Volume
LIPIcs, Volume 138
44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS)
Event
MFCS 2019, August 26-30, 2019, Aachen, GermanyEditors
Publication Details
- published at: 2019-08-20
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-117-7
- DBLP: db/conf/mfcs/mfcs2019
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Document
Complete Volume
LIPIcs, Volume 138, MFCS'19, Complete Volume
Abstract
LIPIcs, Volume 138, MFCS'19, Complete Volume
Cite as
44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@Proceedings{rossmanith_et_al:LIPIcs.MFCS.2019,
title = {{LIPIcs, Volume 138, MFCS'19, Complete Volume}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019},
URN = {urn:nbn:de:0030-drops-112092},
doi = {10.4230/LIPIcs.MFCS.2019},
annote = {Keywords: Theory of computation}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{rossmanith_et_al:LIPIcs.MFCS.2019.0,
author = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.0},
URN = {urn:nbn:de:0030-drops-109444},
doi = {10.4230/LIPIcs.MFCS.2019.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Trustworthy Graph Algorithms (Invited Talk)
Abstract
The goal of the LEDA project was to build an easy-to-use and extendable library of correct and efficient data structures, graph algorithms and geometric algorithms. We report on the use of formal program verification to achieve an even higher level of trustworthiness. Specifically, we report on an ongoing and largely finished verification of the blossom-shrinking algorithm for maximum cardinality matching.
Cite as
Mohammad Abdulaziz, Kurt Mehlhorn, and Tobias Nipkow. Trustworthy Graph Algorithms (Invited Talk). In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{abdulaziz_et_al:LIPIcs.MFCS.2019.1,
author = {Abdulaziz, Mohammad and Mehlhorn, Kurt and Nipkow, Tobias},
title = {{Trustworthy Graph Algorithms}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {1:1--1:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.1},
URN = {urn:nbn:de:0030-drops-109456},
doi = {10.4230/LIPIcs.MFCS.2019.1},
annote = {Keywords: graph algorithms, formal correct proofs, Isabelle, LEDA, certifying algorithms}
}
Document
Invited Talk
Guarded Kleene Algebra with Tests: Verification of Uninterpreted Programs in Nearly Linear Time (Invited Talk)
Abstract
Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union (+) and iteration (*) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa’s axiomatization of Kleene Algebra. We will also discuss how this result has practical implications in the verification of programs, with examples from network and probabilistic programming. This is joint work with Nate Foster, Justin Hsu, Tobias Kappe, Dexter Kozen, and Steffen Smolka.
Cite as
Alexandra Silva. Guarded Kleene Algebra with Tests: Verification of Uninterpreted Programs in Nearly Linear Time (Invited Talk). In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{silva:LIPIcs.MFCS.2019.2,
author = {Silva, Alexandra},
title = {{Guarded Kleene Algebra with Tests: Verification of Uninterpreted Programs in Nearly Linear Time}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {2:1--2:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.2},
URN = {urn:nbn:de:0030-drops-109462},
doi = {10.4230/LIPIcs.MFCS.2019.2},
annote = {Keywords: Kleene algebra, verification, decision procedures}
}
Document
Invited Talk
Picking Random Vertices (Invited Talk)
Abstract
We survey some recent graph algorithms that are based on picking a vertex at random and declaring it to be a part of the solution. This simple idea has been deployed to obtain state-of-the-art parameterized, exact exponential time, and approximation algorithms for a number of problems, such as Feedback Vertex Set and 3-Hitting Set. We will also discuss a recent 2-approximation algorithm for Feedback Vertex Set in Tournaments that is based on picking a vertex at random and declaring it to not be part of the solution.
Cite as
Daniel Lokshtanov. Picking Random Vertices (Invited Talk). In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{lokshtanov:LIPIcs.MFCS.2019.3,
author = {Lokshtanov, Daniel},
title = {{Picking Random Vertices}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {3:1--3:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.3},
URN = {urn:nbn:de:0030-drops-109478},
doi = {10.4230/LIPIcs.MFCS.2019.3},
annote = {Keywords: Graph Algorithm}
}
Document
Invited Talk
Popular Matchings: Good, Bad, and Mixed (Invited Talk)
Abstract
We consider the landscape of popular matchings in a bipartite graph G where every vertex has strict preferences over its neighbors. This is a very well-studied model in two-sided matching markets. A matching M is popular if it does not lose a head-to-head election against any matching, where each vertex casts a vote for the matching where it gets a better assignment. Roughly speaking, a popular matching is one such that there is no matching where more vertices are happier. The notion of popularity is more relaxed than stability: a classical notion studied for the last several decades. Popular matchings always exist in G since stable matchings always exist in a bipartite graph and every stable matching is popular.
Algorithmically speaking, the landscape of popular matching seems to have only a few bright spots. Every stable matching is a min-size popular matching and there are also simple linear time algorithms for computing a max-size popular matching and for the popular edge problem. All these algorithms reduce the popular matching problem to an appropriate question in stable matchings and solve the corresponding stable matching problem.
We now know NP-hardness results for many popular matching problems. These include the min-cost/max-weight popular matching problem and the problem of deciding if G admits a popular matching that is neither a min-size nor a max-size popular matching. For non-bipartite graphs, it is NP-hard to even decide if a popular matching exists or not.
A mixed matching is a probability distribution or a lottery over matchings. A popular mixed matching is one that never loses a head-to-head election against any mixed matching. As an allocation mechanism, a popular mixed matching has several nice properties. Moreover, finding a max-weight or min-cost popular mixed matching in G is easy (by solving a linear program). Interestingly, there is always an optimal popular mixed matching Pi with a simple structure: Pi = {(M_0,1/2),(M_1,1/2)} where M_0 and M_1 are matchings in G. Popular mixed matchings always exist in non-bipartite graphs as well and can be computed in polynomial time.
Cite as
Telikepalli Kavitha. Popular Matchings: Good, Bad, and Mixed (Invited Talk). In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, p. 4:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{kavitha:LIPIcs.MFCS.2019.4,
author = {Kavitha, Telikepalli},
title = {{Popular Matchings: Good, Bad, and Mixed}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {4:1--4:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.4},
URN = {urn:nbn:de:0030-drops-109489},
doi = {10.4230/LIPIcs.MFCS.2019.4},
annote = {Keywords: Matchings under preferences, Algorithms, NP-Hardness}
}
Document
Invited Talk
Petri Net Reachability Problem (Invited Talk)
Abstract
Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. In this presentation, we overview decidability and complexity results over the last fifty years about the Petri net reachability problem.
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Jérôme Leroux. Petri Net Reachability Problem (Invited Talk). In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 5:1-5:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{leroux:LIPIcs.MFCS.2019.5,
author = {Leroux, J\'{e}r\^{o}me},
title = {{Petri Net Reachability Problem}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {5:1--5:3},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.5},
URN = {urn:nbn:de:0030-drops-109493},
doi = {10.4230/LIPIcs.MFCS.2019.5},
annote = {Keywords: Petri net, Reachability problem, Formal verification, Concurrency}
}
Document
An Improved Online Algorithm for the Traveling Repairperson Problem on a Line
Abstract
In the online variant of the traveling repairperson problem (TRP), requests arrive in time at points of a metric space X and must be eventually visited by a server. The server starts at a designated point of X and travels at most at unit speed. Each request has a given weight and once the server visits its position, the request is considered serviced; we call such time completion time of the request. The goal is to minimize the weighted sum of completion times of all requests.
In this paper, we give a 5.429-competitive deterministic algorithm for line metrics improving over 5.829-competitive solution by Krumke et al. (TCS 2003). Our result is obtained by modifying the schedule by serving requests that are close to the origin first. To compute the competitive ratio of our approach, we use a charging scheme, and later evaluate its properties using a factor-revealing linear program which upper-bounds the competitive ratio.
Cite as
Marcin Bienkowski and Hsiang-Hsuan Liu. An Improved Online Algorithm for the Traveling Repairperson Problem on a Line. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bienkowski_et_al:LIPIcs.MFCS.2019.6,
author = {Bienkowski, Marcin and Liu, Hsiang-Hsuan},
title = {{An Improved Online Algorithm for the Traveling Repairperson Problem on a Line}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {6:1--6:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.6},
URN = {urn:nbn:de:0030-drops-109503},
doi = {10.4230/LIPIcs.MFCS.2019.6},
annote = {Keywords: traveling repairperson problem, competitive analysis, minimizing completion time, factor-revealing LP}
}
Document
Query-Competitive Sorting with Uncertainty
Abstract
We study the problem of sorting under incomplete information, when queries are used to resolve uncertainties. Each of n data items has an unknown value, which is known to lie in a given interval. We can pay a query cost to learn the actual value, and we may allow an error threshold in the sorting. The goal is to find a nearly-sorted permutation by performing a minimum-cost set of queries.
We show that an offline optimum query set can be found in polynomial time, and that both oblivious and adaptive problems have simple query-competitive algorithms. The query-competitiveness for the oblivious problem is n for uniform query costs, and unbounded for arbitrary costs; for the adaptive problem, the ratio is 2.
We then present a unified adaptive strategy for uniform query costs that yields: (i) a 3/2-query-competitive randomized algorithm; (ii) a 5/3-query-competitive deterministic algorithm if the dependency graph has no 2-components after some preprocessing, which has query-competitive ratio 3/2 + O(1/k) if the components obtained have size at least k; (iii) an exact algorithm if the intervals constitute a laminar family. The first two results have matching lower bounds, and we have a lower bound of 7/5 for large components.
We also show that the advice complexity of the adaptive problem is floor[n/2] if no error threshold is allowed, and ceil[n/3 * lg 3] for the general case.
Cite as
Magnús M. Halldórsson and Murilo Santos de Lima. Query-Competitive Sorting with Uncertainty. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{halldorsson_et_al:LIPIcs.MFCS.2019.7,
author = {Halld\'{o}rsson, Magn\'{u}s M. and de Lima, Murilo Santos},
title = {{Query-Competitive Sorting with Uncertainty}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {7:1--7:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.7},
URN = {urn:nbn:de:0030-drops-109519},
doi = {10.4230/LIPIcs.MFCS.2019.7},
annote = {Keywords: online algorithms, sorting, randomized algorithms, advice complexity, threshold tolerance graphs}
}
Document
Better Bounds for Online Line Chasing
Abstract
We study online competitive algorithms for the line chasing problem in Euclidean spaces R^d, where the input consists of an initial point P_0 and a sequence of lines X_1, X_2, ..., X_m, revealed one at a time. At each step t, when the line X_t is revealed, the algorithm must determine a point P_t in X_t. An online algorithm is called c-competitive if for any input sequence the path P_0, P_1 , ..., P_m it computes has length at most c times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets X_t are arbitrary convex sets.
To date, the best competitive ratio for the line chasing problem was 28.1, even in the plane. We improve this bound by providing a simple 3-competitive algorithm for any dimension d. We complement this bound by a matching lower bound for algorithms that are memoryless in the sense of our algorithm, and a lower bound of 1.5358 for arbitrary algorithms. The latter bound also improves upon the previous lower bound of sqrt{2}~=1.412 for convex body chasing in 2 dimensions.
Cite as
Marcin Bienkowski, Jarosław Byrka, Marek Chrobak, Christian Coester, Łukasz Jeż, and Elias Koutsoupias. Better Bounds for Online Line Chasing. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bienkowski_et_al:LIPIcs.MFCS.2019.8,
author = {Bienkowski, Marcin and Byrka, Jaros{\l}aw and Chrobak, Marek and Coester, Christian and Je\.{z}, {\L}ukasz and Koutsoupias, Elias},
title = {{Better Bounds for Online Line Chasing}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {8:1--8:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.8},
URN = {urn:nbn:de:0030-drops-109521},
doi = {10.4230/LIPIcs.MFCS.2019.8},
annote = {Keywords: convex body chasing, line chasing, competitive analysis}
}
Document
Nash Equilibria in Games over Graphs Equipped with a Communication Mechanism
Abstract
We study pure Nash equilibria in infinite-duration games on graphs, with partial visibility of actions but communication (based on a graph) among the players. We show that a simple communication mechanism consisting in reporting the deviator when seeing it and propagating this information is sufficient for characterizing Nash equilibria. We propose an epistemic game construction, which conveniently records important information about the knowledge of the players. With this abstraction, we are able to characterize Nash equilibria which follow the simple communication pattern via winning strategies. We finally discuss the size of the construction, which would allow efficient algorithmic solutions to compute Nash equilibria in the original game.
Cite as
Patricia Bouyer and Nathan Thomasset. Nash Equilibria in Games over Graphs Equipped with a Communication Mechanism. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bouyer_et_al:LIPIcs.MFCS.2019.9,
author = {Bouyer, Patricia and Thomasset, Nathan},
title = {{Nash Equilibria in Games over Graphs Equipped with a Communication Mechanism}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {9:1--9:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.9},
URN = {urn:nbn:de:0030-drops-109532},
doi = {10.4230/LIPIcs.MFCS.2019.9},
annote = {Keywords: Multiplayer games, Nash equilibria, partial information}
}
Document
Parity Games: Zielonka’s Algorithm in Quasi-Polynomial Time
Abstract
Calude, Jain, Khoussainov, Li, and Stephan (2017) proposed a quasi-polynomial-time algorithm solving parity games. After this breakthrough result, a few other quasi-polynomial-time algorithms were introduced; none of them is easy to understand. Moreover, it turns out that in practice they operate very slowly. On the other side there is Zielonka’s recursive algorithm, which is very simple, exponential in the worst case, and the fastest in practice. We combine these two approaches: we propose a small modification of Zielonka’s algorithm, which ensures that the running time is at most quasi-polynomial. In effect, we obtain a simple algorithm that solves parity games in quasi-polynomial time. We also hope that our algorithm, after further optimizations, can lead to an algorithm that shares the good performance of Zielonka’s algorithm on typical inputs, while reducing the worst-case complexity on difficult inputs.
Cite as
Paweł Parys. Parity Games: Zielonka’s Algorithm in Quasi-Polynomial Time. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{parys:LIPIcs.MFCS.2019.10,
author = {Parys, Pawe{\l}},
title = {{Parity Games: Zielonka’s Algorithm in Quasi-Polynomial Time}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {10:1--10:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.10},
URN = {urn:nbn:de:0030-drops-109543},
doi = {10.4230/LIPIcs.MFCS.2019.10},
annote = {Keywords: Parity games, Zielonka’s algorithm, quasi-polynomial time}
}
Document
Bidding Mechanisms in Graph Games
Abstract
In two-player games on graphs, the players move a token through a graph to produce a finite or infinite path, which determines the qualitative winner or quantitative payoff of the game. We study bidding games in which the players bid for the right to move the token. Several bidding rules were studied previously. In Richman bidding, in each round, the players simultaneously submit bids, and the higher bidder moves the token and pays the other player. Poorman bidding is similar except that the winner of the bidding pays the "bank" rather than the other player. Taxman bidding spans the spectrum between Richman and poorman bidding. They are parameterized by a constant tau in [0,1]: portion tau of the winning bid is paid to the other player, and portion 1-tau to the bank. While finite-duration (reachability) taxman games have been studied before, we present, for the first time, results on infinite-duration taxman games. It was previously shown that both Richman and poorman infinite-duration games with qualitative objectives reduce to reachability games, and we show a similar result here. Our most interesting results concern quantitative taxman games, namely mean-payoff games, where poorman and Richman bidding differ significantly. A central quantity in these games is the ratio between the two players' initial budgets. While in poorman mean-payoff games, the optimal payoff of a player depends on the initial ratio, in Richman bidding, the payoff depends only on the structure of the game. In both games the optimal payoffs can be found using (different) probabilistic connections with random-turn games in which in each turn, instead of bidding, a coin is tossed to determine which player moves. While the value with Richman bidding equals the value of a random-turn game with an un-biased coin, with poorman bidding, the bias in the coin is the initial ratio of the budgets. We give a complete classification of mean-payoff taxman games that is based on a probabilistic connection: the value of a taxman bidding game with parameter tau and initial ratio r, equals the value of a random-turn game that uses a coin with bias F(tau, r) = (r+tau * (1-r))/(1+tau). Thus, we show that Richman bidding is the exception; namely, for every tau <1, the value of the game depends on the initial ratio. Our proof technique simplifies and unifies the previous proof techniques for both Richman and poorman bidding.
Cite as
Guy Avni, Thomas A. Henzinger, and Đorđe Žikelić. Bidding Mechanisms in Graph Games. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{avni_et_al:LIPIcs.MFCS.2019.11,
author = {Avni, Guy and Henzinger, Thomas A. and \v{Z}ikeli\'{c}, {\D}or{\d}e},
title = {{Bidding Mechanisms in Graph Games}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {11:1--11:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.11},
URN = {urn:nbn:de:0030-drops-109553},
doi = {10.4230/LIPIcs.MFCS.2019.11},
annote = {Keywords: Bidding games, Richman bidding, poorman bidding, taxman bidding, mean-payoff games, random-turn games}
}
Document
Cluster Deletion on Interval Graphs and Split Related Graphs
Abstract
In the Cluster Deletion problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of Cluster Deletion is NP-complete on (P_5-free) chordal graphs, whereas Cluster Deletion is solved in polynomial time on split graphs. However, the existence of a polynomial-time algorithm of Cluster Deletion on interval graphs, a proper subclass of chordal graphs, remained a well-known open problem. Our main contribution is that we settle this problem in the affirmative, by providing a polynomial-time algorithm for Cluster Deletion on interval graphs. Moreover, despite the simple formulation of the algorithm on split graphs, we show that Cluster Deletion remains NP-complete on a natural and slight generalization of split graphs that constitutes a proper subclass of P_5-free chordal graphs. Although the later result arises from the already-known reduction for P_5-free chordal graphs, we give an alternative proof showing an interesting connection between edge-weighted and vertex-weighted variations of the problem. To complement our results, we provide faster and simpler polynomial-time algorithms for Cluster Deletion on subclasses of such a generalization of split graphs.
Cite as
Athanasios L. Konstantinidis and Charis Papadopoulos. Cluster Deletion on Interval Graphs and Split Related Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{konstantinidis_et_al:LIPIcs.MFCS.2019.12,
author = {Konstantinidis, Athanasios L. and Papadopoulos, Charis},
title = {{Cluster Deletion on Interval Graphs and Split Related Graphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {12:1--12:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.12},
URN = {urn:nbn:de:0030-drops-109568},
doi = {10.4230/LIPIcs.MFCS.2019.12},
annote = {Keywords: Cluster deletion, interval graphs, split graphs}
}
Document
Constrained Representations of Map Graphs and Half-Squares
Abstract
The square of a graph H, denoted H^2, is obtained from H by adding new edges between two distinct vertices whenever their distance in H is two. The half-squares of a bipartite graph B=(X,Y,E_B) are the subgraphs of B^2 induced by the color classes X and Y, B^2[X] and B^2[Y]. For a given graph G=(V,E_G), if G=B^2[V] for some bipartite graph B=(V,W,E_B), then B is a representation of G and W is the set of points in B. If in addition B is planar, then G is also called a map graph and B is a witness of G [Chen, Grigni, Papadimitriou. Map graphs. J. ACM , 49 (2) (2002) 127-138].
While Chen, Grigni, Papadimitriou proved that any map graph G=(V,E_G) has a witness with at most 3|V|-6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is NP-complete. As a by-product, we obtain NP-completeness of edge clique partition on planar graphs; until this present paper, the complexity status of edge clique partition for planar graphs was previously unknown.
We also consider half-squares of tree-convex bipartite graphs and prove the following complexity dichotomy: Given a graph G=(V,E_G) and an integer k, deciding if G=B^2[V] for some tree-convex bipartite graph B=(V,W,E_B) with |W|<=k points is NP-complete if G is non-chordal dually chordal and solvable in linear time otherwise. Our proof relies on a characterization of half-squares of tree-convex bipartite graphs, saying that these are precisely the chordal and dually chordal graphs.
Cite as
Hoang-Oanh Le and Van Bang Le. Constrained Representations of Map Graphs and Half-Squares. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{le_et_al:LIPIcs.MFCS.2019.13,
author = {Le, Hoang-Oanh and Le, Van Bang},
title = {{Constrained Representations of Map Graphs and Half-Squares}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.13},
URN = {urn:nbn:de:0030-drops-109574},
doi = {10.4230/LIPIcs.MFCS.2019.13},
annote = {Keywords: map graph, half-square, edge clique cover, edge clique partition, graph classes}
}
Document
Colouring H-Free Graphs of Bounded Diameter
Abstract
The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that Colouring is NP-complete for H-free graphs if H contains a cycle or claw, even for fixed k >= 3. We examine to what extent the situation may change if in addition the input graph has bounded diameter.
Cite as
Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring H-Free Graphs of Bounded Diameter. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{martin_et_al:LIPIcs.MFCS.2019.14,
author = {Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani},
title = {{Colouring H-Free Graphs of Bounded Diameter}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {14:1--14:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.14},
URN = {urn:nbn:de:0030-drops-109584},
doi = {10.4230/LIPIcs.MFCS.2019.14},
annote = {Keywords: vertex colouring, H-free graph, diameter}
}
Document
Distance Labeling Schemes for Cube-Free Median Graphs
Abstract
Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. One of the important problems is finding natural classes of graphs admitting distance labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on n nodes enjoys distance labeling scheme with labels of O(log^3 n) bits.
Cite as
Victor Chepoi, Arnaud Labourel, and Sébastien Ratel. Distance Labeling Schemes for Cube-Free Median Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{chepoi_et_al:LIPIcs.MFCS.2019.15,
author = {Chepoi, Victor and Labourel, Arnaud and Ratel, S\'{e}bastien},
title = {{Distance Labeling Schemes for Cube-Free Median Graphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {15:1--15:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.15},
URN = {urn:nbn:de:0030-drops-109598},
doi = {10.4230/LIPIcs.MFCS.2019.15},
annote = {Keywords: median graphs, labeling schemes, distributed distance computation}
}
Document
One-Dimensional Guarded Fragments
Abstract
We call a first-order formula one-dimensional if every maximal block of existential (or universal) quantifiers in it leaves at most one variable free. We consider the one-dimensional restrictions of the guarded fragment, GF, and the tri-guarded fragment, TGF, the latter being a recent extension of GF in which quantification for subformulas with at most two free variables need not be guarded, and which thus may be seen as a unification of GF and the two-variable fragment, FO^2. We denote the resulting formalisms, resp., GF_1, and TGF_1. We show that GF_1 has an exponential model property and NExpTime-complete satisfiability problem (that is, it is easier than full GF). For TGF_1 we show that it is decidable, has the finite model property, and its satisfiability problem is 2-ExpTime-complete (NExpTime-complete in the absence of equality). All the above-mentioned results are obtained for signatures with no constants. We finally discuss the impact of their addition, observing that constants do not spoil the decidability but increase the complexity of the satisfiability problem.
Cite as
Emanuel Kieroński. One-Dimensional Guarded Fragments. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{kieronski:LIPIcs.MFCS.2019.16,
author = {Kiero\'{n}ski, Emanuel},
title = {{One-Dimensional Guarded Fragments}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {16:1--16:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.16},
URN = {urn:nbn:de:0030-drops-109608},
doi = {10.4230/LIPIcs.MFCS.2019.16},
annote = {Keywords: guarded fragment, two-variable logic, satisfiability, finite model property}
}
Document
Finite Satisfiability of Unary Negation Fragment with Transitivity
Abstract
We show that the finite satisfiability problem for the unary negation fragment with an arbitrary number of transitive relations is decidable and 2-ExpTime-complete. Our result actually holds for a more general setting in which one can require that some binary symbols are interpreted as arbitrary transitive relations, some as partial orders and some as equivalences. We also consider finite satisfiability of various extensions of our primary logic, in particular capturing the concepts of nominals and role hierarchies known from description logic. As the unary negation fragment can express unions of conjunctive queries, our results have interesting implications for the problem of finite query answering, both in the classical scenario and in the description logics setting.
Cite as
Daniel Danielski and Emanuel Kieroński. Finite Satisfiability of Unary Negation Fragment with Transitivity. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{danielski_et_al:LIPIcs.MFCS.2019.17,
author = {Danielski, Daniel and Kiero\'{n}ski, Emanuel},
title = {{Finite Satisfiability of Unary Negation Fragment with Transitivity}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {17:1--17:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.17},
URN = {urn:nbn:de:0030-drops-109612},
doi = {10.4230/LIPIcs.MFCS.2019.17},
annote = {Keywords: unary negation fragment, transitivity, finite satisfiability, finite open-world query answering, description logics}
}
Document
The Fluted Fragment with Transitivity
Abstract
We study the satisfiability problem for the fluted fragment extended with transitive relations. We show that the logic enjoys the finite model property when only one transitive relation is available. On the other hand we show that the satisfiability problem is undecidable already for the two-variable fragment of the logic in the presence of three transitive relations.
Cite as
Ian Pratt-Hartmann and Lidia Tendera. The Fluted Fragment with Transitivity. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{pratthartmann_et_al:LIPIcs.MFCS.2019.18,
author = {Pratt-Hartmann, Ian and Tendera, Lidia},
title = {{The Fluted Fragment with Transitivity}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {18:1--18:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.18},
URN = {urn:nbn:de:0030-drops-109626},
doi = {10.4230/LIPIcs.MFCS.2019.18},
annote = {Keywords: First-Order logic, Decidability, Satisfiability, Transitivity, Complexity}
}
Document
Counting of Teams in First-Order Team Logics
Abstract
We study descriptive complexity of counting complexity classes in the range from #P to #*NP. A corollary of Fagin’s characterization of NP by existential second-order logic is that #P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. In this paper we extend this study to classes beyond #P and extensions of first-order logic with team semantics. These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of first-order logic in Tarski’s semantics. Our results show that the class #*NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of #*NP and #P, respectively. We also study the function class generated by inclusion logic and relate it to the complexity class TotP, which is a subclass of #P. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean Sigma_1-formulae is #*NP-complete with respect to Turing reductions as well as complete for the function class generated by dependence logic with respect to first-order reductions.
Cite as
Anselm Haak, Juha Kontinen, Fabian Müller, Heribert Vollmer, and Fan Yang. Counting of Teams in First-Order Team Logics. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{haak_et_al:LIPIcs.MFCS.2019.19,
author = {Haak, Anselm and Kontinen, Juha and M\"{u}ller, Fabian and Vollmer, Heribert and Yang, Fan},
title = {{Counting of Teams in First-Order Team Logics}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {19:1--19:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.19},
URN = {urn:nbn:de:0030-drops-109634},
doi = {10.4230/LIPIcs.MFCS.2019.19},
annote = {Keywords: team-based logics, counting classes, finite model theory, descriptive complexity}
}
Document
Approximating Activation Edge-Cover and Facility Location Problems
Abstract
What approximation ratio can we achieve for the Facility Location problem if whenever a client u connects to a facility v, the opening cost of v is at most theta times the service cost of u? We show that this and many other problems are a particular case of the Activation Edge-Cover problem. Here we are given a multigraph G=(V,E), a set R subseteq V of terminals, and thresholds {t^e_u,t^e_v} for each uv-edge e in E. The goal is to find an assignment a={a_v:v in V} to the nodes minimizing sum_{v in V} a_v, such that the edge set E_a={e=uv: a_u >= t^e_u, a_v >= t^e_v} activated by a covers R. We obtain ratio 1+max_{x>=1}(ln x)/(1+x/theta)~= ln theta - ln ln theta for the problem, where theta is a problem parameter. This result is based on a simple generic algorithm for the problem of minimizing a sum of a decreasing and a sub-additive set functions, which is of independent interest. As an application, we get the same ratio for the above variant of {Facility Location}. If for each facility all service costs are identical then we show a better ratio 1+max_{k in N}(H_k-1)/(1+k/theta), where H_k=sum_{i=1}^k 1/i. For the Min-Power Edge-Cover problem we improve the ratio 1.406 of [Calinescu et al, 2019] (achieved by iterative randomized rounding) to 1.2785. For unit thresholds we improve the ratio 73/60~=1.217 of [Calinescu et al, 2019] to 1555/1347~=1.155.
Cite as
Zeev Nutov, Guy Kortsarz, and Eli Shalom. Approximating Activation Edge-Cover and Facility Location Problems. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{nutov_et_al:LIPIcs.MFCS.2019.20,
author = {Nutov, Zeev and Kortsarz, Guy and Shalom, Eli},
title = {{Approximating Activation Edge-Cover and Facility Location Problems}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {20:1--20:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.20},
URN = {urn:nbn:de:0030-drops-109642},
doi = {10.4230/LIPIcs.MFCS.2019.20},
annote = {Keywords: generalized min-covering problem, activation edge-cover, facility location, minimum power, approximation algorithm}
}
Document
Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs
Abstract
We give deterministic distributed (1+epsilon)-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in O( (1 / epsilon) log n) rounds, and our independent set algorithm has a runtime of O( (1/epsilon) log(1/epsilon)log^* n) rounds. For coloring, existing lower bounds imply that the dependencies on 1/epsilon and log n are best possible. For independent set, we prove that Omega(1/epsilon) rounds are necessary.
Both our algorithms make use of the tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into O(log n) layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first O(log (1/epsilon)) layers are required as they already contain a large enough independent set. We develop a (1+epsilon)-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers.
This work raises the question as to how useful tree decompositions are for distributed computing.
Cite as
Christian Konrad and Viktor Zamaraev. Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{konrad_et_al:LIPIcs.MFCS.2019.21,
author = {Konrad, Christian and Zamaraev, Viktor},
title = {{Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {21:1--21:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.21},
URN = {urn:nbn:de:0030-drops-109651},
doi = {10.4230/LIPIcs.MFCS.2019.21},
annote = {Keywords: local model, approximation algorithms, minimum vertex coloring, maximum independent set, chordal graphs}
}
Document
Multistage Knapsack
Abstract
Many systems have to be maintained while the underlying constraints, costs and/or profits change over time. Although the state of a system may evolve during time, a non-negligible transition cost is incured for transitioning from one state to another. In order to model such situations, Gupta et al. (ICALP 2014) and Eisenstat et al. (ICALP 2014) introduced a multistage model where the input is a sequence of instances (one for each time step), and the goal is to find a sequence of solutions (one for each time step) that simultaneously (i) have good quality on the time steps and (ii) as stable as possible. We focus on the multistage version of the Knapsack problem where we are given a time horizon t=1,2,...,T, and a sequence of knapsack instances I_1,I_2,...,I_T, one for each time step, defined on a set of n objects. In every time step t we have to choose a feasible knapsack S_t of I_t, which gives a knapsack profit. To measure the stability/similarity of two consecutive solutions S_t and S_{t+1}, we identify the objects for which the decision, to be picked or not, remains the same in S_t and S_{t+1}, giving a transition profit. We are asked to produce a sequence of solutions S_1,S_2,...,S_T so that the total knapsack profit plus the overall transition profit is maximized.
We propose a PTAS for the Multistage Knapsack problem. This is the first approximation scheme for a combinatorial optimization problem in the considered multistage setting, and its existence contrasts with the inapproximability results for other combinatorial optimization problems that are even polynomial-time solvable in the static case (e.g.multistage Spanning Tree, or multistage Bipartite Perfect Matching). Then, we prove that there is no FPTAS for the problem even in the case where T=2, unless P=NP. Furthermore, we give a pseudopolynomial time algorithm for the case where the number of steps is bounded by a fixed constant and we show that otherwise the problem remains NP-hard even in the case where all the weights, profits and capacities are 0 or 1.
Cite as
Evripidis Bampis, Bruno Escoffier, and Alexandre Teiller. Multistage Knapsack. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bampis_et_al:LIPIcs.MFCS.2019.22,
author = {Bampis, Evripidis and Escoffier, Bruno and Teiller, Alexandre},
title = {{Multistage Knapsack}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {22:1--22:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.22},
URN = {urn:nbn:de:0030-drops-109664},
doi = {10.4230/LIPIcs.MFCS.2019.22},
annote = {Keywords: Knapsack, Approximation Algorithms, Multistage Optimization}
}
Document
Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations
Abstract
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several implicit characterizations of complexity and computability classes.
The proposed framework presents an original point of view on complexity and computation classes. It unifies several constructions that have been proposed for characterizing these classes including classical approaches in implicit complexity using restricted recursion schemes, as well as recent characterizations of computability and complexity by classes of continuous ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory.
At a more technical point of view, this paper points out the fundamental role of linear (discrete) ordinary differential equations and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms.
Cite as
Olivier Bournez and Arnaud Durand. Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bournez_et_al:LIPIcs.MFCS.2019.23,
author = {Bournez, Olivier and Durand, Arnaud},
title = {{Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {23:1--23:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.23},
URN = {urn:nbn:de:0030-drops-109670},
doi = {10.4230/LIPIcs.MFCS.2019.23},
annote = {Keywords: Implicit complexity, discrete ordinary differential equations, recursion scheme}
}
Document
On the Coalgebra of Partial Differential Equations
Abstract
We note that the coalgebra of formal power series in commutative variables is final in a certain subclass of coalgebras. Moreover, a system Sigma of polynomial PDEs, under a coherence condition, naturally induces such a coalgebra over differential polynomial expressions. As a result, we obtain a clean coinductive proof of existence and uniqueness of solutions of initial value problems for PDEs. Based on this characterization, we give complete algorithms for checking equivalence of differential polynomial expressions, given Sigma.
Cite as
Michele Boreale. On the Coalgebra of Partial Differential Equations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{boreale:LIPIcs.MFCS.2019.24,
author = {Boreale, Michele},
title = {{On the Coalgebra of Partial Differential Equations}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {24:1--24:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.24},
URN = {urn:nbn:de:0030-drops-109689},
doi = {10.4230/LIPIcs.MFCS.2019.24},
annote = {Keywords: coalgebra, partial differential equations, polynomials}
}
Document
Random Subgroups of Rationals
Abstract
This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup (G,+) of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of (G,+); second, what learnability properties can one extract from G and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of (G,+) coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for G with respect to any generating sequence for G is not even semi-decidable, one can build a generating sequence beta such that the word problem for G with respect to beta is co-recursively enumerable (assuming that the set of generators of G is limit-recursive). In regard to the second question, it is proven that there is a generating sequence beta for G such that every non-trivial finitely generated subgroup of G is recursively enumerable and the class of all such subgroups of G is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of G is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of G cannot be syntactically identified in the limit with respect to any generating sequence for G. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.
Cite as
Ziyuan Gao, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, Alexander Melnikov, Karen Seidel, and Frank Stephan. Random Subgroups of Rationals. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{gao_et_al:LIPIcs.MFCS.2019.25,
author = {Gao, Ziyuan and Jain, Sanjay and Khoussainov, Bakhadyr and Li, Wei and Melnikov, Alexander and Seidel, Karen and Stephan, Frank},
title = {{Random Subgroups of Rationals}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {25:1--25:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.25},
URN = {urn:nbn:de:0030-drops-109693},
doi = {10.4230/LIPIcs.MFCS.2019.25},
annote = {Keywords: Martin-L\"{o}f randomness, subgroups of rationals, finitely generated subgroups of rationals, learning in the limit, behaviourally correct learning}
}
Document
Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness
Abstract
We study the problem #IndSub(Phi) of counting all induced subgraphs of size k in a graph G that satisfy the property Phi. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties Phi including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(Phi), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], allowing them to prove hardness of a wide range of properties Phi. In this work, we refine this technique for graph properties that are non-trivial on edge-transitive graphs with a prime power number of edges. In particular, we fully classify the case of monotone bipartite graph properties: It is shown that, given any graph property Phi that is closed under the removal of vertices and edges, and that is non-trivial for bipartite graphs, the problem #IndSub(Phi) is #W[1]-hard and cannot be solved in time f(k)* n^{o(k)} for any computable function f, unless the Exponential Time Hypothesis fails. This holds true even if the input graph is restricted to be bipartite and counting is done modulo a fixed prime. A similar result is shown for properties that are closed under the removal of edges only.
Cite as
Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{dorfler_et_al:LIPIcs.MFCS.2019.26,
author = {D\"{o}rfler, Julian and Roth, Marc and Schmitt, Johannes and Wellnitz, Philip},
title = {{Counting Induced Subgraphs: An Algebraic Approach to #W\lbrack1\rbrack-hardness}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {26:1--26:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.26},
URN = {urn:nbn:de:0030-drops-109703},
doi = {10.4230/LIPIcs.MFCS.2019.26},
annote = {Keywords: counting complexity, edge-transitive graphs, graph homomorphisms, induced subgraphs, parameterized complexity}
}
Document
Packing Arc-Disjoint Cycles in Tournaments
Abstract
A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the linear programming dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT and ATT are fixed-parameter tractable, they can be solved in 2^{O(k log k)} n^{O(1)} time and 2^{O(k)} n^{O(1)} time respectively. Moreover, they both admit a kernel with O(k) vertices. We also prove that ACT and ATT cannot be solved in 2^{o(sqrt{k})} n^{O(1)} time under the Exponential-Time Hypothesis.
Cite as
Stéphane Bessy, Marin Bougeret, R. Krithika, Abhishek Sahu, Saket Saurabh, Jocelyn Thiebaut, and Meirav Zehavi. Packing Arc-Disjoint Cycles in Tournaments. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bessy_et_al:LIPIcs.MFCS.2019.27,
author = {Bessy, St\'{e}phane and Bougeret, Marin and Krithika, R. and Sahu, Abhishek and Saurabh, Saket and Thiebaut, Jocelyn and Zehavi, Meirav},
title = {{Packing Arc-Disjoint Cycles in Tournaments}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {27:1--27:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.27},
URN = {urn:nbn:de:0030-drops-109714},
doi = {10.4230/LIPIcs.MFCS.2019.27},
annote = {Keywords: arc-disjoint cycle packing, tournaments, parameterized algorithms, kernelization}
}
Document
A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs
Abstract
Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that |L| and |R| differ by at most 1 and the number of arcs from R to L is at most k. This problem, in general, is W-hard as it is known to be NP-hard even when k=0. We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that Minimum Directed Bisection on semicomplete digraphs is one of a handful of problems that admit sub-exponential time fixed-parameter tractable algorithms. That is, we show that the problem admits a 2^{O(sqrt{k} log k)}n^{O(1)} time algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use (n,k,k^2)-splitters. To the best of our knowledge, this is the first time such pseudorandom objects have been used in the design of kernels. We believe that the framework of designing kernels using splitters could be applied to more problems that admit sub-exponential time algorithms via chromatic coding. To complement the above mentioned results, we prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.
Cite as
Jayakrishnan Madathil, Roohani Sharma, and Meirav Zehavi. A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{madathil_et_al:LIPIcs.MFCS.2019.28,
author = {Madathil, Jayakrishnan and Sharma, Roohani and Zehavi, Meirav},
title = {{A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {28:1--28:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.28},
URN = {urn:nbn:de:0030-drops-109721},
doi = {10.4230/LIPIcs.MFCS.2019.28},
annote = {Keywords: bisection, semicomplete digraph, tournament, fpt algorithm, chromatic coding, polynomial kernel, splitters}
}
Document
The Quantifier Alternation Hierarchy of Synchronous Relations
Abstract
The class of synchronous relations, also known as automatic or regular, is one of the most studied subclasses of rational relations. It enjoys many desirable closure properties and is known to be logically characterized: the synchronous relations are exactly those that are defined by a first-order formula on the structure of all finite words, with the prefix, equal-length and last-letter predicates. Here, we study the quantifier alternation hierarchy of this logic. We show that it collapses at level Sigma_3 and that all levels below admit decidable characterizations. Our results reveal the connections between this hierarchy and the well-known hierarchy of first-order defined languages of finite words.
Cite as
Diego Figueira, Varun Ramanathan, and Pascal Weil. The Quantifier Alternation Hierarchy of Synchronous Relations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{figueira_et_al:LIPIcs.MFCS.2019.29,
author = {Figueira, Diego and Ramanathan, Varun and Weil, Pascal},
title = {{The Quantifier Alternation Hierarchy of Synchronous Relations}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {29:1--29:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.29},
URN = {urn:nbn:de:0030-drops-109735},
doi = {10.4230/LIPIcs.MFCS.2019.29},
annote = {Keywords: synchronous relations, automatic relations, first-order logic, characterization, quantifier alternation}
}
Document
Two variable fragment of Term Modal Logic
Abstract
Term modal logics (TML) are modal logics with unboundedly many modalities, with quantification over modal indices, so that we can have formulas of the form Exists y Forall x (Box_x P(x,y) implies Diamond_y P(y,x)). Like First order modal logic, TML is also "notoriously" undecidable, in the sense that even very simple fragments are undecidable. In this paper, we show the decidability of one interesting fragment, that of two variable TML. This is in contrast to two-variable First order modal logic, which is undecidable.
Cite as
Anantha Padmanabha and R. Ramanujam. Two variable fragment of Term Modal Logic. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{padmanabha_et_al:LIPIcs.MFCS.2019.30,
author = {Padmanabha, Anantha and Ramanujam, R.},
title = {{Two variable fragment of Term Modal Logic}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {30:1--30:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.30},
URN = {urn:nbn:de:0030-drops-109741},
doi = {10.4230/LIPIcs.MFCS.2019.30},
annote = {Keywords: Term modal logic, satisfiability problem, two variable fragment, decidability}
}
Document
Choiceless Logarithmic Space
Abstract
One of the most important open problems in finite model theory is the question whether there is a logic characterising efficient computation. While this question usually concerns Ptime, it can also be applied to other complexity classes, and in particular to Logspace which can be seen as a formalisation of efficient computation for big data. One of the strongest candidates for a logic capturing Ptime is Choiceless Polynomial Time (CPT). It is based on the idea of choiceless algorithms, a general model of symmetric computation over abstract structures (rather than their encodings by finite strings). However, there is currently neither a comparably strong candidate for a logic for Logspace, nor a logic transferring the idea of choiceless computation to Logspace.
We propose here a notion of Choiceless Logarithmic Space which overcomes some of the obstacles posed by Logspace as a less robust complexity class. The resulting logic is contained in both Logspace and CPT, and is strictly more expressive than all logics for Logspace that have been known so far. Further, we address the question whether this logic can define all Logspace-queries, and prove that this is not the case.
Cite as
Erich Grädel and Svenja Schalthöfer. Choiceless Logarithmic Space. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{gradel_et_al:LIPIcs.MFCS.2019.31,
author = {Gr\"{a}del, Erich and Schalth\"{o}fer, Svenja},
title = {{Choiceless Logarithmic Space}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {31:1--31:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.31},
URN = {urn:nbn:de:0030-drops-109758},
doi = {10.4230/LIPIcs.MFCS.2019.31},
annote = {Keywords: Finite Model Theory, Logics for Logspace, Choiceless Computation}
}
Document
Faster FPT Algorithm for 5-Path Vertex Cover
Abstract
The problem of d-Path Vertex Cover, d-PVC lies in determining a subset F of vertices of a given graph G=(V,E) such that G \ F does not contain a path on d vertices. The paths we aim to cover need not to be induced. It is known that the d-PVC problem is NP-complete for any d >= 2. When parameterized by the size of the solution k, 5-PVC has direct trivial algorithm with O(5^kn^{O(1)}) running time and, since d-PVC is a special case of d-Hitting Set, an algorithm running in O(4.0755^kn^{O(1)}) time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in O(4^kn^{O(1)}) time.
Cite as
Radovan Červený and Ondřej Suchý. Faster FPT Algorithm for 5-Path Vertex Cover. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2019.32,
author = {\v{C}erven\'{y}, Radovan and Such\'{y}, Ond\v{r}ej},
title = {{Faster FPT Algorithm for 5-Path Vertex Cover}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {32:1--32:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.32},
URN = {urn:nbn:de:0030-drops-109761},
doi = {10.4230/LIPIcs.MFCS.2019.32},
annote = {Keywords: graph algorithms, Hitting Set, iterative compression, parameterized complexity, d-Path Vertex Cover}
}
Document
Parameterized Complexity of Fair Vertex Evaluation Problems
Abstract
A prototypical graph problem is centered around a graph-theoretic property for a set of vertices and a solution to it is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of a certain quality, where we use size as a quality measure. In this work, we are changing the measure to the fair measure (cf. Lin and Sahni [Li-Shin Lin and Sartaj Sahni, 1989]). The fair measure of a set of vertices S is (at most) k if the number of neighbors in the set S of any vertex (in the input graph) does not exceed k. One possible way to study graph problems is by defining the property in a certain logic. For a given objective, an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. More formally, we study the {MSO} Fair Vertex Evaluation, where the graph-theoretic property is described by an {MSO} formula.
In the presented paper we show that there is an FPT algorithm for the {MSO} Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph and the size of the formula. One may define an extended variant of {MSO} Fair Vertex Evaluation for formulas with l free variables; here we measure a maximum number of neighbors in each of the l sets. However, such variant is {W[1]}-hard for parameter l even on graphs with twin cover one.
Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is {W[1]}-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width.
Cite as
Dušan Knop, Tomáš Masařík, and Tomáš Toufar. Parameterized Complexity of Fair Vertex Evaluation Problems. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{knop_et_al:LIPIcs.MFCS.2019.33,
author = {Knop, Du\v{s}an and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s}},
title = {{Parameterized Complexity of Fair Vertex Evaluation Problems}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {33:1--33:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.33},
URN = {urn:nbn:de:0030-drops-109775},
doi = {10.4230/LIPIcs.MFCS.2019.33},
annote = {Keywords: Fair objective, metatheorem, fair vertex cover, twin cover, modular width}
}
Document
A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs
Abstract
A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by chi_b(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on chi_b(G): The maximum degree Delta(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i-1. We obtain a dichotomy result for all fixed k in N when k is close to one of the two above mentioned upper bounds. Concretely, we show that if k in {Delta(G) + 1 - p, m(G) - p}, the problem is polynomial-time solvable whenever p in {0, 1} and, even when k = 3, it is NP-complete whenever p >= 2. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Delta(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Delta(G). Second, we show that b-Coloring{} is FPT parameterized by Delta(G) + l_k(G), where l_k(G) denotes the number of vertices of degree at least k.
Cite as
Lars Jaffke and Paloma T. Lima. A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2019.34,
author = {Jaffke, Lars and Lima, Paloma T.},
title = {{A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {34:1--34:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.34},
URN = {urn:nbn:de:0030-drops-109784},
doi = {10.4230/LIPIcs.MFCS.2019.34},
annote = {Keywords: b-Coloring, b-Chromatic Number}
}
Document
Parameterized Complexity of Conflict-Free Matchings and Paths
Abstract
An input to a conflict-free variant of a classical problem Gamma, called Conflict-Free Gamma, consists of an instance I of Gamma coupled with a graph H, called the conflict graph. A solution to Conflict-Free Gamma in (I,H) is a solution to I in Gamma, which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Matching (CF-Matching) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-Matching and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-Matching holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-Matching when the conflict graph is chordal. Also, we give FPT algorithms for both CF-Matching and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-Matching and CF-SP, where the conflicting conditions are given by a (representable) matroid.
Cite as
Akanksha Agrawal, Pallavi Jain, Lawqueen Kanesh, and Saket Saurabh. Parameterized Complexity of Conflict-Free Matchings and Paths. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{agrawal_et_al:LIPIcs.MFCS.2019.35,
author = {Agrawal, Akanksha and Jain, Pallavi and Kanesh, Lawqueen and Saurabh, Saket},
title = {{Parameterized Complexity of Conflict-Free Matchings and Paths}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {35:1--35:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.35},
URN = {urn:nbn:de:0030-drops-109798},
doi = {10.4230/LIPIcs.MFCS.2019.35},
annote = {Keywords: Conflict-free, Matching, Shortest Path, FPT algorithm, W\lbrack1\rbrack-hard, Matroid}
}
Document
On the Strength of Uniqueness Quantification in Primitive Positive Formulas
Abstract
Uniqueness quantification (Exists!) is a quantifier in first-order logic where one requires that exactly one element exists satisfying a given property. In this paper we investigate the strength of uniqueness quantification when it is used in place of existential quantification in conjunctive formulas over a given set of relations Gamma, so-called primitive positive definitions (pp-definitions). We fully classify the Boolean sets of relations where uniqueness quantification has the same strength as existential quantification in pp-definitions and give several results valid for arbitrary finite domains. We also consider applications of Exists!-quantified pp-definitions in computer science, which can be used to study the computational complexity of problems where the number of solutions is important. Using our classification we give a new and simplified proof of the trichotomy theorem for the unique satisfiability problem, and prove a general result for the unique constraint satisfaction problem. Studying these problems in a more rigorous framework also turns out to be advantageous in the context of lower bounds, and we relate the complexity of these problems to the exponential-time hypothesis.
Cite as
Victor Lagerkvist and Gustav Nordh. On the Strength of Uniqueness Quantification in Primitive Positive Formulas. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{lagerkvist_et_al:LIPIcs.MFCS.2019.36,
author = {Lagerkvist, Victor and Nordh, Gustav},
title = {{On the Strength of Uniqueness Quantification in Primitive Positive Formulas}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {36:1--36:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.36},
URN = {urn:nbn:de:0030-drops-109808},
doi = {10.4230/LIPIcs.MFCS.2019.36},
annote = {Keywords: Primitive positive definitions, clone theory, constraint satisfaction problems}
}
Document
Resolution Lower Bounds for Refutation Statements
Abstract
For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three applications. (1) An open question in [Atserias and Müller, 2019] asks whether a certain natural propositional encoding of the above statement is hard for Resolution. We answer by giving an exponential size lower bound. (2) We show exponential resolution size lower bounds for reflection principles, thereby improving a result in [Albert Atserias and María Luisa Bonet, 2004]. (3) We provide new examples of CNFs that exponentially separate Res(2) from Resolution (an exponential separation of these two proof systems was originally proved in [Nathan Segerlind et al., 2004]).
Cite as
Michal Garlík. Resolution Lower Bounds for Refutation Statements. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{garlik:LIPIcs.MFCS.2019.37,
author = {Garl{\'\i}k, Michal},
title = {{Resolution Lower Bounds for Refutation Statements}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {37:1--37:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.37},
URN = {urn:nbn:de:0030-drops-109817},
doi = {10.4230/LIPIcs.MFCS.2019.37},
annote = {Keywords: reflection principles, refutation statements, Resolution, proof complexity}
}
Document
Tangles and Single Linkage Hierarchical Clustering
Abstract
We establish a connection between tangles, a concept from structural graph theory that plays a central role in Robertson and Seymour’s graph minor project, and hierarchical clustering. Tangles cannot only be defined for graphs, but in fact for arbitrary connectivity functions, which are functions defined on the subsets of some finite universe, which in typical clustering applications consists of points in some metric space.
Connectivity functions are usually required to be submodular. It is our first contribution to show that the central duality theorem connecting tangles with hierarchical decompositions (so-called branch decompositions) also holds if submodularity is replaced by a different property that we call maximum-submodular.
We then define a natural, though somewhat unusual connectivity function on finite data sets in an arbitrary metric space and prove that its tangles are in one-to-one correspondence with the clusters obtained by applying the well-known single linkage clustering algorithms to the same data set.
The idea of viewing tangles as clusters has first been proposed by Diestel and Whittle [Reinhard Diestel et al., 2019] as an approach to image segmentation. To the best of our knowledge, our result is the first that establishes a precise technical connection between tangles and clusters.
Cite as
Eva Fluck. Tangles and Single Linkage Hierarchical Clustering. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 38:1-38:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{fluck:LIPIcs.MFCS.2019.38,
author = {Fluck, Eva},
title = {{Tangles and Single Linkage Hierarchical Clustering}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {38:1--38:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.38},
URN = {urn:nbn:de:0030-drops-109826},
doi = {10.4230/LIPIcs.MFCS.2019.38},
annote = {Keywords: Tangles, Branch Decomposition, Duality, Hierarchical Clustering, Single Linkage}
}
Document
Approximating the Orthogonality Dimension of Graphs and Hypergraphs
Abstract
A t-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in R^t to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph H, denoted by overline{xi}(H), is the smallest integer t for which there exists a t-dimensional orthogonal representation of H. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We prove that for every k >= 4, it is NP-hard (resp. quasi-NP-hard) to distinguish n-vertex k-uniform hypergraphs H with overline{xi}(H) <= 2 from those satisfying overline{xi}(H) >= Omega(log^delta n) for some constant delta>0 (resp. overline{xi}(H) >= Omega(log^{1-o(1)} n)). For graphs, we relate the NP-hardness of approximating the orthogonality dimension to a variant of a long-standing conjecture of Stahl. We also consider the algorithmic problem in which given a graph G with overline{xi}(G) <= 3 the goal is to find an orthogonal representation of G of as low dimension as possible, and provide a polynomial time approximation algorithm based on semidefinite programming.
Cite as
Ishay Haviv. Approximating the Orthogonality Dimension of Graphs and Hypergraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{haviv:LIPIcs.MFCS.2019.39,
author = {Haviv, Ishay},
title = {{Approximating the Orthogonality Dimension of Graphs and Hypergraphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {39:1--39:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.39},
URN = {urn:nbn:de:0030-drops-109836},
doi = {10.4230/LIPIcs.MFCS.2019.39},
annote = {Keywords: orthogonal representations of hypergraphs, orthogonality dimension, hardness of approximation, Kneser and Schrijver graphs, semidefinite programming}
}
Document
Domination Above r-Independence: Does Sparseness Help?
Abstract
Inspired by the potential of improving tractability via gap- or above-guarantee parametrisations, we investigate the complexity of Dominating Set when given a suitable lower-bound witness. Concretely, we consider being provided with a maximal r-independent set X (a set in which all vertices have pairwise distance at least r+1) along the input graph G which, for r >= 2, lower-bounds the minimum size of any dominating set of G. In the spirit of gap-parameters, we consider a parametrisation by the size of the "residual" set R := V(G) \ N[X].
Our work aims to answer two questions: How does the constant r affect the tractability of the problem and does the restriction to sparse graph classes help here? For the base case r = 2, we find that the problem is paraNP-complete even in apex- and bounded-degree graphs. For r = 3, the problem is W[2]-hard for general graphs but in FPT for nowhere dense classes and it admits a linear kernel for bounded expansion classes. For r >= 4, the parametrisation becomes essentially equivalent to the natural parameter, the size of the dominating set.
Cite as
Carl Einarson and Felix Reidl. Domination Above r-Independence: Does Sparseness Help?. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{einarson_et_al:LIPIcs.MFCS.2019.40,
author = {Einarson, Carl and Reidl, Felix},
title = {{Domination Above r-Independence: Does Sparseness Help?}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {40:1--40:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.40},
URN = {urn:nbn:de:0030-drops-109840},
doi = {10.4230/LIPIcs.MFCS.2019.40},
annote = {Keywords: Dominating Set, Above Guarantee, Kernel, Bounded Expansion, Nowhere Dense}
}
Document
Reducing the Domination Number of Graphs via Edge Contractions
Abstract
In this paper, we study the following problem: given a connected graph G, can we reduce the domination number of G by at least one using k edge contractions, for some fixed integer k >= 0? We show that for k <= 2, the problem is coNP-hard. We further prove that for k=1, the problem is W[1]-hard parameterized by the size of a minimum dominating set plus the mim-width of the input graph, and that it remains NP-hard when restricted to P_9-free graphs, bipartite graphs and {C_3,...,C_{l}}-free graphs for any l >= 3. Finally, we show that for any k >= 1, the problem is polynomial-time solvable for P_5-free graphs and that it can be solved in FPT-time and XP-time when parameterized by tree-width and mim-width, respectively.
Cite as
Esther Galby, Paloma T. Lima, and Bernard Ries. Reducing the Domination Number of Graphs via Edge Contractions. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{galby_et_al:LIPIcs.MFCS.2019.41,
author = {Galby, Esther and Lima, Paloma T. and Ries, Bernard},
title = {{Reducing the Domination Number of Graphs via Edge Contractions}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {41:1--41:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.41},
URN = {urn:nbn:de:0030-drops-109856},
doi = {10.4230/LIPIcs.MFCS.2019.41},
annote = {Keywords: domination number, blocker problem, graph classes}
}
Document
Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth
Abstract
We develop a framework for applying treewidth-based dynamic programming on graphs with "hybrid structure", i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.
Cite as
Eduard Eiben, Robert Ganian, Thekla Hamm, and O-joung Kwon. Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{eiben_et_al:LIPIcs.MFCS.2019.42,
author = {Eiben, Eduard and Ganian, Robert and Hamm, Thekla and Kwon, O-joung},
title = {{Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {42:1--42:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.42},
URN = {urn:nbn:de:0030-drops-109867},
doi = {10.4230/LIPIcs.MFCS.2019.42},
annote = {Keywords: Parameterized complexity, treewidth, rank-width, fixed-parameter algorithms}
}
Document
The Power Word Problem
Abstract
In this work we introduce a new succinct variant of the word problem in a finitely generated group G, which we call the power word problem: the input word may contain powers p^x, where p is a finite word over generators of G and x is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G). The main result of the paper states that the power word problem for a finitely generated free group F is AC^0-Turing-reducible to the word problem for F. Moreover, the following hardness result is shown: For a wreath product G Wr Z, where G is either free of rank at least two or finite non-solvable, the power word problem is complete for coNP. This contrasts with the situation where G is abelian: then the power word problem is shown to be in TC^0.
Cite as
Markus Lohrey and Armin Weiß. The Power Word Problem. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2019.43,
author = {Lohrey, Markus and Wei{\ss}, Armin},
title = {{The Power Word Problem}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {43:1--43:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.43},
URN = {urn:nbn:de:0030-drops-109871},
doi = {10.4230/LIPIcs.MFCS.2019.43},
annote = {Keywords: word problem, compressed word problem, free groups}
}
Document
Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations
Abstract
It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP. It has also been conjectured that the length of a minimal solution to a quadratic equation is at most exponential in the length of the equation, with the latter conjecture implying the former. We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations. We also discuss a connection of our techniques to the topic of unavoidable patterns, and the possibility of exploiting this connection to produce further similar results.
Cite as
Joel D. Day, Florin Manea, and Dirk Nowotka. Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{day_et_al:LIPIcs.MFCS.2019.44,
author = {Day, Joel D. and Manea, Florin and Nowotka, Dirk},
title = {{Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {44:1--44:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.44},
URN = {urn:nbn:de:0030-drops-109889},
doi = {10.4230/LIPIcs.MFCS.2019.44},
annote = {Keywords: Quadratic Word Equations, Length Upper Bounds, NP, Unavoidable Patterns}
}
Document
The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs
Abstract
The Weisfeiler-Leman procedure is a widely-used approach for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into bi- and triconnected components.
We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its triconnected components. Thus, the dimension of the algorithm needed to distinguish two given graphs is at most the dimension required to distinguish the corresponding decompositions into 3-connected components (assuming dimension at least 2).
This result implies that for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes.
In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a construction by Cai, Fürer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.
Cite as
Sandra Kiefer and Daniel Neuen. The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{kiefer_et_al:LIPIcs.MFCS.2019.45,
author = {Kiefer, Sandra and Neuen, Daniel},
title = {{The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {45:1--45:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.45},
URN = {urn:nbn:de:0030-drops-109893},
doi = {10.4230/LIPIcs.MFCS.2019.45},
annote = {Keywords: Weisfeiler-Leman, separators, first-order logic, counting quantifiers}
}
Document
The Domino Problem is Undecidable on Surface Groups
Abstract
We show that the domino problem is undecidable on orbit graphs of non-deterministic substitutions which satisfy a technical property. As an application, we prove that the domino problem is undecidable for the fundamental group of any closed orientable surface of genus at least 2.
Cite as
Nathalie Aubrun, Sebastián Barbieri, and Etienne Moutot. The Domino Problem is Undecidable on Surface Groups. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{aubrun_et_al:LIPIcs.MFCS.2019.46,
author = {Aubrun, Nathalie and Barbieri, Sebasti\'{a}n and Moutot, Etienne},
title = {{The Domino Problem is Undecidable on Surface Groups}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {46:1--46:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.46},
URN = {urn:nbn:de:0030-drops-109900},
doi = {10.4230/LIPIcs.MFCS.2019.46},
annote = {Keywords: tilings, substitutions, SFTs, decidability, domino problem}
}
Document
P-Optimal Proof Systems for Each NP-Set but no Complete Disjoint NP-Pairs Relative to an Oracle
Abstract
Pudlák [P. Pudlák, 2017] lists several major conjectures from the field of proof complexity and asks for oracles that separate corresponding relativized conjectures. Among these conjectures are:
- DisjNP: The class of all disjoint NP-pairs has no many-one complete elements.
- SAT: NP contains no many-one complete sets that have P-optimal proof systems.
- UP: UP has no many-one complete problems.
- NP cap coNP: NP cap coNP has no many-one complete problems.
As one answer to this question, we construct an oracle relative to which DisjNP, neg SAT, UP, and NP cap coNP hold, i.e., there is no relativizable proof for the implication DisjNP wedge UP wedge NP cap coNP ==> SAT. In particular, regarding the conjectures by Pudlák this extends a result by Khaniki [Khaniki, 2019]. Since Khaniki [Khaniki, 2019] constructs an oracle showing that the implication SAT ==> DisjNP has no relativizable proof, we obtain that the conjectures DisjNP and SAT are independent in relativized worlds, i.e., none of the implications DisjNP ==> SAT and SAT ==> DisjNP can be proven relativizably.
Cite as
Titus Dose. P-Optimal Proof Systems for Each NP-Set but no Complete Disjoint NP-Pairs Relative to an Oracle. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{dose:LIPIcs.MFCS.2019.47,
author = {Dose, Titus},
title = {{P-Optimal Proof Systems for Each NP-Set but no Complete Disjoint NP-Pairs Relative to an Oracle}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {47:1--47:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.47},
URN = {urn:nbn:de:0030-drops-109918},
doi = {10.4230/LIPIcs.MFCS.2019.47},
annote = {Keywords: NP-complete, proof systems, disjoint NP-pair, oracle, UP}
}
Document
Semicomputable Points in Euclidean Spaces
Abstract
We introduce the notion of a semicomputable point in R^n, defined as a point having left-c.e. projections. We study the range of such a point, which is the set of directions on which its projections are left-c.e., and is a convex cone. We provide a thorough study of these notions, proving along the way new results on the computability of convex sets. We prove realization results, by identifying computability properties of convex cones that make them ranges of semicomputable points. We give two applications of the theory. The first one provides a better understanding of the Solovay derivatives. The second one is the investigation of left-c.e. quadratic polynomials. We show that this is, in fact, a particular case of the general theory of semicomputable points.
Cite as
Mathieu Hoyrup and Donald M. Stull. Semicomputable Points in Euclidean Spaces. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{hoyrup_et_al:LIPIcs.MFCS.2019.48,
author = {Hoyrup, Mathieu and Stull, Donald M.},
title = {{Semicomputable Points in Euclidean Spaces}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {48:1--48:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.48},
URN = {urn:nbn:de:0030-drops-109928},
doi = {10.4230/LIPIcs.MFCS.2019.48},
annote = {Keywords: Semicomputable point, Left-c.e. real, Convex cone, Solovay reducibility, Genericity}
}
Document
Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs
Abstract
We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2^{tw(G)^{Omega(1/d)}} in depth-d Frege systems for d<(K log n)/(log log n), where tw(G) is the treewidth of G. This extends Håstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2^{tw(G)^{O(1/d)}} poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.
Cite as
Nicola Galesi, Dmitry Itsykson, Artur Riazanov, and Anastasia Sofronova. Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{galesi_et_al:LIPIcs.MFCS.2019.49,
author = {Galesi, Nicola and Itsykson, Dmitry and Riazanov, Artur and Sofronova, Anastasia},
title = {{Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {49:1--49:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.49},
URN = {urn:nbn:de:0030-drops-109932},
doi = {10.4230/LIPIcs.MFCS.2019.49},
annote = {Keywords: Tseitin formula, treewidth, AC0-Frege}
}
Document
On the Expressivity of Linear Recursion Schemes
Abstract
We investigate the expressive power of higher-order recursion schemes (HORS) restricted to linear types. Two formalisms are considered: multiplicative additive HORS (MAHORS), which feature both linear function types and products, and multiplicative HORS (MHORS), based on linear function types only.
For MAHORS, we establish an equi-expressivity result with a variant of tree-stack automata. Consequently, we can show that MAHORS are strictly more expressive than first-order HORS, that they are incomparable with second-order HORS, and that the associated branch languages lie at the third level of the collapsible pushdown hierarchy.
In the multiplicative case, we show that MHORS are equivalent to a special kind of pushdown automata. It follows that any MHORS can be translated to an equivalent first-order MHORS in polynomial time. Further, we show that MHORS generate regular trees and can be translated to equivalent order-0 HORS in exponential time. Consequently, MHORS turn out to have the same expressive power as 0-HORS but they can be exponentially more concise.
Our results are obtained through a combination of techniques from game semantics, the geometry of interaction and automata theory.
Cite as
Pierre Clairambault and Andrzej S. Murawski. On the Expressivity of Linear Recursion Schemes. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{clairambault_et_al:LIPIcs.MFCS.2019.50,
author = {Clairambault, Pierre and Murawski, Andrzej S.},
title = {{On the Expressivity of Linear Recursion Schemes}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {50:1--50:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.50},
URN = {urn:nbn:de:0030-drops-109945},
doi = {10.4230/LIPIcs.MFCS.2019.50},
annote = {Keywords: higher-order recursion schemes, linear logic, game semantics, geometry of interaction}
}
Document
Uniform Random Expressions Lack Expressivity
Abstract
In this article, we question the relevance of uniform random models for algorithms that use expressions as inputs. Using a general framework to describe expressions, we prove that if there is a subexpression that is absorbing for a given operator, then, after repeatedly applying the induced simplification to a uniform random expression of size n, we obtain an equivalent expression of constant expected size. This proves that uniform random expressions lack expressivity, as soon as there is an absorbing pattern. For instance, (a+b)^* is absorbing for the union for regular expressions on {a,b}, hence random regular expressions can be drastically reduced using the induced simplification.
Cite as
Florent Koechlin, Cyril Nicaud, and Pablo Rotondo. Uniform Random Expressions Lack Expressivity. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{koechlin_et_al:LIPIcs.MFCS.2019.51,
author = {Koechlin, Florent and Nicaud, Cyril and Rotondo, Pablo},
title = {{Uniform Random Expressions Lack Expressivity}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {51:1--51:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.51},
URN = {urn:nbn:de:0030-drops-109957},
doi = {10.4230/LIPIcs.MFCS.2019.51},
annote = {Keywords: Random expressions, simplification algorithms, analytic combinatorics}
}
Document
Lower Bounds for Multilinear Order-Restricted ABPs
Abstract
Proving super-polynomial lower bounds on the size of syntactic multilinear Algebraic Branching Programs (smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in {x_1,...,x_n} appear along any source to sink path in an smABP can be viewed as a permutation in S_n. In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted:
1) Strict circular-interval ABPs: For every sub-program the index set of variables occurring in it is contained in some circular interval of {1,..., n}.
2) L-ordered ABPs: There is a set of L permutations (orders) of variables such that every source to sink path in the smABP reads variables in one of these L orders, where L <=2^{n^{1/2 -epsilon}} for some epsilon>0.
We prove exponential (i.e., 2^{Omega(n^delta)}, delta>0) lower bounds on the size of above models computing an explicit multilinear 2n-variate polynomial in VP.
As a main ingredient in our lower bounds, we show that any polynomial that can be computed by an smABP of size S, can be written as a sum of O(S) many multilinear polynomials where each summand is a product of two polynomials in at most 2n/3 variables, computable by smABPs. As a corollary, we show that any size S syntactic multilinear ABP can be transformed into a size S^{O(sqrt{n})} depth four syntactic multilinear Sigma Pi Sigma Pi circuit where the bottom Sigma gates compute polynomials on at most O(sqrt{n}) variables.
Finally, we compare the above models with other standard models for computing multilinear polynomials.
Cite as
C. Ramya and B. V. Raghavendra Rao. Lower Bounds for Multilinear Order-Restricted ABPs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{ramya_et_al:LIPIcs.MFCS.2019.52,
author = {Ramya, C. and Rao, B. V. Raghavendra},
title = {{Lower Bounds for Multilinear Order-Restricted ABPs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {52:1--52:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.52},
URN = {urn:nbn:de:0030-drops-109963},
doi = {10.4230/LIPIcs.MFCS.2019.52},
annote = {Keywords: Computational complexity, Algebraic complexity theory, Polynomials}
}
Document
On the Symmetries of and Equivalence Test for Design Polynomials
Abstract
In a Nisan-Wigderson design polynomial (in short, a design polynomial), every pair of monomials share a few common variables. A useful example of such a polynomial, introduced in [Neeraj Kayal et al., 2014], is the following: NW_{d,k}({x}) = sum_{h in F_d[z], deg(h) <= k}{ prod_{i=0}^{d-1}{x_{i, h(i)}}}, where d is a prime, F_d is the finite field with d elements, and k << d. The degree of the gcd of every pair of monomials in NW_{d,k} is at most k. For concreteness, we fix k = ceil[sqrt{d}]. The family of polynomials NW := {NW_{d,k} : d is a prime} and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various structural and algorithmic/complexity aspects of NW beyond the fact that NW in VNP. Is NW_{d,k} characterized by its symmetries? Is it circuit-testable, i.e., given a circuit C can we check efficiently if C computes NW_{d,k}? What is the complexity of equivalence test for NW, i.e., given black-box access to a f in F[{x}], can we check efficiently if there exists an invertible linear transformation A such that f = NW_{d,k}(A * {x})? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the first two questions and partially answer the third.
We show that NW_{d,k} is characterized by its group of symmetries over C, but not over R. We also show that NW_{d,k} is characterized by circuit identities which implies that NW_{d,k} is circuit-testable in randomized polynomial time. As another application of this characterization, we obtain the "flip theorem" for NW.
We give an efficient equivalence test for NW in the case where the transformation A is a block-diagonal permutation-scaling matrix. The design of this algorithm is facilitated by an almost complete understanding of the group of symmetries of NW_{d,k}: We show that if A is in the group of symmetries of NW_{d,k} then A = D * P, where D and P are diagonal and permutation matrices respectively. This is proved by completely characterizing the Lie algebra of NW_{d,k}, and using an interplay between the Hessian of NW_{d,k} and the evaluation dimension.
Cite as
Nikhil Gupta and Chandan Saha. On the Symmetries of and Equivalence Test for Design Polynomials. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{gupta_et_al:LIPIcs.MFCS.2019.53,
author = {Gupta, Nikhil and Saha, Chandan},
title = {{On the Symmetries of and Equivalence Test for Design Polynomials}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {53:1--53:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.53},
URN = {urn:nbn:de:0030-drops-109979},
doi = {10.4230/LIPIcs.MFCS.2019.53},
annote = {Keywords: Nisan-Wigderson design polynomial, characterization by symmetries, Lie algebra, group of symmetries, circuit testability, flip theorem, equivalence test}
}
Document
The Complexity of Homomorphism Indistinguishability
Abstract
For every graph class {F}, let HomInd({F}) be the problem of deciding whether two given graphs are homomorphism-indistinguishable over {F}, i.e., for every graph F in {F}, the number hom(F, G) of homomorphisms from F to G equals the corresponding number hom(F, H) for H. For several natural graph classes (such as paths, trees, bounded treewidth graphs), homomorphism-indistinguishability over the class has an efficient structural characterization, resulting in polynomial time solvability [H. Dell et al., 2018].
In particular, it is known that two non-isomorphic graphs are homomorphism-indistinguishable over the class {T}_k of graphs of treewidth k if and only if they are not distinguished by k-dimensional Weisfeiler-Leman algorithm, a central heuristic for isomorphism testing: this characterization implies a polynomial time algorithm for HomInd({T}_k), for every fixed k in N. In this paper, we show that there is a polynomial-time-decidable class {F} of undirected graphs of bounded treewidth such that HomInd({F}) is undecidable.
Our second hardness result concerns the class {K} of complete graphs. We show that HomInd({K}) is co-NP-hard, and in fact, we show completeness for the class C_=P (under P-time Turing reductions). On the algorithmic side, we show that HomInd({P}) can be solved in polynomial time for the class {P} of directed paths. We end with a brief study of two variants of the HomInd({F}) problem: (a) the problem of lexographic-comparison of homomorphism numbers of two graphs, and (b) the problem of computing certain distance-measures (defined via homomorphism numbers) between two graphs.
Cite as
Jan Böker, Yijia Chen, Martin Grohe, and Gaurav Rattan. The Complexity of Homomorphism Indistinguishability. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{boker_et_al:LIPIcs.MFCS.2019.54,
author = {B\"{o}ker, Jan and Chen, Yijia and Grohe, Martin and Rattan, Gaurav},
title = {{The Complexity of Homomorphism Indistinguishability}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {54:1--54:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.54},
URN = {urn:nbn:de:0030-drops-109980},
doi = {10.4230/LIPIcs.MFCS.2019.54},
annote = {Keywords: graph homomorphism numbers, counting complexity, treewidth}
}
Document
SZX-Calculus: Scalable Graphical Quantum Reasoning
Abstract
We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representation of sub-diagrams via binary matrices. We show soundness and completeness of the SZX-calculus and provide two examples of applications, for graph states and error correcting codes.
Cite as
Titouan Carette, Dominic Horsman, and Simon Perdrix. SZX-Calculus: Scalable Graphical Quantum Reasoning. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{carette_et_al:LIPIcs.MFCS.2019.55,
author = {Carette, Titouan and Horsman, Dominic and Perdrix, Simon},
title = {{SZX-Calculus: Scalable Graphical Quantum Reasoning}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {55:1--55:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.55},
URN = {urn:nbn:de:0030-drops-109999},
doi = {10.4230/LIPIcs.MFCS.2019.55},
annote = {Keywords: Quantum computing, categorical quantum mechanics, completeness, scalability}
}
Document
On the Stretch Factor of Polygonal Chains
Abstract
Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i<j<k <= n. The stretch factor is a global property: it measures how close P is to a straight line, and it involves all the vertices of P; being a c-chain, on the other hand, is a fingerprint-property: it only depends on subsets of O(1) vertices of the chain.
We investigate how the c-chain property influences the stretch factor in the plane: (i) we show that for every epsilon > 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space.
Cite as
Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth. On the Stretch Factor of Polygonal Chains. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{chen_et_al:LIPIcs.MFCS.2019.56,
author = {Chen, Ke and Dumitrescu, Adrian and Mulzer, Wolfgang and T\'{o}th, Csaba D.},
title = {{On the Stretch Factor of Polygonal Chains}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {56:1--56:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.56},
URN = {urn:nbn:de:0030-drops-110005},
doi = {10.4230/LIPIcs.MFCS.2019.56},
annote = {Keywords: polygonal chain, vertex dilation, Koch curve, recursive construction}
}
Document
Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks
Abstract
Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information or behaviour spread over social networks, biological diseases spreading over contact or trade networks, and the potential flow of goods over logistical infrastructure. Often, the networks over which these processes spread are dynamic in nature, and can be modeled with graphs whose structure is subject to discrete changes over time, i.e. with temporal graphs. Here, we consider temporal graphs in which edges are available at specified timesteps, and study the problem of deleting edges from a given temporal graph in order to reduce the number of vertices (temporally) reachable from a given starting point. This could be used to control the spread of a disease, rumour, etc. in a temporal graph. In particular, our aim is to find a temporal subgraph in which a process starting at any single vertex can be transferred to only a limited number of other vertices using a temporally-feasible path (i.e. a path, along which the times of the edge availabilities increase). We introduce a natural deletion problem for temporal graphs and we provide positive and negative results on its computational complexity, both in the traditional and the parameterised sense (subject to various natural parameters), as well as addressing the approximability of this problem.
Cite as
Jessica Enright, Kitty Meeks, George B. Mertzios, and Viktor Zamaraev. Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{enright_et_al:LIPIcs.MFCS.2019.57,
author = {Enright, Jessica and Meeks, Kitty and Mertzios, George B. and Zamaraev, Viktor},
title = {{Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {57:1--57:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.57},
URN = {urn:nbn:de:0030-drops-110010},
doi = {10.4230/LIPIcs.MFCS.2019.57},
annote = {Keywords: Temporal networks, spreading processes, graph modification, parameterised complexity}
}
Document
Constant Delay Enumeration with FPT-Preprocessing for Conjunctive Queries of Bounded Submodular Width
Abstract
Marx (STOC 2010, J. ACM 2013) introduced the notion of submodular width of a conjunctive query (CQ) and showed that for any class Phi of Boolean CQs of bounded submodular width, the model-checking problem for Phi on the class of all finite structures is fixed-parameter tractable (FPT). Note that for non-Boolean queries, the size of the query result may be far too large to be computed entirely within FPT time. We investigate the free-connex variant of submodular width and generalise Marx’s result to non-Boolean queries as follows: For every class Phi of CQs of bounded free-connex submodular width, within FPT-preprocessing time we can build a data structure that allows to enumerate, without repetition and with constant delay, all tuples of the query result. Our proof builds upon Marx’s splitting routine to decompose the query result into a union of results; but we have to tackle the additional technical difficulty to ensure that these can be enumerated efficiently.
Cite as
Christoph Berkholz and Nicole Schweikardt. Constant Delay Enumeration with FPT-Preprocessing for Conjunctive Queries of Bounded Submodular Width. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{berkholz_et_al:LIPIcs.MFCS.2019.58,
author = {Berkholz, Christoph and Schweikardt, Nicole},
title = {{Constant Delay Enumeration with FPT-Preprocessing for Conjunctive Queries of Bounded Submodular Width}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {58:1--58:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.58},
URN = {urn:nbn:de:0030-drops-110021},
doi = {10.4230/LIPIcs.MFCS.2019.58},
annote = {Keywords: conjunctive queries, constant delay enumeration, hypertree decompositions, submodular width, fixed-parameter tractability}
}
Document
Counting Homomorphisms Modulo a Prime Number
Abstract
Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is #GraphHom(H) - the problem of finding the number of homomorphisms from a given graph G to the graph H. Not only the complexity of this basic problem is known, but also of its many variants for digraphs, more general relational structures, graphs with weights, and others. In this paper we consider a modification of #GraphHom(H), the #_{p}GraphHom(H) problem, p a prime number: Given a graph G, find the number of homomorphisms from G to H modulo p. In a series of papers Faben and Jerrum, and Göbel et al. determined the complexity of #_{2}GraphHom(H) in the case H (or, in fact, a certain graph derived from H) is square-free, that is, does not contain a 4-cycle. Also, Göbel et al. found the complexity of #_{p}GraphHom(H) when H is a tree for an arbitrary prime p. Here we extend the above result to show that the #_{p}GraphHom(H) problem is #_{p}P-hard whenever the derived graph associated with H is square-free and is not a star, which completely classifies the complexity of #_{p}GraphHom(H) for square-free graphs H.
Cite as
Amirhossein Kazeminia and Andrei A. Bulatov. Counting Homomorphisms Modulo a Prime Number. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 59:1-59:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{kazeminia_et_al:LIPIcs.MFCS.2019.59,
author = {Kazeminia, Amirhossein and Bulatov, Andrei A.},
title = {{Counting Homomorphisms Modulo a Prime Number}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {59:1--59:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.59},
URN = {urn:nbn:de:0030-drops-110032},
doi = {10.4230/LIPIcs.MFCS.2019.59},
annote = {Keywords: graph homomorphism, modular counting, computational hardness}
}
Document
Approximate Counting CSP Seen from the Other Side
Abstract
In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C,-), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C,-) is solvable in polynomial time if C has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT != #W[1], there are no other cases of #CSP(C,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04].
We show that, assuming FPT != W[1] (under randomised parametrised reductions) and for C satisfying certain general conditions, #CSP(C,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C,-). In particular, our condition generalises the case when C is closed under taking minors.
Cite as
Andrei A. Bulatov and Stanislav Živný. Approximate Counting CSP Seen from the Other Side. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bulatov_et_al:LIPIcs.MFCS.2019.60,
author = {Bulatov, Andrei A. and \v{Z}ivn\'{y}, Stanislav},
title = {{Approximate Counting CSP Seen from the Other Side}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {60:1--60:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.60},
URN = {urn:nbn:de:0030-drops-110041},
doi = {10.4230/LIPIcs.MFCS.2019.60},
annote = {Keywords: constraint satisfaction, approximate counting, homomorphisms}
}
Document
Uniformisation Gives the Full Strength of Regular Languages
Abstract
Given R a binary relation between words (which we treat as a language over a product alphabet AxB), a uniformisation of it is another relation L included in R which chooses a single word over B, for each word over A whenever there exists one. It is known that MSO, the full class of regular languages, is strong enough to define a uniformisation for each of its relations. The quest of this work is to see which other formalisms, weaker than MSO, also have this property. In this paper, we solve this problem for pseudo-varieties of semigroups: we show that no nonempty pseudo-variety weaker than MSO can provide uniformisations for its relations.
Cite as
Nathan Lhote, Vincent Michielini, and Michał Skrzypczak. Uniformisation Gives the Full Strength of Regular Languages. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{lhote_et_al:LIPIcs.MFCS.2019.61,
author = {Lhote, Nathan and Michielini, Vincent and Skrzypczak, Micha{\l}},
title = {{Uniformisation Gives the Full Strength of Regular Languages}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {61:1--61:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.61},
URN = {urn:nbn:de:0030-drops-110053},
doi = {10.4230/LIPIcs.MFCS.2019.61},
annote = {Keywords: pseudo-variety, finite word, semigroup, uniformisation, regular language}
}
Document
New Pumping Technique for 2-Dimensional VASS
Abstract
We propose a new pumping technique for 2-dimensional vector addition systems with states (2-VASS) building on natural geometric properties of runs. We illustrate its applicability by reproving an exponential bound on the length of the shortest accepting run, and by proving a new pumping lemma for languages of 2-VASS. The technique is expected to be useful for settling questions concerning languages of 2-VASS, e.g., for establishing decidability status of the regular separability problem.
Cite as
Wojciech Czerwiński, Sławomir Lasota, Christof Löding, and Radosław Piórkowski. New Pumping Technique for 2-Dimensional VASS. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{czerwinski_et_al:LIPIcs.MFCS.2019.62,
author = {Czerwi\'{n}ski, Wojciech and Lasota, S{\l}awomir and L\"{o}ding, Christof and Pi\'{o}rkowski, Rados{\l}aw},
title = {{New Pumping Technique for 2-Dimensional VASS}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {62:1--62:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.62},
URN = {urn:nbn:de:0030-drops-110066},
doi = {10.4230/LIPIcs.MFCS.2019.62},
annote = {Keywords: vector addition systems with states, pumping, decidability}
}
Document
Computational Complexity of Synchronization under Regular Constraints
Abstract
Many variations of synchronization of finite automata have been studied in the previous decades. Here, we suggest studying the question if synchronizing words exist that belong to some fixed constraint language, given by some partial finite automaton called constraint automaton. We show that this synchronization problem becomes PSPACE-complete even for some constraint automata with two states and a ternary alphabet. In addition, we characterize constraint automata with arbitrarily many states for which the constrained synchronization problem is polynomial-time solvable. We classify the complexity of the constrained synchronization problem for constraint automata with two states and two or three letters completely and lift those results to larger classes of finite automata.
Cite as
Henning Fernau, Vladimir V. Gusev, Stefan Hoffmann, Markus Holzer, Mikhail V. Volkov, and Petra Wolf. Computational Complexity of Synchronization under Regular Constraints. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{fernau_et_al:LIPIcs.MFCS.2019.63,
author = {Fernau, Henning and Gusev, Vladimir V. and Hoffmann, Stefan and Holzer, Markus and Volkov, Mikhail V. and Wolf, Petra},
title = {{Computational Complexity of Synchronization under Regular Constraints}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {63:1--63:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.63},
URN = {urn:nbn:de:0030-drops-110078},
doi = {10.4230/LIPIcs.MFCS.2019.63},
annote = {Keywords: Finite automata, synchronization, computational complexity}
}
Document
A Constant-Time Colored Choice Dictionary with Almost Robust Iteration
Abstract
A (colored) choice dictionary is a data structure that is initialized with positive integers n and c and subsequently maintains a sequence of n elements of {0,...,c-1}, called colors, under operations to inspect and to update the color in a given position and to return the position of an occurrence of a given color. Choice dictionaries are fundamental in space-efficient computing. Some applications call for the additional operation of dynamic iteration, i.e., enumeration of the positions containing a given color while the sequence of colors may change. An iteration is robust if it enumerates every position that contains the relevant color throughout the iteration but never enumerates a position more than once or when it does not contain the color in question. We describe the first choice dictionary that executes every operation in constant amortized time and almost robust iteration in constant amortized time per element enumerated. The iteration is robust, except that it may enumerate some elements a second time. The data structure occupies n log_2 c+O((log n)^2) bits. The time and space bounds assume that c=O((log n)^{1/2}(log log n)^{-{3/2}}).
Cite as
Torben Hagerup. A Constant-Time Colored Choice Dictionary with Almost Robust Iteration. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{hagerup:LIPIcs.MFCS.2019.64,
author = {Hagerup, Torben},
title = {{A Constant-Time Colored Choice Dictionary with Almost Robust Iteration}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {64:1--64:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.64},
URN = {urn:nbn:de:0030-drops-110083},
doi = {10.4230/LIPIcs.MFCS.2019.64},
annote = {Keywords: Succinct data structures, space efficiency, in-place chain technique, bounded universes, amortization}
}
Document
Fault Tolerant and Fully Dynamic DFS in Undirected Graphs: Simple Yet Efficient
Abstract
We present an algorithm for a fault tolerant Depth First Search (DFS) Tree in an undirected graph. This algorithm is drastically simpler than the current state-of-the-art algorithms for this problem, uses optimal space and optimal preprocessing time, and still achieves better time complexity. This algorithm also leads to a better time complexity for maintaining a DFS tree in a fully dynamic environment.
Cite as
Surender Baswana, Shiv Gupta, and Ayush Tulsyan. Fault Tolerant and Fully Dynamic DFS in Undirected Graphs: Simple Yet Efficient. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 65:1-65:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{baswana_et_al:LIPIcs.MFCS.2019.65,
author = {Baswana, Surender and Gupta, Shiv and Tulsyan, Ayush},
title = {{Fault Tolerant and Fully Dynamic DFS in Undirected Graphs: Simple Yet Efficient}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {65:1--65:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.65},
URN = {urn:nbn:de:0030-drops-110096},
doi = {10.4230/LIPIcs.MFCS.2019.65},
annote = {Keywords: Depth first search, DFS, Dynamic graph algorithms, Fault tolerant}
}
Document
RLE Edit Distance in Near Optimal Time
Abstract
We show that the edit distance between two run-length encoded strings of compressed lengths m and n respectively, can be computed in O(mn log(mn)) time. This improves the previous record by a factor of O(n/log(mn)). The running time of our algorithm is within subpolynomial factors of being optimal, subject to the standard SETH-hardness assumption. This effectively closes a line of algorithmic research first started in 1993.
Cite as
Raphaël Clifford, Paweł Gawrychowski, Tomasz Kociumaka, Daniel P. Martin, and Przemysław Uznański. RLE Edit Distance in Near Optimal Time. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 66:1-66:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{clifford_et_al:LIPIcs.MFCS.2019.66,
author = {Clifford, Rapha\"{e}l and Gawrychowski, Pawe{\l} and Kociumaka, Tomasz and Martin, Daniel P. and Uzna\'{n}ski, Przemys{\l}aw},
title = {{RLE Edit Distance in Near Optimal Time}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {66:1--66:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.66},
URN = {urn:nbn:de:0030-drops-110109},
doi = {10.4230/LIPIcs.MFCS.2019.66},
annote = {Keywords: String algorithms, Compression, Pattern matching, Run-length encoding}
}
Document
Indexing Graph Search Trees and Applications
Abstract
We consider the problem of compactly representing the Depth First Search (DFS) tree of a given undirected or directed graph having n vertices and m edges while supporting various DFS related queries efficiently in the RAM with logarithmic word size. We study this problem in two well-known models: indexing and encoding models. While most of these queries can be supported easily in constant time using O(n lg n) bits of extra space, our goal here is, more specifically, to beat this trivial O(n lg n) bit space bound, yet not compromise too much on the running time of these queries. In the indexing model, the space bound of our solution involves the quantity m, hence, we obtain different bounds for sparse and dense graphs respectively. In the encoding model, we first give a space lower bound, followed by an almost optimal data structure with extremely fast query time. Central to our algorithm is a partitioning of the DFS tree into connected subtrees, and a compact way to store these connections. Finally, we also apply these techniques to compactly index the shortest path structure, biconnectivity structures among others.
Cite as
Sankardeep Chakraborty and Kunihiko Sadakane. Indexing Graph Search Trees and Applications. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2019.67,
author = {Chakraborty, Sankardeep and Sadakane, Kunihiko},
title = {{Indexing Graph Search Trees and Applications}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {67:1--67:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.67},
URN = {urn:nbn:de:0030-drops-110112},
doi = {10.4230/LIPIcs.MFCS.2019.67},
annote = {Keywords: Depth First Search Tree, Compact Data Structures, Encoding Schemes}
}
Document
Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties
Abstract
We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over Z_m^n, i.e. , with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map.
As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case.
Cite as
Alberto Dennunzio, Enrico Formenti, Darij Grinberg, and Luciano Margara. Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{dennunzio_et_al:LIPIcs.MFCS.2019.68,
author = {Dennunzio, Alberto and Formenti, Enrico and Grinberg, Darij and Margara, Luciano},
title = {{Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {68:1--68:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.68},
URN = {urn:nbn:de:0030-drops-110126},
doi = {10.4230/LIPIcs.MFCS.2019.68},
annote = {Keywords: Cellular Automata, Symbolic Dynamics, Complex Systems}
}
Document
On Synthesis of Resynchronizers for Transducers
Abstract
We study two formalisms that allow to compare transducers over words under origin semantics: rational and regular resynchronizers, and show that the former are captured by the latter. We then consider some instances of the following synthesis problem: given transducers T_1,T_2, construct a rational (resp. regular) resynchronizer R, if it exists, such that T_1 is contained in R(T_2) under the origin semantics. We show that synthesis of rational resynchronizers is decidable for functional, and even finite-valued, one-way transducers, and undecidable for relational one-way transducers. In the two-way setting, synthesis of regular resynchronizers is shown to be decidable for unambiguous two-way transducers. For larger classes of two-way transducers, the decidability status is open.
Cite as
Sougata Bose, Shankara Narayanan Krishna, Anca Muscholl, Vincent Penelle, and Gabriele Puppis. On Synthesis of Resynchronizers for Transducers. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bose_et_al:LIPIcs.MFCS.2019.69,
author = {Bose, Sougata and Krishna, Shankara Narayanan and Muscholl, Anca and Penelle, Vincent and Puppis, Gabriele},
title = {{On Synthesis of Resynchronizers for Transducers}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {69:1--69:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.69},
URN = {urn:nbn:de:0030-drops-110133},
doi = {10.4230/LIPIcs.MFCS.2019.69},
annote = {Keywords: String transducers, resynchronizers, synthesis}
}
Document
Acceptance Ambiguity for Quantum Automata
Abstract
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post’s correspondence problem.
Cite as
Paul C. Bell and Mika Hirvensalo. Acceptance Ambiguity for Quantum Automata. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bell_et_al:LIPIcs.MFCS.2019.70,
author = {Bell, Paul C. and Hirvensalo, Mika},
title = {{Acceptance Ambiguity for Quantum Automata}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {70:1--70:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.70},
URN = {urn:nbn:de:0030-drops-110149},
doi = {10.4230/LIPIcs.MFCS.2019.70},
annote = {Keywords: Quantum finite automata, matrix freeness, undecidability, Post’s correspondence problem, quaternions}
}
Document
From Regular Expression Matching to Parsing
Abstract
Given a regular expression R and a string Q, the regular expression parsing problem is to determine if Q matches R and if so, determine how it matches, e.g., by a mapping of the characters of Q to the characters in R. Regular expression parsing makes finding matches of a regular expression even more useful by allowing us to directly extract subpatterns of the match, e.g., for extracting IP-addresses from internet traffic analysis or extracting subparts of genomes from genetic data bases. We present a new general techniques for efficiently converting a large class of algorithms that determine if a string Q matches regular expression R into algorithms that can construct a corresponding mapping. As a consequence, we obtain the first efficient linear space solutions for regular expression parsing.
Cite as
Philip Bille and Inge Li Gørtz. From Regular Expression Matching to Parsing. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bille_et_al:LIPIcs.MFCS.2019.71,
author = {Bille, Philip and G{\o}rtz, Inge Li},
title = {{From Regular Expression Matching to Parsing}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {71:1--71:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.71},
URN = {urn:nbn:de:0030-drops-110150},
doi = {10.4230/LIPIcs.MFCS.2019.71},
annote = {Keywords: regular expressions, finite automata, regular expression parsing, algorithms}
}
Document
Solving Systems of Equations in Supernilpotent Algebras
Abstract
Recently, M. Kompatscher proved that for each finite supernilpotent algebra A in a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this algebra. Let mu be the maximal arity of the fundamental operations of A, and let d := |A|^{log_2 mu + log_2 |A| + 1}. Applying a method that G. Károlyi and C. Szabó had used to solve equations over finite nilpotent rings, we show that for A, there is c in N such that a solution of every system of s equations in n variables can be found by testing at most c n^{sd} (instead of all |A|^n possible) assignments to the variables. This also yields new information on some circuit satisfiability problems.
Cite as
Erhard Aichinger. Solving Systems of Equations in Supernilpotent Algebras. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 72:1-72:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{aichinger:LIPIcs.MFCS.2019.72,
author = {Aichinger, Erhard},
title = {{Solving Systems of Equations in Supernilpotent Algebras}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {72:1--72:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.72},
URN = {urn:nbn:de:0030-drops-110162},
doi = {10.4230/LIPIcs.MFCS.2019.72},
annote = {Keywords: Supernilpotent algebras, polynomial equations, polynomial mappings, circuit satisfiability}
}
Document
Listing Induced Steiner Subgraphs as a Compact Way to Discover Steiner Trees in Graphs
Abstract
This paper investigates induced Steiner subgraphs as a variant of the classical Steiner trees, so as to compactly represent the (exponentially many) Steiner trees sharing the same underlying induced subgraph. We prove that the enumeration of all (inclusion-minimal) induced Steiner subgraphs is harder than the well-known Hypergraph Transversal enumeration problem if the number of terminals is not fixed. When the number of terminals is fixed, we propose a polynomial delay algorithm for listing all induced Steiner subgraphs of minimum size. We also propose a polynomial delay algorithm for listing the set of minimal induced Steiner subgraphs when the number of terminals is 3.
Cite as
Alessio Conte, Roberto Grossi, Mamadou Moustapha Kanté, Andrea Marino, Takeaki Uno, and Kunihiro Wasa. Listing Induced Steiner Subgraphs as a Compact Way to Discover Steiner Trees in Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{conte_et_al:LIPIcs.MFCS.2019.73,
author = {Conte, Alessio and Grossi, Roberto and Kant\'{e}, Mamadou Moustapha and Marino, Andrea and Uno, Takeaki and Wasa, Kunihiro},
title = {{Listing Induced Steiner Subgraphs as a Compact Way to Discover Steiner Trees in Graphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {73:1--73:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.73},
URN = {urn:nbn:de:0030-drops-110174},
doi = {10.4230/LIPIcs.MFCS.2019.73},
annote = {Keywords: Graph algorithms, enumeration, listing and counting, Steiner trees, induced subgraphs}
}
Document
Enumeration of Preferred Extensions in Almost Oriented Digraphs
Abstract
In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on n vertices, all preferred extensions can be enumerated in O^*(3^{n/3}) time and there are directed graphs with Omega(3^{n/3}) preferred extensions. We give faster enumeration algorithms for directed graphs with at most 0.8004 * n vertices occurring in 2-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time O(1.2321^n), and we show that there are oriented graphs with Omega(3^{n/6}) > Omega(1.2009^n) preferred extensions.
A combination of three algorithms leads to the fastest enumeration times for various proportions of the number of vertices in 2-cycles. The most innovative one is a new 2-stage sampling algorithm, combined with a new parameterized enumeration algorithm, analyzed with a combination of the recent monotone local search technique (STOC 2016) and an extension thereof (ICALP 2017).
Cite as
Serge Gaspers and Ray Li. Enumeration of Preferred Extensions in Almost Oriented Digraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 74:1-74:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{gaspers_et_al:LIPIcs.MFCS.2019.74,
author = {Gaspers, Serge and Li, Ray},
title = {{Enumeration of Preferred Extensions in Almost Oriented Digraphs}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {74:1--74:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.74},
URN = {urn:nbn:de:0030-drops-110188},
doi = {10.4230/LIPIcs.MFCS.2019.74},
annote = {Keywords: abstract argumentation, exact algorithms, exponential time algorithms, parameterized algorithms, enumeration algorithms, semikernels in digraphs}
}
Document
Determinisation of Finitely-Ambiguous Copyless Cost Register Automata
Abstract
Cost register automata (CRA) are machines reading an input word while computing values using write-only registers: values from registers are combined using the two operations, as well as the constants, of a semiring. Particularly interesting is the subclass of copyless CRAs where the content of a register cannot be used twice for updating the registers. Originally deterministic, non-deterministic variant of CRA may also be defined: the semantics is then obtained by combining the values of all accepting runs with the additive operation of the semiring (as for weighted automata). We show that finitely-ambiguous copyless non-deterministic CRAs (i.e. the ones that admit a bounded number of accepting runs on every input word) can be effectively transformed into an equivalent copyless (deterministic) CRA, without requiring any specific property on the semiring. As a corollary, this also shows that regular look-ahead can effectively be removed from copyless CRAs.
Cite as
Théodore Lopez, Benjamin Monmege, and Jean-Marc Talbot. Determinisation of Finitely-Ambiguous Copyless Cost Register Automata. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 75:1-75:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{lopez_et_al:LIPIcs.MFCS.2019.75,
author = {Lopez, Th\'{e}odore and Monmege, Benjamin and Talbot, Jean-Marc},
title = {{Determinisation of Finitely-Ambiguous Copyless Cost Register Automata}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {75:1--75:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.75},
URN = {urn:nbn:de:0030-drops-110190},
doi = {10.4230/LIPIcs.MFCS.2019.75},
annote = {Keywords: Cost-register automata, Look-ahead removal, Unambiguity, Determinisation}
}
Document
Aperiodic Weighted Automata and Weighted First-Order Logic
Abstract
By fundamental results of Schützenberger, McNaughton and Papert from the 1970s, the classes of first-order definable and aperiodic languages coincide. Here, we extend this equivalence to a quantitative setting. For this, weighted automata form a general and widely studied model. We define a suitable notion of a weighted first-order logic. Then we show that this weighted first-order logic and aperiodic polynomially ambiguous weighted automata have the same expressive power. Moreover, we obtain such equivalence results for suitable weighted sublogics and finitely ambiguous or unambiguous aperiodic weighted automata. Our results hold for general weight structures, including all semirings, average computations of costs, bounded lattices, and others.
Cite as
Manfred Droste and Paul Gastin. Aperiodic Weighted Automata and Weighted First-Order Logic. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 76:1-76:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{droste_et_al:LIPIcs.MFCS.2019.76,
author = {Droste, Manfred and Gastin, Paul},
title = {{Aperiodic Weighted Automata and Weighted First-Order Logic}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {76:1--76:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.76},
URN = {urn:nbn:de:0030-drops-110203},
doi = {10.4230/LIPIcs.MFCS.2019.76},
annote = {Keywords: Weighted automata, weighted logic, aperiodic automata, first-order logic, unambiguous, finitely ambiguous, polynomially ambiguous}
}
Document
A Congruence-based Perspective on Automata Minimization Algorithms
Abstract
In this work we use a framework of finite-state automata constructions based on equivalences over words to provide new insights on the relation between well-known methods for computing the minimal deterministic automaton of a language.
Cite as
Pierre Ganty, Elena Gutiérrez, and Pedro Valero. A Congruence-based Perspective on Automata Minimization Algorithms. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{ganty_et_al:LIPIcs.MFCS.2019.77,
author = {Ganty, Pierre and Guti\'{e}rrez, Elena and Valero, Pedro},
title = {{A Congruence-based Perspective on Automata Minimization Algorithms}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {77:1--77:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.77},
URN = {urn:nbn:de:0030-drops-110214},
doi = {10.4230/LIPIcs.MFCS.2019.77},
annote = {Keywords: Double-Reversal Method, Minimization, Automata, Congruences, Regular Languages}
}
Document
Finding Optimal Solutions With Neighborly Help
Abstract
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted).
We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level.
Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class.
Cite as
Elisabet Burjons, Fabian Frei, Edith Hemaspaandra, Dennis Komm, and David Wehner. Finding Optimal Solutions With Neighborly Help. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 78:1-78:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{burjons_et_al:LIPIcs.MFCS.2019.78,
author = {Burjons, Elisabet and Frei, Fabian and Hemaspaandra, Edith and Komm, Dennis and Wehner, David},
title = {{Finding Optimal Solutions With Neighborly Help}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {78:1--78:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.78},
URN = {urn:nbn:de:0030-drops-110221},
doi = {10.4230/LIPIcs.MFCS.2019.78},
annote = {Keywords: Critical Graphs, Computational Complexity, Structural Self-Reducibility, Minimality Problems, Colorability, Vertex Cover, Satisfiability, Reoptimization, Advice}
}
Document
Reconfiguration of Minimum Steiner Trees via Vertex Exchanges
Abstract
In this paper, we study the problem of deciding if there is a transformation between two given minimum Steiner trees of an unweighted graph such that each transformation step respects a prescribed reconfiguration rule and results in another minimum Steiner tree of the graph. We consider two reconfiguration rules, both of which exchange a single vertex at a time, and generalize the known reconfiguration problem for shortest paths in an unweighted graph. This generalization implies that our problems under both reconfiguration rules are PSPACE-complete for bipartite graphs. We thus study the problems with respect to graph classes, and give some boundaries between the polynomial-time solvable and PSPACE-complete cases.
Cite as
Haruka Mizuta, Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. Reconfiguration of Minimum Steiner Trees via Vertex Exchanges. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 79:1-79:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{mizuta_et_al:LIPIcs.MFCS.2019.79,
author = {Mizuta, Haruka and Hatanaka, Tatsuhiko and Ito, Takehiro and Zhou, Xiao},
title = {{Reconfiguration of Minimum Steiner Trees via Vertex Exchanges}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {79:1--79:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.79},
URN = {urn:nbn:de:0030-drops-110234},
doi = {10.4230/LIPIcs.MFCS.2019.79},
annote = {Keywords: Combinatorial reconfiguration, Graph algorithms, Steiner tree}
}
Document
The Perfect Matching Reconfiguration Problem
Abstract
We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time.
Cite as
Marthe Bonamy, Nicolas Bousquet, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Arnaud Mary, Moritz Mühlenthaler, and Kunihiro Wasa. The Perfect Matching Reconfiguration Problem. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bonamy_et_al:LIPIcs.MFCS.2019.80,
author = {Bonamy, Marthe and Bousquet, Nicolas and Heinrich, Marc and Ito, Takehiro and Kobayashi, Yusuke and Mary, Arnaud and M\"{u}hlenthaler, Moritz and Wasa, Kunihiro},
title = {{The Perfect Matching Reconfiguration Problem}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {80:1--80:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.80},
URN = {urn:nbn:de:0030-drops-110248},
doi = {10.4230/LIPIcs.MFCS.2019.80},
annote = {Keywords: Combinatorial Reconfiguration, Graph Algorithms, Perfect Matching}
}
Document
Spectral Aspects of Symmetric Matrix Signings
Abstract
The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of finding symmetric signings of matrices with natural spectral properties. Our results are the following:
1) We characterize matrices that have an invertible signing: a symmetric matrix has an invertible symmetric signing if and only if the support graph of the matrix contains a perfect 2-matching. Further, we present an efficient algorithm to search for an invertible symmetric signing.
2) We use the above-mentioned characterization to give an algorithm to find a minimum increase in the support of a given symmetric matrix so that it has an invertible symmetric signing.
3) We show NP-completeness of the following problems: verifying whether a given matrix has a symmetric signing that is singular or has bounded eigenvalues. However, we also illustrate that the complexity could differ substantially for input matrices that are adjacency matrices of graphs.
We use combinatorial techniques in addition to classic results from matching theory.
Cite as
Charles Carlson, Karthekeyan Chandrasekaran, Hsien-Chih Chang, Naonori Kakimura, and Alexandra Kolla. Spectral Aspects of Symmetric Matrix Signings. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{carlson_et_al:LIPIcs.MFCS.2019.81,
author = {Carlson, Charles and Chandrasekaran, Karthekeyan and Chang, Hsien-Chih and Kakimura, Naonori and Kolla, Alexandra},
title = {{Spectral Aspects of Symmetric Matrix Signings}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {81:1--81:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.81},
URN = {urn:nbn:de:0030-drops-110258},
doi = {10.4230/LIPIcs.MFCS.2019.81},
annote = {Keywords: Spectral Graph Theory, Matrix Signing, Matchings}
}
Document
Efficient Analysis of Unambiguous Automata Using Matrix Semigroup Techniques
Abstract
We introduce a novel technique to analyse unambiguous Büchi automata quantitatively, and apply this to the model checking problem. It is based on linear-algebra arguments that originate from the analysis of matrix semigroups with constant spectral radius. This method can replace a combinatorial procedure that dominates the computational complexity of the existing procedure by Baier et al. We analyse the complexity in detail, showing that, in terms of the set Q of states of the automaton, the new algorithm runs in time O(|Q|^4), improving on an efficient implementation of the combinatorial algorithm by a factor of |Q|.
Cite as
Stefan Kiefer and Cas Widdershoven. Efficient Analysis of Unambiguous Automata Using Matrix Semigroup Techniques. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 82:1-82:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{kiefer_et_al:LIPIcs.MFCS.2019.82,
author = {Kiefer, Stefan and Widdershoven, Cas},
title = {{Efficient Analysis of Unambiguous Automata Using Matrix Semigroup Techniques}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {82:1--82:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.82},
URN = {urn:nbn:de:0030-drops-110269},
doi = {10.4230/LIPIcs.MFCS.2019.82},
annote = {Keywords: Algorithms, Automata, Markov Chains, Matrix Semigroups}
}
Document
On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond
Abstract
We consider the following variant of the Mortality Problem: given k x k matrices A_1, A_2, ...,A_{t}, does there exist nonnegative integers m_1, m_2, ...,m_t such that the product A_1^{m_1} A_2^{m_2} * ... * A_{t}^{m_{t}} is equal to the zero matrix? It is known that this problem is decidable when t <= 2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k even for integral matrices.
In this paper, we prove the first decidability results for t>2. We show as one of our central results that for t=3 this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for t=3 and k <= 3 for matrices over algebraic numbers and for t=3 and k=4 for matrices over real algebraic numbers. Another consequence is that the set of triples (m_1,m_2,m_3) for which the equation A_1^{m_1} A_2^{m_2} A_3^{m_3} equals the zero matrix is equal to a finite union of direct products of semilinear sets.
For t=4 we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular 2 x 2 rational matrices by employing powerful tools from transcendence theory such as Baker’s theorem and S-unit equations.
Cite as
Paul C. Bell, Igor Potapov, and Pavel Semukhin. On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 83:1-83:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
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@InProceedings{bell_et_al:LIPIcs.MFCS.2019.83,
author = {Bell, Paul C. and Potapov, Igor and Semukhin, Pavel},
title = {{On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {83:1--83:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-117-7},
ISSN = {1868-8969},
year = {2019},
volume = {138},
editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.83},
URN = {urn:nbn:de:0030-drops-110279},
doi = {10.4230/LIPIcs.MFCS.2019.83},
annote = {Keywords: Linear recurrence sequences, Skolem’s problem, mortality problem, matrix equations, primary decomposition theorem, Baker’s theorem}
}