author = {Ganty, Pierre and Guti\'{e}rrez, Elena and Valero, Pedro},
title = {{A Quasiorder-Based Perspective on Residual Automata}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {40:1--40:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.40},
URN = {urn:nbn:de:0030-drops-127071},
doi = {10.4230/LIPIcs.MFCS.2020.40},
annote = {Keywords: Residual Automata, Quasiorders, Double-Reversal Method, Canonical RFA, Regular Languages}
}
Document
Fractional Covers of Hypergraphs with Bounded Multi-Intersection
Abstract
Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself. This allows us to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is ≤ k for some constant k. We also show how our results translate to fractional vertex covers.
Cite as
Georg Gottlob, Matthias Lanzinger, Reinhard Pichler, and Igor Razgon. Fractional Covers of Hypergraphs with Bounded Multi-Intersection. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gottlob_et_al:LIPIcs.MFCS.2020.41,
author = {Gottlob, Georg and Lanzinger, Matthias and Pichler, Reinhard and Razgon, Igor},
title = {{Fractional Covers of Hypergraphs with Bounded Multi-Intersection}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {41:1--41:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.41},
URN = {urn:nbn:de:0030-drops-127317},
doi = {10.4230/LIPIcs.MFCS.2020.41},
annote = {Keywords: Fractional graph theory, fractional edge cover, fractional hypertree width, bounded multi-intersection, fractional cover, fractional vertex cover}
}
Document
Factoring Polynomials over Finite Fields with Linear Galois Groups: An Additive Combinatorics Approach
Abstract
Let f̃(X) ∈ ℤ[X] be a degree-n polynomial such that f(X): = f̃(X)od p factorizes into n distinct linear factors over 𝔽_p. We study the problem of deterministically factoring f(X) over 𝔽_p given f̃(X). Under the generalized Riemann hypothesis (GRH), we give an improved deterministic algorithm that computes the complete factorization of f(X) in the case that the Galois group of f̃(X) is (permutation isomorphic to) a linear group G ≤ GL(V) on the set S of roots of f̃(X), where V is a finite-dimensional vector space over a finite field 𝔽 and S is identified with a subset of V. In particular, when |S| = |V|^{Ω(1)}, the algorithm runs in time polynomial in n^{log n/(log log log log n)^{1/3}} and the size of the input, improving Evdokimov’s algorithm. Our result also applies to a general Galois group G when combined with a recent algorithm of the author.
To prove our main result, we introduce a family of objects called linear m-schemes and reduce the problem of factoring f(X) to a combinatorial problem about these objects. We then apply techniques from additive combinatorics to obtain an improved bound. Our techniques may be of independent interest.
Cite as
Zeyu Guo. Factoring Polynomials over Finite Fields with Linear Galois Groups: An Additive Combinatorics Approach. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{guo:LIPIcs.MFCS.2020.42,
author = {Guo, Zeyu},
title = {{Factoring Polynomials over Finite Fields with Linear Galois Groups: An Additive Combinatorics Approach}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {42:1--42:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.42},
URN = {urn:nbn:de:0030-drops-127082},
doi = {10.4230/LIPIcs.MFCS.2020.42},
annote = {Keywords: polynomial factoring, permutation group, finite field, algebraic combinatorics, additive combinatorics, derandomization}
}
Document
Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs
Abstract
We show that given an embedding of an O(log n) genus bipartite graph, one can construct an edge weight function in logarithmic space, with respect to which the minimum weight perfect matching in the graph is unique, if one exists.
As a consequence, we obtain that deciding whether such a graph has a perfect matching or not is in SPL. In 1999, Reinhardt, Allender and Zhou proved that if one can construct a polynomially bounded weight function for a graph in logspace such that it isolates a minimum weight perfect matching in the graph, then the perfect matching problem can be solved in SPL. In this paper, we give a deterministic logspace construction of such a weight function for O(log n) genus bipartite graphs.
Cite as
Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari. Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 43:1-43:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gupta_et_al:LIPIcs.MFCS.2020.43,
author = {Gupta, Chetan and Sharma, Vimal Raj and Tewari, Raghunath},
title = {{Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {43:1--43:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.43},
URN = {urn:nbn:de:0030-drops-127099},
doi = {10.4230/LIPIcs.MFCS.2020.43},
annote = {Keywords: Logspace computation, High genus, Matching isolation}
}
Document
Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory
Abstract
It is known that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal to the length of the longest shared secret key that two parties can establish via a probabilistic protocol with interaction on a public channel, assuming that the parties hold as their inputs x and y respectively. We determine the worst-case communication complexity of this problem for the setting where the parties can use private sources of random bits.
We show that for some x, y the communication complexity of the secret key agreement does not decrease even if the parties have to agree on a secret key the size of which is much smaller than the mutual information between x and y. On the other hand, we provide examples of x, y such that the communication complexity of the protocol declines gradually with the size of the derived secret key.
The proof of the main result uses spectral properties of appropriate graphs and the expander mixing lemma as well as various information theoretic techniques.
Cite as
Emirhan Gürpınar and Andrei Romashchenko. Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gurpinar_et_al:LIPIcs.MFCS.2020.44,
author = {G\"{u}rp{\i}nar, Emirhan and Romashchenko, Andrei},
title = {{Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {44:1--44:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.44},
URN = {urn:nbn:de:0030-drops-127102},
doi = {10.4230/LIPIcs.MFCS.2020.44},
annote = {Keywords: Kolmogorov complexity, mutual information, communication complexity, expander mixing lemma, finite geometry}
}
Document
∃ℝ-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games
Abstract
We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is ∃ℝ-complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known gadget games without any stationary Nash equilibrium, we obtain that for cyclic games, just deciding existence of any stationary Nash equilibrium is ∃ℝ-complete. This holds for reach-a-set games, stay-in-a-set games, and for deterministic recursive games.
Cite as
Kristoffer Arnsfelt Hansen and Steffan Christ Sølvsten. ∃ℝ-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{hansen_et_al:LIPIcs.MFCS.2020.45,
author = {Hansen, Kristoffer Arnsfelt and S{\o}lvsten, Steffan Christ},
title = {{\exists\mathbb{R}-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {45:1--45:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.45},
URN = {urn:nbn:de:0030-drops-127113},
doi = {10.4230/LIPIcs.MFCS.2020.45},
annote = {Keywords: Existential Theory of the Reals, Stationary Nash Equilibrium, Perfect Information Stochastic Games}
}
Document
Minimum 0-Extension Problems on Directed Metrics
Abstract
For a metric μ on a finite set T, the minimum 0-extension problem 0-Ext[μ] is defined as follows: Given V ⊇ T and c:(V 2) → ℚ+, minimize ∑ c(xy)μ(γ(x),γ(y)) subject to γ:V → T, γ(t) = t (∀ t ∈ T), where the sum is taken over all unordered pairs in V. This problem generalizes several classical combinatorial optimization problems such as the minimum cut problem or the multiterminal cut problem. The complexity dichotomy of 0-Ext[μ] was established by Karzanov and Hirai, which is viewed as a manifestation of the dichotomy theorem for finite-valued CSPs due to Thapper and Živný.
In this paper, we consider a directed version 0→-Ext[μ] of the minimum 0-extension problem, where μ and c are not assumed to be symmetric. We extend the NP-hardness condition of 0-Ext[μ] to 0→-Ext[μ]: If μ cannot be represented as the shortest path metric of an orientable modular graph with an orbit-invariant "directed" edge-length, then 0→-Ext[μ] is NP-hard. We also show a partial converse: If μ is a directed metric of a modular lattice with an orbit-invariant directed edge-length, then 0→-Ext[μ] is tractable. We further provide a new NP-hardness condition characteristic of 0→-Ext[μ], and establish a dichotomy for the case where μ is a directed metric of a star.
Cite as
Hiroshi Hirai and Ryuhei Mizutani. Minimum 0-Extension Problems on Directed Metrics. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{hirai_et_al:LIPIcs.MFCS.2020.46,
author = {Hirai, Hiroshi and Mizutani, Ryuhei},
title = {{Minimum 0-Extension Problems on Directed Metrics}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {46:1--46:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.46},
URN = {urn:nbn:de:0030-drops-127120},
doi = {10.4230/LIPIcs.MFCS.2020.46},
annote = {Keywords: Minimum 0-extension problems, Directed metrics, Valued constraint satisfaction problems, Computational complexity}
}
Document
Hierarchical Clusterings of Unweighted Graphs
Abstract
We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph G and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs H having the property that for any k the graph H^{(k)} being the join of k copies of H has an optimal hierarchical clustering that splits each copy of H in the same optimal way. To optimally cluster such a graph H^{(k)} we thus only need to optimally cluster the smaller graph H. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.
Cite as
Svein Høgemo, Christophe Paul, and Jan Arne Telle. Hierarchical Clusterings of Unweighted Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{hgemo_et_al:LIPIcs.MFCS.2020.47,
author = {H{\o}gemo, Svein and Paul, Christophe and Telle, Jan Arne},
title = {{Hierarchical Clusterings of Unweighted Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {47:1--47:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.47},
URN = {urn:nbn:de:0030-drops-127139},
doi = {10.4230/LIPIcs.MFCS.2020.47},
annote = {Keywords: Hierarchical Clustering}
}
Document
On Affine Reachability Problems
Abstract
We analyze affine reachability problems in dimensions 1 and 2. We show that the reachability problem for 1-register machines over the integers with affine updates is PSPACE-hard, hence PSPACE-complete, strengthening a result by Finkel et al. that required polynomial updates. Building on recent results on two-dimensional integer matrices, we prove NP-completeness of the mortality problem for 2-dimensional integer matrices with determinants +1 and 0. Motivated by tight connections with 1-dimensional affine reachability problems without control states, we also study the complexity of a number of reachability problems in finitely generated semigroups of 2-dimensional upper-triangular integer matrices.
Cite as
Stefan Jaax and Stefan Kiefer. On Affine Reachability Problems. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{jaax_et_al:LIPIcs.MFCS.2020.48,
author = {Jaax, Stefan and Kiefer, Stefan},
title = {{On Affine Reachability Problems}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {48:1--48:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.48},
URN = {urn:nbn:de:0030-drops-127148},
doi = {10.4230/LIPIcs.MFCS.2020.48},
annote = {Keywords: Counter Machines, Matrix Semigroups, Reachability}
}
Document
Structural Parameterizations of Clique Coloring
Abstract
A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an 𝒪^⋆(q^{tw})-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width.
Cite as
Lars Jaffke, Paloma T. Lima, and Geevarghese Philip. Structural Parameterizations of Clique Coloring. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2020.49,
author = {Jaffke, Lars and Lima, Paloma T. and Philip, Geevarghese},
title = {{Structural Parameterizations of Clique Coloring}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {49:1--49:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.49},
URN = {urn:nbn:de:0030-drops-127157},
doi = {10.4230/LIPIcs.MFCS.2020.49},
annote = {Keywords: clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis}
}
Document
Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach
Abstract
It can be shown that each permutation group G ⊑ 𝕊_n can be embedded, in a well defined sense, in a connected graph with O(n+|G|) vertices. Some groups, however, require much fewer vertices. For instance, 𝕊_n itself can be embedded in the n-clique K_n, a connected graph with n vertices.
In this work, we show that the minimum size of a context-free grammar generating a finite permutation group G⊑ 𝕊_n can be upper bounded by three structural parameters of connected graphs embedding G: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group G ⊑ 𝕊_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree Δ, can also be generated by a context-free grammar of size 2^{O(kΔlogΔ)}⋅ m^{O(k)}. By combining our upper bound with a connection established by Pesant, Quimper, Rousseau and Sellmann [Gilles Pesant et al., 2009] between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity 2^{O(kΔlogΔ)}⋅ m^{O(k)}.
The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated 2^{Ω(n)} lower bound on the grammar complexity of the symmetric group 𝕊_n due to Glaister and Shallit [Glaister and Shallit, 1996] we have that connected graphs of treewidth o(n/log n) and maximum degree o(n/log n) embedding subgroups of 𝕊_n of index 2^{cn} for some small constant c must have n^{ω(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{ε} for ε < 1 and maximum degree o(n/log n).
Cite as
Lars Jaffke, Mateus de Oliveira Oliveira, and Hans Raj Tiwary. Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2020.50,
author = {Jaffke, Lars and de Oliveira Oliveira, Mateus and Tiwary, Hans Raj},
title = {{Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {50:1--50:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.50},
URN = {urn:nbn:de:0030-drops-127161},
doi = {10.4230/LIPIcs.MFCS.2020.50},
annote = {Keywords: Permutation Groups, Context Free Grammars, Extension Complexity, Graph Embedding Complexity}
}
Document
Unary Prime Languages
Abstract
A regular language L of finite words is composite if there are regular languages L₁,L₂,…,L_t such that L = ⋂_{i = 1}^t L_i and the index (number of states in a minimal DFA) of every language L_i is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [O. Kupferman and J. Mosheiff, 2015], where the complexity of deciding the primality of the language of a given DFA was left open, with a doubly-exponential gap between the upper and lower bounds. We study primality for unary regular languages, namely regular languages with a singleton alphabet. A unary language corresponds to a subset of ℕ, making the study of unary prime languages closer to that of primality in number theory. We show that the setting of languages is richer. In particular, while every composite number is the product of two smaller numbers, the number t of languages necessary to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness for a unary language L, namely a word that is not in L but is in all products of languages that contain L and have an index smaller than L’s, may be of exponential length. Still, we are able to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace algorithm for primality checking of unary DFAs.
Cite as
Ismaël Jecker, Orna Kupferman, and Nicolas Mazzocchi. Unary Prime Languages. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{jecker_et_al:LIPIcs.MFCS.2020.51,
author = {Jecker, Isma\"{e}l and Kupferman, Orna and Mazzocchi, Nicolas},
title = {{Unary Prime Languages}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {51:1--51:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.51},
URN = {urn:nbn:de:0030-drops-127177},
doi = {10.4230/LIPIcs.MFCS.2020.51},
annote = {Keywords: Deterministic Finite Automata (DFA), Regular Languages, Primality}
}
Document
A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes
Abstract
Permutation Pattern Matching (PPM) is the problem of deciding for a given pair of permutations π and τ whether the pattern π is contained in the text τ. Bose, Buss and Lubiw showed that PPM is NP-complete. In view of this result, it is natural to ask how the situation changes when we restrict the pattern π to a fixed permutation class 𝒞; this is known as the 𝒞-Pattern PPM problem. There have been several results in this direction, namely the work of Jelínek and Kynčl who completely resolved the hardness of 𝒞-Pattern PPM when 𝒞 is taken to be the class of σ-avoiding permutations for some σ.
Grid classes are special kind of permutation classes, consisting of permutations admitting a grid-like decomposition into simpler building blocks. Of particular interest are the so-called monotone grid classes, in which each building block is a monotone sequence. Recently, it has been discovered that grid classes, especially the monotone ones, play a fundamental role in the understanding of the structure of general permutation classes. This motivates us to study the hardness of 𝒞-Pattern PPM for a (monotone) grid class 𝒞.
We provide a complexity dichotomy for 𝒞-Pattern PPM when 𝒞 is taken to be a monotone grid class. Specifically, we show that the problem is polynomial-time solvable if a certain graph associated with 𝒞, called the cell graph, is a forest, and it is NP-complete otherwise. We further generalize our results to grid classes whose blocks belong to classes of bounded grid-width. We show that the 𝒞-Pattern PPM for such a grid class 𝒞 is polynomial-time solvable if the cell graph of 𝒞 avoids a cycle or a certain special type of path, and it is NP-complete otherwise.
Cite as
Vít Jelínek, Michal Opler, and Jakub Pekárek. A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 52:1-52:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{jelinek_et_al:LIPIcs.MFCS.2020.52,
author = {Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Pek\'{a}rek, Jakub},
title = {{A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {52:1--52:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.52},
URN = {urn:nbn:de:0030-drops-127186},
doi = {10.4230/LIPIcs.MFCS.2020.52},
annote = {Keywords: permutations, pattern matching, grid classes}
}
Document
Quantum-Inspired Classical Algorithms for Singular Value Transformation
Abstract
A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS 2017). The resulting algorithm, classical but "quantum-inspired", efficiently computes a low-rank approximation of the users' preference matrix. Subsequent works have shown how to construct efficient quantum-inspired algorithms for approximating the pseudo-inverse of a low-rank matrix as well, which can be used to (approximately) solve low-rank linear systems of equations. In the present paper, we pursue this line of research and develop quantum-inspired algorithms for a large class of matrix transformations that are defined via the singular value decomposition of the matrix. In particular, we obtain classical algorithms with complexity polynomially related (in most parameters) to the complexity of the best quantum algorithms for singular value transformation recently developed by Chakraborty, Gilyén and Jeffery (ICALP 2019) and Gilyén, Su, Low and Wiebe (STOC 2019).
Cite as
Dhawal Jethwani, François Le Gall, and Sanjay K. Singh. Quantum-Inspired Classical Algorithms for Singular Value Transformation. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{jethwani_et_al:LIPIcs.MFCS.2020.53,
author = {Jethwani, Dhawal and Le Gall, Fran\c{c}ois and Singh, Sanjay K.},
title = {{Quantum-Inspired Classical Algorithms for Singular Value Transformation}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {53:1--53:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.53},
URN = {urn:nbn:de:0030-drops-127193},
doi = {10.4230/LIPIcs.MFCS.2020.53},
annote = {Keywords: Sampling algorithms, quantum-inspired algorithms, linear algebra}
}
Document
On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems
Abstract
Consider a discrete dynamical system given by a square matrix M ∈ ℚ^{d × d} and a starting point s ∈ ℚ^d. The orbit of such a system is the infinite trajectory ⟨ s, Ms, M²s, …⟩. Given a collection T₁, T₂, …, T_m ⊆ ℝ^d of semialgebraic sets, we can associate with each T_i an atomic proposition P_i which evaluates to true at time n if, and only if, M^ns ∈ T_i. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system (M,s) and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.
Cite as
Toghrul Karimov, Joël Ouaknine, and James Worrell. On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{karimov_et_al:LIPIcs.MFCS.2020.54,
author = {Karimov, Toghrul and Ouaknine, Jo\"{e}l and Worrell, James},
title = {{On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {54:1--54:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.54},
URN = {urn:nbn:de:0030-drops-127215},
doi = {10.4230/LIPIcs.MFCS.2020.54},
annote = {Keywords: Linear dynamical systems, Orbit Problem, LTL model checking}
}
Document
Even Faster Algorithms for CSAT Over supernilpotent Algebras
Abstract
Recently, a few papers considering the polynomial equation satisfiability problem and the circuit satisfiability problem over finite supernilpotent algebras from so called congruence modular varieties were published. All the algorithms considered in these papers are quite similar and rely on checking a not too big set of potential solutions. Two of these algorithms achieving the lowest time complexity up to now, were presented in [Aichinger, 2019] (algorithm working for finite supernilpotent algebras) and in [Földvári, 2018] (algorithm working in the group case). In this paper we show a deterministic algorithm of the same type solving the considered problems for finite supernilpotent algebras which has lower computational complexity than the algorithm presented in [Aichinger, 2019] and in most cases even lower than the group case algorithm from [Földvári, 2018]. We also present a linear time Monte Carlo algorithm solving the same problem. This, together with the algorithm for nilpotent but not supernilpotent algebras presented in [Paweł M. Idziak et al., 2020], is the very first attempt to solving the circuit satisfiability problem using probabilistic algorithms.
Cite as
Piotr Kawałek and Jacek Krzaczkowski. Even Faster Algorithms for CSAT Over supernilpotent Algebras. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kawalek_et_al:LIPIcs.MFCS.2020.55,
author = {Kawa{\l}ek, Piotr and Krzaczkowski, Jacek},
title = {{Even Faster Algorithms for CSAT Over supernilpotent Algebras}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {55:1--55:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.55},
URN = {urn:nbn:de:0030-drops-127225},
doi = {10.4230/LIPIcs.MFCS.2020.55},
annote = {Keywords: circuit satisfiability, solving equations, supernilpotent algebras, satisfiability in groups}
}
Document
Continuous and Monotone Machines
Abstract
We investigate a variant of the fuel-based approach to modeling diverging computation in type theories and use it to abstractly capture the essence of oracle Turing machines. The resulting objects we call continuous machines. We prove that it is possible to translate back and forth between such machines and names in the standard function encoding used in computable analysis. Put differently, among the operators on Baire space, exactly the partial continuous ones are implementable by continuous machines and the data that such a machine provides is a description of the operator as a sequentially realizable functional. Continuous machines are naturally formulated in type theories and we have formalized our findings in Coq as part of Incone, a Coq library for computable analysis.
The correctness proofs use a classical meta-theory with countable choice. Along the way we formally prove some known results such as the existence of a self-modulating modulus of continuity for partial continuous operators on Baire space. To illustrate their versatility we use continuous machines to specify some algorithms that operate on objects that cannot be fully described by finite means, such as real numbers and functions. We present particularly simple algorithms for finding the multiplicative inverse of a real number and for composition of partial continuous operators on Baire space. Some of the simplicity is achieved by utilizing the fact that continuous machines are compatible with multivalued semantics.
Cite as
Michal Konečný, Florian Steinberg, and Holger Thies. Continuous and Monotone Machines. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 56:1-56:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{konecny_et_al:LIPIcs.MFCS.2020.56,
author = {Kone\v{c}n\'{y}, Michal and Steinberg, Florian and Thies, Holger},
title = {{Continuous and Monotone Machines}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {56:1--56:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.56},
URN = {urn:nbn:de:0030-drops-127230},
doi = {10.4230/LIPIcs.MFCS.2020.56},
annote = {Keywords: Computable Analysis, exact real computation, formal proofs, proof assistant, Coq}
}
Document
U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited
Abstract
Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs - a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs.
Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.
Cite as
Jan Kratochvíl, Tomáš Masařík, and Jana Novotná. U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 57:1-57:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kratochvil_et_al:LIPIcs.MFCS.2020.57,
author = {Kratochv{\'\i}l, Jan and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Novotn\'{a}, Jana},
title = {{U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {57:1--57:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.57},
URN = {urn:nbn:de:0030-drops-127247},
doi = {10.4230/LIPIcs.MFCS.2020.57},
annote = {Keywords: Interval Graphs, Mixed Unit Interval Graphs, MaxCut Problem, Clique Width, Subexponential Algorithm, Bubble Model}
}
Document
Regular Resynchronizability of Origin Transducers Is Undecidable
Abstract
We study the relation of containment up to unknown regular resynchronization between two-way non-deterministic transducers. We show that it constitutes a preorder, and that the corresponding equivalence relation is properly intermediate between origin equivalence and classical equivalence. We give a syntactical characterization for containment of two transducers up to resynchronization, and use it to show that this containment relation is undecidable already for one-way non-deterministic transducers, and for simple classes of resynchronizations. This answers the open problem stated in recent works, asking whether this relation is decidable for two-way non-deterministic transducers.
Cite as
Denis Kuperberg and Jan Martens. Regular Resynchronizability of Origin Transducers Is Undecidable. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kuperberg_et_al:LIPIcs.MFCS.2020.58,
author = {Kuperberg, Denis and Martens, Jan},
title = {{Regular Resynchronizability of Origin Transducers Is Undecidable}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {58:1--58:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.58},
URN = {urn:nbn:de:0030-drops-127255},
doi = {10.4230/LIPIcs.MFCS.2020.58},
annote = {Keywords: transducers, origin, resynchronisation, MSO, one-way, two-way, undecidability}
}
Document
On Repetition Languages
Abstract
A regular language R of finite words induces three repetition languages of infinite words: the language lim(R), which contains words with infinitely many prefixes in R, the language ∞ R, which contains words with infinitely many disjoint subwords in R, and the language R^ω, which contains infinite concatenations of words in R. Specifying behaviors, the three repetition languages provide three different ways of turning a specification of a finite behavior into an infinite one. We study the expressive power required for recognizing repetition languages, in particular whether they can always be recognized by a deterministic Büchi word automaton (DBW), the blow up in going from an automaton for R to automata for the repetition languages, and the complexity of related decision problems. For lim R and ∞ R, most of these problems have already been studied or are easy. We focus on R^ω. Its study involves some new and interesting results about additional repetition languages, in particular R^#, which contains exactly all words with unboundedly many concatenations of words in R. We show that R^ω is DBW-recognizable iff R^# is ω-regular iff R^# = R^ω, and there are languages for which these criteria do not hold. Thus, R^ω need not be DBW-recognizable. In addition, when exists, the construction of a DBW for R^ω may involve a 2^{O(n log n)} blow-up, and deciding whether R^ω is DBW-recognizable, for R given by a nondeterministic automaton, is PSPACE-complete. Finally, we lift the difference between R^# and R^ω to automata on finite words and study a variant of Büchi automata where a word is accepted if (possibly different) runs on it visit accepting states unboundedly many times.
Cite as
Orna Kupferman and Ofer Leshkowitz. On Repetition Languages. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kupferman_et_al:LIPIcs.MFCS.2020.59,
author = {Kupferman, Orna and Leshkowitz, Ofer},
title = {{On Repetition Languages}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {59:1--59:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.59},
URN = {urn:nbn:de:0030-drops-127268},
doi = {10.4230/LIPIcs.MFCS.2020.59},
annote = {Keywords: B\"{u}chi automata, Expressive power, Succinctness}
}
Document
Efficient Enumerations for Minimal Multicuts and Multiway Cuts
Abstract
Let G = (V, E) be an undirected graph and let B ⊆ V × V be a set of terminal pairs. A node/edge multicut is a subset of vertices/edges of G whose removal destroys all the paths between every terminal pair in B. The problem of computing a minimum node/edge multicut is NP-hard and extensively studied from several viewpoints. In this paper, we study the problem of enumerating all minimal node multicuts. We give an incremental polynomial delay enumeration algorithm for minimal node multicuts, which extends an enumeration algorithm due to Khachiyan et al. (Algorithmica, 2008) for minimal edge multicuts.
Important special cases of node/edge multicuts are node/edge multiway cuts, where the set of terminal pairs contains every pair of vertices in some subset T ⊆ V, that is, B = T × T. We improve the running time bound for this special case: We devise a polynomial delay and exponential space enumeration algorithm for minimal node multiway cuts and a polynomial delay and space enumeration algorithm for minimal edge multiway cuts.
Cite as
Kazuhiro Kurita and Yasuaki Kobayashi. Efficient Enumerations for Minimal Multicuts and Multiway Cuts. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kurita_et_al:LIPIcs.MFCS.2020.60,
author = {Kurita, Kazuhiro and Kobayashi, Yasuaki},
title = {{Efficient Enumerations for Minimal Multicuts and Multiway Cuts}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {60:1--60:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.60},
URN = {urn:nbn:de:0030-drops-127272},
doi = {10.4230/LIPIcs.MFCS.2020.60},
annote = {Keywords: Multicuts, Multiway cuts, Enumeration algorithms}
}
Document
Complexity of Counting First-Order Logic for the Subword Order
Abstract
This paper considers the structure consisting of the set of all words over a given alphabet together with the subword relation, regular predicates, and constants for every word. We are interested in the counting extension of first-order logic by threshold counting quantifiers. The main result shows that the two-variable fragment of this logic can be decided in two-fold exponential space provided the regular predicates are restricted to piecewise testable ones. This result improves prior insights by Karandikar and Schnoebelen by extending the logic and saving one exponent. Its proof consists of two main parts: First, we provide a quantifier elimination procedure that results in a formula with constants of bounded length (this generalizes the procedure by Karandikar and Schnoebelen for first-order logic). From this, it follows that quantification in formulas can be restricted to words of bounded length, i.e., the second part of the proof is an adaptation of the method by Ferrante and Rackoff to counting logic and deviates significantly from the path of reasoning by Karandikar and Schnoebelen.
Cite as
Dietrich Kuske and Christian Schwarz. Complexity of Counting First-Order Logic for the Subword Order. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kuske_et_al:LIPIcs.MFCS.2020.61,
author = {Kuske, Dietrich and Schwarz, Christian},
title = {{Complexity of Counting First-Order Logic for the Subword Order}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {61:1--61:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.61},
URN = {urn:nbn:de:0030-drops-127297},
doi = {10.4230/LIPIcs.MFCS.2020.61},
annote = {Keywords: Counting logic, piecewise testable languages}
}
Document
Sensitivity Lower Bounds from Linear Dependencies
Abstract
Recently, using spectral techniques, H. Huang proved that every subgraph of the hypercube of dimension n induced on more than half the vertices has maximum degree at least √n. Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work we show how to derive a proof of Huang’s result using only linear dependency and independence of vectors associated with the vertices of the hypercube. Our approach leads to several improvements of the result. In particular we prove that in any induced subgraph of H_n with more than half the number of vertices, there are two vertices, one of odd parity and the other of even parity, each with at least n vertices at distance at most 2. As an application we show that for any Boolean function f, the polynomial degree of f is bounded above by s₀(f) s₁(f), a strictly stronger statement which implies the sensitivity conjecture.
Cite as
Sophie Laplante, Reza Naserasr, and Anupa Sunny. Sensitivity Lower Bounds from Linear Dependencies. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{laplante_et_al:LIPIcs.MFCS.2020.62,
author = {Laplante, Sophie and Naserasr, Reza and Sunny, Anupa},
title = {{Sensitivity Lower Bounds from Linear Dependencies}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {62:1--62:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.62},
URN = {urn:nbn:de:0030-drops-127320},
doi = {10.4230/LIPIcs.MFCS.2020.62},
annote = {Keywords: Boolean Functions, Polynomial Degree, Sensitivity}
}
Document
Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes
Abstract
Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed p ≥ 3 both variants of the problem become NP-complete when restricted to split (S₃,…,S_p)-free graphs, where S_q denotes the q-sun graph.
Cite as
Paloma T. Lima, Erik Jan van Leeuwen, and Marieke van der Wegen. Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{lima_et_al:LIPIcs.MFCS.2020.63,
author = {Lima, Paloma T. and van Leeuwen, Erik Jan and van der Wegen, Marieke},
title = {{Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {63:1--63:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.63},
URN = {urn:nbn:de:0030-drops-127331},
doi = {10.4230/LIPIcs.MFCS.2020.63},
annote = {Keywords: rainbow vertex coloring, permutation graphs, powers of trees}
}
Document
Reducing Graph Transversals via Edge Contractions
Abstract
For a graph parameter π, the Contraction(π) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where π is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ℋ according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ℋ, which in particular imply that Contraction(π) is co-NP-hard even for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.
Cite as
Paloma T. Lima, Vinicius F. dos Santos, Ignasi Sau, and Uéverton S. Souza. Reducing Graph Transversals via Edge Contractions. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 64:1-64:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{lima_et_al:LIPIcs.MFCS.2020.64,
author = {Lima, Paloma T. and dos Santos, Vinicius F. and Sau, Ignasi and Souza, U\'{e}verton S.},
title = {{Reducing Graph Transversals via Edge Contractions}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {64:1--64:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.64},
URN = {urn:nbn:de:0030-drops-127346},
doi = {10.4230/LIPIcs.MFCS.2020.64},
annote = {Keywords: blocker problem, edge contraction, graph transversal, parameterized complexity, vertex cover, feedback vertex set, odd cycle transversal}
}
Document
Elimination Distance to Bounded Degree on Planar Graphs
Abstract
We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d and k decides in time f(k,d)⋅ n^c for a computable function f and constant c whether the elimination distance of G to the class of degree d graphs is at most k.
Cite as
Alexander Lindermayr, Sebastian Siebertz, and Alexandre Vigny. Elimination Distance to Bounded Degree on Planar Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 65:1-65:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{lindermayr_et_al:LIPIcs.MFCS.2020.65,
author = {Lindermayr, Alexander and Siebertz, Sebastian and Vigny, Alexandre},
title = {{Elimination Distance to Bounded Degree on Planar Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {65:1--65:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.65},
URN = {urn:nbn:de:0030-drops-127557},
doi = {10.4230/LIPIcs.MFCS.2020.65},
annote = {Keywords: Elimination distance, parameterized complexity, structural graph theory}
}
Document
Analysing Spatial Properties on Neighbourhood Spaces
Abstract
We present a bisimulation relation for neighbourhood spaces, a generalisation of topological spaces. We show that this notion, path preserving bisimulation, preserves formulas of the spatial logic SLCS. We then use this preservation result to show that SLCS cannot express standard topological properties such as separation and connectedness. Furthermore, we compare the bisimulation relation with standard modal bisimulation and modal bisimulation with converse on graphs and prove it coincides with the latter.
Cite as
Sven Linker, Fabio Papacchini, and Michele Sevegnani. Analysing Spatial Properties on Neighbourhood Spaces. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{linker_et_al:LIPIcs.MFCS.2020.66,
author = {Linker, Sven and Papacchini, Fabio and Sevegnani, Michele},
title = {{Analysing Spatial Properties on Neighbourhood Spaces}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {66:1--66:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.66},
URN = {urn:nbn:de:0030-drops-127352},
doi = {10.4230/LIPIcs.MFCS.2020.66},
annote = {Keywords: spatial logic, topology, bisimulation}
}
Document
Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups
Abstract
We prove that the power word problem for the solvable Baumslag-Solitar groups BS(1,q) = ⟨ a,t ∣ t a t^{-1} = a^q ⟩ can be solved in TC⁰. In the power word problem, the input consists of group elements g₁, …, g_d and binary encoded integers n₁, …, n_d and it is asked whether g₁^{n₁} ⋯ g_d^{n_d} = 1 holds. Moreover, we prove that the knapsack problem for BS(1,q) is NP-complete. In the knapsack problem, the input consists of group elements g₁, …, g_d,h and it is asked whether the equation g₁^{x₁} ⋯ g_d^{x_d} = h has a solution in ℕ^d.
Cite as
Markus Lohrey and Georg Zetzsche. Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2020.67,
author = {Lohrey, Markus and Zetzsche, Georg},
title = {{Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {67:1--67:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.67},
URN = {urn:nbn:de:0030-drops-127364},
doi = {10.4230/LIPIcs.MFCS.2020.67},
annote = {Keywords: computational group theory, matrix problems, Baumslag-Solitar groups}
}
Document
A Relaxation of the Directed Disjoint Paths Problem: A Global Congestion Metric Helps
Abstract
In the Directed Disjoint Paths problem, we are given a digraph D and a set of requests {(s₁, t₁), …, (s_k, t_k)}, and the task is to find a collection of pairwise vertex-disjoint paths {P₁, …, P_k} such that each P_i is a path from s_i to t_i in D. This problem is NP-complete for fixed k = 2 and W[1]-hard with parameter k in DAGs. A few positive results are known under restrictions on the input digraph, such as being planar or having bounded directed tree-width, or under relaxations of the problem, such as allowing for vertex congestion. Good news are scarce, however, for general digraphs. In this article we propose a novel global congestion metric for the problem: we only require the paths to be "disjoint enough", in the sense that they must behave properly not in the whole graph, but in an unspecified large part of it. Namely, in the Disjoint Enough Directed Paths problem, given an n-vertex digraph D, a set of k requests, and non-negative integers d and s, the task is to find a collection of paths connecting the requests such that at least d vertices of D occur in at most s paths of the collection. We study the parameterized complexity of this problem for a number of choices of the parameter, including the directed tree-width of D. Among other results, we show that the problem is W[1]-hard in DAGs with parameter d and, on the positive side, we give an algorithm in time 𝒪(n^{d+2} ⋅ k^{d⋅ s}) and a kernel of size d ⋅ 2^{k-s}⋅ binom(k,s) + 2k in general digraphs. This latter result has consequences for the Steiner Network problem: we show that it is FPT parameterized by the number k of terminals and d, where d = n - c and c is the size of the solution.
Cite as
Raul Lopes and Ignasi Sau. A Relaxation of the Directed Disjoint Paths Problem: A Global Congestion Metric Helps. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{lopes_et_al:LIPIcs.MFCS.2020.68,
author = {Lopes, Raul and Sau, Ignasi},
title = {{A Relaxation of the Directed Disjoint Paths Problem: A Global Congestion Metric Helps}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {68:1--68:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.68},
URN = {urn:nbn:de:0030-drops-127378},
doi = {10.4230/LIPIcs.MFCS.2020.68},
annote = {Keywords: Parameterized complexity, directed disjoint paths, congestion, dual parameterization, kernelization, directed tree-width}
}
Document
Regular Choice Functions and Uniformisations For countable Domains
Abstract
We view languages of words over a product alphabet A x B as relations between words over A and words over B. This leads to the notion of regular relations - relations given by a regular language. We ask when it is possible to find regular uniformisations of regular relations. The answer depends on the structure or shape of the underlying model: it is true e.g. for ω-words, while false for words over ℤ or for infinite trees.
In this paper we focus on countable orders. Our main result characterises, which countable linear orders D have the property that every regular relation between words over D has a regular uniformisation. As it turns out, the only obstacle for uniformisability is the one displayed in the case of ℤ - non-trivial automorphisms of the given structure. Thus, we show that either all regular relations over D have regular uniformisations, or there is a non-trivial automorphism of D and even the simple relation of choice cannot be uniformised. Moreover, this dichotomy is effective.
Cite as
Vincent Michielini and Michał Skrzypczak. Regular Choice Functions and Uniformisations For countable Domains. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 69:1-69:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{michielini_et_al:LIPIcs.MFCS.2020.69,
author = {Michielini, Vincent and Skrzypczak, Micha{\l}},
title = {{Regular Choice Functions and Uniformisations For countable Domains}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {69:1--69:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.69},
URN = {urn:nbn:de:0030-drops-127386},
doi = {10.4230/LIPIcs.MFCS.2020.69},
annote = {Keywords: Uniformisation, Monadic Second-order logic, Countable words}
}
Document
Quick Separation in Chordal and Split Graphs
Abstract
In this paper we study two classical cut problems, namely Multicut and Multiway Cut on chordal graphs and split graphs. In the Multicut problem, the input is a graph G, a collection of 𝓁 vertex pairs (s_i, t_i), i ∈ [𝓁], and a positive integer k and the goal is to decide if there exists a vertex subset S ⊆ V(G)⧵ {s_i,t_i : i ∈ [𝓁]} of size at most k such that for every vertex pair (s_i,t_i), s_i and t_i are in two different connected components of G-S. In Unrestricted Multicut, the solution S can possibly pick the vertices in the vertex pairs {(s_i,t_i): i ∈ [𝓁]}. An important special case of the Multicut problem is the Multiway Cut problem, where instead of vertex pairs, we are given a set T of terminal vertices, and the goal is to separate every pair of distinct vertices in T× T. The fixed parameter tractability (FPT) of these problems was a long-standing open problem and has been resolved fairly recently. Multicut and Multiway Cut now admit algorithms with running times 2^{{𝒪}(k³)}n^{{𝒪}(1)} and 2^k n^{{𝒪}(1)}, respectively. However, the kernelization complexity of both these problems is not fully resolved: while Multicut cannot admit a polynomial kernel under reasonable complexity assumptions, it is a well known open problem to construct a polynomial kernel for Multiway Cut. Towards designing faster FPT algorithms and polynomial kernels for the above mentioned problems, we study them on chordal and split graphs. In particular we obtain the following results.
1) Multicut on chordal graphs admits a polynomial kernel with {𝒪}(k³ 𝓁⁷) vertices. Multiway Cut on chordal graphs admits a polynomial kernel with {𝒪}(k^{13}) vertices.
2) Multicut on chordal graphs can be solved in time min {𝒪(2^{k} ⋅ (k³+𝓁) ⋅ (n+m)), 2^{𝒪(𝓁 log k)} ⋅ (n+m) + 𝓁 (n+m)}. Hence Multicut on chordal graphs parameterized by the number of terminals is in XP.
3) Multicut on split graphs can be solved in time min {𝒪(1.2738^k + kn+𝓁(n+m), 𝒪(2^{𝓁} ⋅ 𝓁 ⋅ (n+m))}. Unrestricted Multicut on split graphs can be solved in time 𝒪(4^{𝓁}⋅ 𝓁 ⋅ (n+m)).
Cite as
Pranabendu Misra, Fahad Panolan, Ashutosh Rai, Saket Saurabh, and Roohani Sharma. Quick Separation in Chordal and Split Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{misra_et_al:LIPIcs.MFCS.2020.70,
author = {Misra, Pranabendu and Panolan, Fahad and Rai, Ashutosh and Saurabh, Saket and Sharma, Roohani},
title = {{Quick Separation in Chordal and Split Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {70:1--70:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.70},
URN = {urn:nbn:de:0030-drops-127391},
doi = {10.4230/LIPIcs.MFCS.2020.70},
annote = {Keywords: chordal graphs, multicut, multiway cut, FPT, kernel}
}
Document
A Timecop’s Work Is Harder Than You Think
Abstract
We consider the (parameterized) complexity of a cop and robber game on periodic, temporal graphs and a problem on periodic sequences to which these games relate intimately. In particular, we show that it is NP-hard to decide (a) whether there is some common index at which all given periodic, binary sequences are 0, and (b) whether a single cop can catch a single robber on an edge-periodic temporal graph. We further present results for various parameterizations of both problems and show that hardness not only applies in general, but also for highly limited instances. As one main result we show that even if the graph has a size-2 vertex cover and is acyclic in each time step, the cop and robber game on periodic, temporal graphs is NP-hard and W[1]-hard when parameterized by the size of the underlying input graph.
Cite as
Nils Morawietz, Carolin Rehs, and Mathias Weller. A Timecop’s Work Is Harder Than You Think. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{morawietz_et_al:LIPIcs.MFCS.2020.71,
author = {Morawietz, Nils and Rehs, Carolin and Weller, Mathias},
title = {{A Timecop’s Work Is Harder Than You Think}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {71:1--71:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.71},
URN = {urn:nbn:de:0030-drops-127404},
doi = {10.4230/LIPIcs.MFCS.2020.71},
annote = {Keywords: edge-periodic temporal graphs, cops and robbers, tally-intersection, congruence satisfyability}
}
Document
Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests
Abstract
Equivalence testing for a polynomial family {g_m}_{m ∈ ℕ} over a field 𝔽 is the following problem: Given black-box access to an n-variate polynomial f({𝐱}), where n is the number of variables in g_m for some m ∈ ℕ, check if there exists an A ∈ GL(n,𝔽) such that f({𝐱}) = g_m(A{𝐱}). If yes, then output such an A. The complexity of equivalence testing has been studied for a number of important polynomial families, including the determinant (Det) and the family of iterated matrix multiplication polynomials. Two popular variants of the iterated matrix multiplication polynomial are: IMM_{w,d} (the (1,1) entry of the product of d many w× w symbolic matrices) and Tr-IMM_{w,d} (the trace of the product of d many w× w symbolic matrices). The families - Det, IMM and Tr-IMM - are VBP-complete under p-projections, and so, in this sense, they have the same complexity. But, do they have the same equivalence testing complexity? We show that the answer is "yes" for Det and Tr-IMM (modulo the use of randomness). The above result may appear a bit surprising as the complexity of equivalence testing for IMM and that for Det are quite different over ℚ: a randomized poly-time equivalence testing for IMM over ℚ is known [Neeraj Kayal et al., 2019], whereas [Ankit Garg et al., 2019] showed that equivalence testing for Det over ℚ is integer factoring hard (under randomized reductions and assuming GRH). To our knowledge, the complexity of equivalence testing for Tr-IMM was not known before this work. We show that, despite the syntactic similarity between IMM and Tr-IMM, equivalence testing for Tr-IMM and that for Det are randomized poly-time Turing reducible to each other over any field of characteristic zero or sufficiently large. The result is obtained by connecting the two problems via another well-studied problem in computer algebra, namely the full matrix algebra isomorphism problem (FMAI). In particular, we prove the following:
1) Testing equivalence of polynomials to Tr-IMM_{w,d}, for d ≥ 3 and w ≥ 2, is randomized polynomial-time Turing reducible to testing equivalence of polynomials to Det_w, the determinant of the w × w matrix of formal variables. (Here, d need not be a constant.)
2) FMAI is randomized polynomial-time Turing reducible to equivalence testing (in fact, to tensor isomorphism testing) for the family of matrix multiplication tensors {Tr-IMM_{w,3}}_{w ∈ ℕ}. These results, in conjunction with the randomized poly-time reduction (shown in [Ankit Garg et al., 2019]) from determinant equivalence testing to FMAI, imply that the four problems - FMAI, equivalence testing for Tr-IMM and for Det, and the 3-tensor isomorphism problem for the family of matrix multiplication tensors - are randomized poly-time equivalent under Turing reductions.
Cite as
Janaky Murthy, Vineet Nair, and Chandan Saha. Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{murthy_et_al:LIPIcs.MFCS.2020.72,
author = {Murthy, Janaky and Nair, Vineet and Saha, Chandan},
title = {{Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {72:1--72:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.72},
URN = {urn:nbn:de:0030-drops-127419},
doi = {10.4230/LIPIcs.MFCS.2020.72},
annote = {Keywords: equivalence testing, determinant, trace of the matrix product, full-matrix algebra isomorphism}
}
Document
Randomness and Effective Dimension of Continued Fractions
Abstract
Recently, Scheerer [Adrian-Maria Scheerer, 2017] and Vandehey [Vandehey, 2016] showed that normality for continued fraction expansions and base-b expansions are incomparable notions. This shows that at some level, randomness for continued fractions and binary expansion are different statistical concepts. In contrast, we show that the continued fraction expansion of a real is computably random if and only if its binary expansion is computably random.
To quantify the degree to which a continued fraction fails to be effectively random, we define the effective Hausdorff dimension of individual continued fractions, explicitly constructing continued fractions with dimension 0 and 1.
Cite as
Satyadev Nandakumar and Prateek Vishnoi. Randomness and Effective Dimension of Continued Fractions. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 73:1-73:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2020.73,
author = {Nandakumar, Satyadev and Vishnoi, Prateek},
title = {{Randomness and Effective Dimension of Continued Fractions}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {73:1--73:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.73},
URN = {urn:nbn:de:0030-drops-127424},
doi = {10.4230/LIPIcs.MFCS.2020.73},
annote = {Keywords: Continued fractions, Martin-L\"{o}f randomness, Computable randomness, effective Fractal dimension}
}
Document
Optimally Resilient Strategies in Pushdown Safety Games
Abstract
Infinite-duration games with disturbances extend the classical framework of infinite-duration games, which captures the reactive synthesis problem, with a discrete measure of resilience against non-antagonistic external influence. This concerns events where the observed system behavior differs from the intended one prescribed by the controller. For games played on finite arenas it is known that computing optimally resilient strategies only incurs a polynomial overhead over solving classical games.
This paper studies safety games with disturbances played on infinite arenas induced by pushdown systems. We show how to compute optimally resilient strategies in triply-exponential time. For the subclass of safety games played on one-counter configuration graphs, we show that determining the degree of resilience of the initial configuration is PSPACE-complete and that optimally resilient strategies can be computed in doubly-exponential time.
Cite as
Daniel Neider, Patrick Totzke, and Martin Zimmermann. Optimally Resilient Strategies in Pushdown Safety Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 74:1-74:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{neider_et_al:LIPIcs.MFCS.2020.74,
author = {Neider, Daniel and Totzke, Patrick and Zimmermann, Martin},
title = {{Optimally Resilient Strategies in Pushdown Safety Games}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {74:1--74:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.74},
URN = {urn:nbn:de:0030-drops-127432},
doi = {10.4230/LIPIcs.MFCS.2020.74},
annote = {Keywords: Controller Synthesis, Infinite Games, Resilient Strategies, Pushdown Games}
}
Document
On the Parameterized Complexity of Deletion to ℋ-Free Strong Components
Abstract
Directed Feedback Vertex Set (DFVS) is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the ℋ-SCC Deletion problem. Here, one is given a digraph D, an integer k and the objective is to decide whether there is a vertex set of size at most k whose deletion leaves a digraph where every strong component excludes graphs in the fixed finite family ℋ as (not necessarily induced) subgraphs. When ℋ comprises only the digraph with a single arc, then this problem is precisely DFVS.
Our main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if ℋ only contains rooted graphs or if ℋ contains at least one directed path. Along with generalizing the fixed-parameter tractability result for DFVS, our result also generalizes the recent results of Göke et al. [CIAC 2019] for the 1-Out-Regular Vertex Deletion and Bounded Size Strong Component Vertex Deletion problems. Moreover, we design algorithms for the two above mentioned problems, whose running times are better and match with the best bounds for DFVS, without using the heavy machinery of shadow removal as is done by Göke et al. [CIAC 2019].
Cite as
Rian Neogi, M. S. Ramanujan, Saket Saurabh, and Roohani Sharma. On the Parameterized Complexity of Deletion to ℋ-Free Strong Components. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 75:1-75:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{neogi_et_al:LIPIcs.MFCS.2020.75,
author = {Neogi, Rian and Ramanujan, M. S. and Saurabh, Saket and Sharma, Roohani},
title = {{On the Parameterized Complexity of Deletion to ℋ-Free Strong Components}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {75:1--75:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.75},
URN = {urn:nbn:de:0030-drops-127444},
doi = {10.4230/LIPIcs.MFCS.2020.75},
annote = {Keywords: Directed Cut Problems, Fixed-parameter Tractability, DFVS}
}
Document
Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions
Abstract
In this paper, we investigate the complexity of a number of computational problems defined on a synchronous boolean finite dynamical system, where update functions are chosen from a template set of exclusive-or and its negation. We first show that the reachability and path-intersection problems are solvable in logarithmic space-uniform AC¹ if the objects execute permutations, while the reachability problem is known to be in P and the path-intersection problem to be in UP in general. We also explore the case where the reachability or intersection are tested on a subset of objects, and show that this hardens complexity of the problems: both problems become NP-complete, and even Π^p₂-complete if we further require universality of the intersection. We next consider the exact cycle length problem, that is, determining whether there exists an initial configuration that yields a cycle in the configuration space having exactly a given length, and show that this problem is NP-complete. Lastly, we consider the t-predecessor and t-Garden of Eden problem, and prove that these are solvable in polynomial time even if the value of t is also given in binary as part of instance, and the two problems are in logarithmic space-uniform NC² if the value of t is given in unary as part of instance.
Cite as
Mitsunori Ogihara and Kei Uchizawa. Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 76:1-76:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ogihara_et_al:LIPIcs.MFCS.2020.76,
author = {Ogihara, Mitsunori and Uchizawa, Kei},
title = {{Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {76:1--76:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.76},
URN = {urn:nbn:de:0030-drops-127543},
doi = {10.4230/LIPIcs.MFCS.2020.76},
annote = {Keywords: Computational complexity, dynamical systems, Garden of Eden, predecessor, reachability}
}
Document
Preservation of Equations by Monoidal Monads
Abstract
If a monad T is monoidal, then operations on a set X can be lifted canonically to operations on TX. In this paper we study structural properties under which T preserves equations between those operations. It has already been shown that any monoidal monad preserves linear equations; affine monads preserve drop equations (where some variable appears only on one side, such as x⋅ y = y) and relevant monads preserve dup equations (where some variable is duplicated, such as x ⋅ x = x). We start the paper by showing a converse: if the monad at hand preserves a drop equation, then it must be affine. From this, we show that the problem whether a given (drop) equation is preserved is undecidable. A converse for relevance turns out to be more subtle: preservation of certain dup equations implies a weaker notion which we call n-relevance. Finally, we identify a subclass of equations such that their preservation is equivalent to relevance.
Cite as
Louis Parlant, Jurriaan Rot, Alexandra Silva, and Bas Westerbaan. Preservation of Equations by Monoidal Monads. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{parlant_et_al:LIPIcs.MFCS.2020.77,
author = {Parlant, Louis and Rot, Jurriaan and Silva, Alexandra and Westerbaan, Bas},
title = {{Preservation of Equations by Monoidal Monads}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {77:1--77:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.77},
URN = {urn:nbn:de:0030-drops-127460},
doi = {10.4230/LIPIcs.MFCS.2020.77},
annote = {Keywords: monoidal monads, algebraic theories, preservation of equations}
}
Document
VC Density of Set Systems Definable in Tree-Like Graphs
Abstract
We study set systems definable in graphs using variants of logic with different expressive power. Our focus is on the notion of Vapnik-Chervonenkis density: the smallest possible degree of a polynomial bounding the cardinalities of restrictions of such set systems. On one hand, we prove that if phi(x,y) is a fixed CMSO_1 formula and C is a class of graphs with uniformly bounded cliquewidth, then the set systems defined by phi in graphs from C have VC density at most |y|, which is the smallest bound that one could expect. We also show an analogous statement for the case when phi(x,y) is a CMSO_2 formula and C is a class of graphs with uniformly bounded treewidth. We complement these results by showing that if C has unbounded cliquewidth (respectively, treewidth), then, under some mild technical assumptions on C, the set systems definable by CMSO_1 (respectively, CMSO_2) formulas in graphs from C may have unbounded VC dimension, hence also unbounded VC density.
Cite as
Adam Paszke and Michał Pilipczuk. VC Density of Set Systems Definable in Tree-Like Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 78:1-78:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{paszke_et_al:LIPIcs.MFCS.2020.78,
author = {Paszke, Adam and Pilipczuk, Micha{\l}},
title = {{VC Density of Set Systems Definable in Tree-Like Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {78:1--78:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.78},
URN = {urn:nbn:de:0030-drops-127473},
doi = {10.4230/LIPIcs.MFCS.2020.78},
annote = {Keywords: treewidth, cliquewidth, definable sets, Vapnik-Chervonenkis density}
}
Document
All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words
Abstract
We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word. Given an increasing, unbounded function f: ℕ → ℝ, we construct an infinite binary word whose abelian exponents have limit superior growth rate f. As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word.
Cite as
Jarkko Peltomäki and Markus A. Whiteland. All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 79:1-79:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{peltomaki_et_al:LIPIcs.MFCS.2020.79,
author = {Peltom\"{a}ki, Jarkko and Whiteland, Markus A.},
title = {{All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {79:1--79:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.79},
URN = {urn:nbn:de:0030-drops-127481},
doi = {10.4230/LIPIcs.MFCS.2020.79},
annote = {Keywords: abelian equivalence, abelian power, abelian critical exponent}
}
Document
Ambiguity Hierarchy of Regular Infinite Tree Languages
Abstract
An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k > 0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if there is k ∈ ℕ, such that for every input it has at most k accepting computations. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations.
The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words every regular language is accepted by a deterministic automaton. Over finite trees every regular language is accepted by an unambiguous automaton. Over ω-words every regular language is accepted by an unambiguous Büchi automaton [Arnold, 1983] and by a deterministic parity automaton. Over infinite trees there are ambiguous languages [Carayol et al., 2010].
We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > 1 there are k-ambiguous languages which are not k-1 ambiguous; there are finitely (respectively countably, uncountably) ambiguous languages which are not boundedly (respectively finitely, countably) ambiguous.
Cite as
Alexander Rabinovich and Doron Tiferet. Ambiguity Hierarchy of Regular Infinite Tree Languages. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{rabinovich_et_al:LIPIcs.MFCS.2020.80,
author = {Rabinovich, Alexander and Tiferet, Doron},
title = {{Ambiguity Hierarchy of Regular Infinite Tree Languages}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {80:1--80:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.80},
URN = {urn:nbn:de:0030-drops-127495},
doi = {10.4230/LIPIcs.MFCS.2020.80},
annote = {Keywords: automata on infinite trees, ambiguous automata, monadic second-order logic}
}
Document
Palindromic k-Factorization in Pure Linear Time
Abstract
Given a string s of length n over a general alphabet and an integer k, the problem is to decide whether s is a concatenation of k nonempty palindromes. Two previously known solutions for this problem work in time O(kn) and O(nlog n) respectively. Here we settle the complexity of this problem in the word-RAM model, presenting an O(n)-time online deciding algorithm. The algorithm simultaneously finds the minimum odd number of factors and the minimum even number of factors in a factorization of a string into nonempty palindromes. We also demonstrate how to get an explicit factorization of s into k palindromes with an O(n)-time offline postprocessing.
Cite as
Mikhail Rubinchik and Arseny M. Shur. Palindromic k-Factorization in Pure Linear Time. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 81:1-81:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{rubinchik_et_al:LIPIcs.MFCS.2020.81,
author = {Rubinchik, Mikhail and Shur, Arseny M.},
title = {{Palindromic k-Factorization in Pure Linear Time}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {81:1--81:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.81},
URN = {urn:nbn:de:0030-drops-127508},
doi = {10.4230/LIPIcs.MFCS.2020.81},
annote = {Keywords: stringology, palindrome, palindromic factorization, online algorithm}
}
Document
Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs
Abstract
For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ⊆ V(G) such that G⧵ S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) ⋅ n^{𝒪(1)} assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively.
We show that f_H(t) = 2^{𝒪(t^{h-2})} for every graph H on h ≥ 3 vertices, and that f_H(t) = 2^{𝒪(t)} if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^{Θ(t^{h-2})}. We also show that f_H(t) = 2^{Ω(t^{h})} when H = K_{h,h}, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_{C₄}(t) for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h ≤ 2 the problem can be solved in polynomial time; if h ≥ 3, f_H(t) = 2^{Θ(t)} if H is a clique, and f_H(t) = 2^{Θ(t^{h-2})} otherwise.
Cite as
Ignasi Sau and Uéverton dos Santos Souza. Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 82:1-82:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{sau_et_al:LIPIcs.MFCS.2020.82,
author = {Sau, Ignasi and Souza, U\'{e}verton dos Santos},
title = {{Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {82:1--82:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.82},
URN = {urn:nbn:de:0030-drops-127511},
doi = {10.4230/LIPIcs.MFCS.2020.82},
annote = {Keywords: parameterized complexity, induced subgraphs, treewidth, hitting subgraphs, dynamic programming, lower bound, Exponential Time Hypothesis}
}
Document
Classically Simulating Quantum Circuits with Local Depolarizing Noise
Abstract
We study the effect of noise on the classical simulatability of quantum circuits defined by computationally tractable (CT) states and efficiently computable sparse (ECS) operations. Examples of such circuits, which we call CT-ECS circuits, are IQP, Clifford Magic, and conjugated Clifford circuits. This means that there exist various CT-ECS circuits such that their output probability distributions are anti-concentrated and not classically simulatable in the noise-free setting (under plausible assumptions). First, we consider a noise model where a depolarizing channel with an arbitrarily small constant rate is applied to each qubit at the end of computation. We show that, under this noise model, if an approximate value of the noise rate is known, any CT-ECS circuit with an anti-concentrated output probability distribution is classically simulatable. This indicates that the presence of small noise drastically affects the classical simulatability of CT-ECS circuits. Then, we consider an extension of the noise model where the noise rate can vary with each qubit, and provide a similar sufficient condition for classically simulating CT-ECS circuits with anti-concentrated output probability distributions.
Cite as
Yasuhiro Takahashi, Yuki Takeuchi, and Seiichiro Tani. Classically Simulating Quantum Circuits with Local Depolarizing Noise. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 83:1-83:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{takahashi_et_al:LIPIcs.MFCS.2020.83,
author = {Takahashi, Yasuhiro and Takeuchi, Yuki and Tani, Seiichiro},
title = {{Classically Simulating Quantum Circuits with Local Depolarizing Noise}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {83:1--83:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.83},
URN = {urn:nbn:de:0030-drops-127533},
doi = {10.4230/LIPIcs.MFCS.2020.83},
annote = {Keywords: Classical Simulation, Quantum Circuit, Local Depolarizing Noise}
}
Document
An Improved Approximation Algorithm for Scheduling Under Arborescence Precedence Constraints
Abstract
We consider a scheduling problem on unrelated machines with precedence constraints. There are m unrelated machines and n jobs and every job has to be processed non-preemptively in some machine. Moreover, jobs have precedence constraints; specifically, a precedence constraint j ≺ j' requires that job j' can only be started whenever job j has been completed. The objective is to minimize the total completion time.
The problem has been widely studied in more restricted machine environments such as identical or related machines. However, for unrelated machines, much less is known. In the paper, we study the problem where the precedence constraints form a forest of arborescences. We present a O((log n)² / (log log n)³)-approximation algorithm - that improves the best-known guarantee of O((log n)² / log log n) due to Kumar et al. a decade ago. The analysis relies on a dual-fitting method in analyzing the Lagrangian function of non-convex programs.
Cite as
Nguyễn Kim Thắng. An Improved Approximation Algorithm for Scheduling Under Arborescence Precedence Constraints. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 84:1-84:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{thang:LIPIcs.MFCS.2020.84,
author = {Thắng, Nguy\~{ê}n Kim},
title = {{An Improved Approximation Algorithm for Scheduling Under Arborescence Precedence Constraints}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {84:1--84:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.84},
URN = {urn:nbn:de:0030-drops-127459},
doi = {10.4230/LIPIcs.MFCS.2020.84},
annote = {Keywords: Scheduling, Precedence Constraints, Lagrangian Duality}
}
Document
The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains
Abstract
Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on finite domains).
Cite as
Caterina Viola and Stanislav Živný. The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{viola_et_al:LIPIcs.MFCS.2020.85,
author = {Viola, Caterina and \v{Z}ivn\'{y}, Stanislav},
title = {{The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {85:1--85:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Esparza, Javier and Kr\'{a}l', Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.85},
URN = {urn:nbn:de:0030-drops-127566},
doi = {10.4230/LIPIcs.MFCS.2020.85},
annote = {Keywords: promise constraint satisfaction, valued constraint satisfaction, convex relaxations, polymorphisms}
}