Volume
LIPIcs, Volume 241
47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS)
Event
MFCS 2022, August 22-26, 2022, Vienna, AustriaEditors
Publication Details
- published at: 2022-08-22
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-256-3
- DBLP: db/conf/mfcs/mfcs2022
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Complete Volume
LIPIcs, Volume 241, MFCS 2022, Complete Volume
Abstract
LIPIcs, Volume 241, MFCS 2022, Complete Volume
Cite as
47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 1-1236, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@Proceedings{szeider_et_al:LIPIcs.MFCS.2022,
title = {{LIPIcs, Volume 241, MFCS 2022, Complete Volume}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {1--1236},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022},
URN = {urn:nbn:de:0030-drops-167975},
doi = {10.4230/LIPIcs.MFCS.2022},
annote = {Keywords: LIPIcs, Volume 241, MFCS 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{szeider_et_al:LIPIcs.MFCS.2022.0,
author = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {0:i--0:xviii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.0},
URN = {urn:nbn:de:0030-drops-167981},
doi = {10.4230/LIPIcs.MFCS.2022.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Long Cycles in Graphs: Extremal Combinatorics Meets Parameterized Algorithms (Invited Talk)
Abstract
We discuss recent algorithmic extensions of two classic results of extremal combinatorics about long paths in graphs. First, the theorem of Dirac from 1952 asserts that a 2-connected graph G with the minimum vertex degree d > 1, is either Hamiltonian or contains a cycle of length at least 2d. Second, the theorem of Erdős-Gallai from 1959, states that a graph G with the average vertex degree D > 1, contains a cycle of length at least D. The proofs of these theorems are constructive, they provide polynomial-time algorithms constructing cycles of lengths 2d and D. We extend these algorithmic results by showing that each of the problems, to decide whether a 2-connected graph contains a cycle of length at least 2d+k or of a cycle of length at least D+k, is fixed-parameter tractable parameterized by k.
Cite as
Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. Long Cycles in Graphs: Extremal Combinatorics Meets Parameterized Algorithms (Invited Talk). In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 1:1-1:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{fomin_et_al:LIPIcs.MFCS.2022.1,
author = {Fomin, Fedor V. and Golovach, Petr A. and Sagunov, Danil and Simonov, Kirill},
title = {{Long Cycles in Graphs: Extremal Combinatorics Meets Parameterized Algorithms}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {1:1--1:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.1},
URN = {urn:nbn:de:0030-drops-167999},
doi = {10.4230/LIPIcs.MFCS.2022.1},
annote = {Keywords: Longest path, longest cycle, fixed-parameter tractability, above guarantee parameterization, average degree, dense graph, Dirac theorem, Erd\H{o}s-Gallai theorem}
}
Document
Invited Talk
Modern Dynamic Data Structures (Invited Talk)
Abstract
We give an overview of differentially private dynamic data structure, aka differentially private algorithms under continual release.
Cite as
Monika Henzinger. Modern Dynamic Data Structures (Invited Talk). In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{henzinger:LIPIcs.MFCS.2022.2,
author = {Henzinger, Monika},
title = {{Modern Dynamic Data Structures}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {2:1--2:2},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.2},
URN = {urn:nbn:de:0030-drops-168009},
doi = {10.4230/LIPIcs.MFCS.2022.2},
annote = {Keywords: Differential privacy, data structures}
}
Document
Invited Talk
An Updated Survey of Bidding Games on Graphs (Invited Talk)
Abstract
A graph game is a two-player zero-sum game in which the players move a token throughout a graph to produce an infinite path, which determines the winner or payoff of the game. In bidding games, both players have budgets, and in each turn, we hold an "auction" (bidding) to determine which player moves the token. In this survey, we consider several bidding mechanisms and their effect on the properties of the game. Specifically, bidding games, and in particular bidding games of infinite duration, have an intriguing equivalence with random-turn games in which in each turn, the player who moves is chosen randomly. We summarize how minor changes in the bidding mechanism lead to unexpected differences in the equivalence with random-turn games.
Cite as
Guy Avni and Thomas A. Henzinger. An Updated Survey of Bidding Games on Graphs (Invited Talk). In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 3:1-3:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{avni_et_al:LIPIcs.MFCS.2022.3,
author = {Avni, Guy and Henzinger, Thomas A.},
title = {{An Updated Survey of Bidding Games on Graphs}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {3:1--3:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.3},
URN = {urn:nbn:de:0030-drops-168017},
doi = {10.4230/LIPIcs.MFCS.2022.3},
annote = {Keywords: Bidding games, Richman bidding, poorman bidding, mean-payoff, parity}
}
Document
Invited Talk
Probabilistic Model Checking for Strategic Equilibria-Based Decision Making: Advances and Challenges (Invited Talk)
Abstract
Game-theoretic concepts have been extensively studied in economics to provide insight into competitive behaviour and strategic decision making. As computing systems increasingly involve concurrently acting autonomous agents, game-theoretic approaches are becoming widespread in computer science as a faithful modelling abstraction. These techniques can be used to reason about the competitive or collaborative behaviour of multiple rational agents with distinct goals or objectives. This paper provides an overview of recent advances in developing a modelling, verification and strategy synthesis framework for concurrent stochastic games implemented in the probabilistic model checker PRISM-games. This is based on a temporal logic that supports finite- and infinite-horizon temporal properties in both a zero-sum and nonzero-sum setting, the latter using Nash and correlated equilibria with respect to two optimality criteria, social welfare and social fairness. We summarise the key concepts, logics and algorithms and the currently available tool support. Future challenges and recent progress in adapting the framework and algorithmic solutions to continuous environments and neural networks are also outlined.
Cite as
Marta Kwiatkowska, Gethin Norman, David Parker, Gabriel Santos, and Rui Yan. Probabilistic Model Checking for Strategic Equilibria-Based Decision Making: Advances and Challenges (Invited Talk). In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kwiatkowska_et_al:LIPIcs.MFCS.2022.4,
author = {Kwiatkowska, Marta and Norman, Gethin and Parker, David and Santos, Gabriel and Yan, Rui},
title = {{Probabilistic Model Checking for Strategic Equilibria-Based Decision Making: Advances and Challenges}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {4:1--4:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.4},
URN = {urn:nbn:de:0030-drops-168026},
doi = {10.4230/LIPIcs.MFCS.2022.4},
annote = {Keywords: Probabilistic model checking, stochastic games, equilibria}
}
Document
Invited Talk
Online Bipartite Matching and Adwords (Invited Talk)
Abstract
The purpose of this paper is to give a "textbook quality" proof of the optimal algorithm, called Ranking, for the online bipartite matching problem (OBM) and to highlight its role in matching-based market design. In particular, we discuss a generalization of OBM, called the adwords problem, which has had a significant impact in the ad auctions marketplace.
Cite as
Vijay V. Vazirani. Online Bipartite Matching and Adwords (Invited Talk). In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 5:1-5:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{vazirani:LIPIcs.MFCS.2022.5,
author = {Vazirani, Vijay V.},
title = {{Online Bipartite Matching and Adwords}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {5:1--5:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.5},
URN = {urn:nbn:de:0030-drops-168031},
doi = {10.4230/LIPIcs.MFCS.2022.5},
annote = {Keywords: matching-based market design, online algorithms, ad auctions, competitive analysis}
}
Document
Parameterized Complexity of Non-Separating and Non-Disconnecting Paths and Sets
Abstract
For a connected graph G = (V, E) and s, t ∈ V, a non-separating s-t path is a path P between s and t such that the set of vertices of P does not separate G, that is, G - V(P) is connected. An s-t path P is non-disconnecting if G - E(P) is connected. The problems of finding shortest non-separating and non-disconnecting paths are both known to be NP-hard. In this paper, we consider the problems from the viewpoint of parameterized complexity. We show that the problem of finding a non-separating s-t path of length at most k is W[1]-hard parameterized by k, while the non-disconnecting counterpart is fixed-parameter tractable (FPT) parameterized by k. We also consider the shortest non-separating path problem on several classes of graphs and show that this problem is NP-hard even on bipartite graphs, split graphs, and planar graphs. As for positive results, the shortest non-separating path problem is FPT parameterized by k on planar graphs and on unit disk graphs (where no s, t is given). Further, we give a polynomial-time algorithm on chordal graphs if k is the distance of the shortest path between s and t.
Cite as
Ankit Abhinav, Susobhan Bandopadhyay, Aritra Banik, Yasuaki Kobayashi, Shunsuke Nagano, Yota Otachi, and Saket Saurabh. Parameterized Complexity of Non-Separating and Non-Disconnecting Paths and Sets. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{abhinav_et_al:LIPIcs.MFCS.2022.6,
author = {Abhinav, Ankit and Bandopadhyay, Susobhan and Banik, Aritra and Kobayashi, Yasuaki and Nagano, Shunsuke and Otachi, Yota and Saurabh, Saket},
title = {{Parameterized Complexity of Non-Separating and Non-Disconnecting Paths and Sets}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {6:1--6:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.6},
URN = {urn:nbn:de:0030-drops-168041},
doi = {10.4230/LIPIcs.MFCS.2022.6},
annote = {Keywords: Non-separating path, Parameterized complexity}
}
Document
Comonadic semantics for hybrid logic
Abstract
Hybrid logic is a widely-studied extension of basic modal logic, which corresponds to the bounded fragment of first-order logic. We study it from two novel perspectives: (1) We apply the recently introduced paradigm of comonadic semantics, which provides a new set of tools drawing on ideas from categorical semantics which can be applied to finite model theory, descriptive complexity and combinatorics. (2) We give a novel semantic characterization of hybrid logic in terms of invariance under disjoint extensions, a minimal form of locality. A notable feature of this result is that we give a uniform proof, valid for both the finite and infinite cases.
Cite as
Samson Abramsky and Dan Marsden. Comonadic semantics for hybrid logic. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{abramsky_et_al:LIPIcs.MFCS.2022.7,
author = {Abramsky, Samson and Marsden, Dan},
title = {{Comonadic semantics for hybrid logic}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {7:1--7:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.7},
URN = {urn:nbn:de:0030-drops-168055},
doi = {10.4230/LIPIcs.MFCS.2022.7},
annote = {Keywords: comonads, model comparison games, semantic characterizations, hybrid logic, bounded fragment}
}
Document
The Complexity of Periodic Energy Minimisation
Abstract
The computational complexity of pairwise energy minimisation of N points in real space is a long-standing open problem. The idea of the potential intractability of the problem was supported by a lack of progress in finding efficient algorithms, even when restricted the integer grid approximation. In this paper we provide a firm answer to the problem on ℤ^d by showing that for a large class of pairwise energy functions the problem of periodic energy minimisation is NP-hard if the size of the period (known as a unit cell) is fixed, and is undecidable otherwise. We do so by introducing an abstraction of pairwise average energy minimisation as a mathematical problem, which covers many existing models. The most influential aspects of this work are showing for the first time: 1) undecidability of average pairwise energy minimisation in general 2) computational hardness for the most natural model with periodic boundary conditions, and 3) novel reductions for a large class of generic pairwise energy functions covering many physical abstractions at once. In particular, we develop a new tool of overlapping digital rhombuses to incorporate the properties of the physical force fields, and we connect it with classical tiling problems. Moreover, we illustrate the power of such reductions by incorporating more physical properties such as charge neutrality, and we show an inapproximability result for the extreme case of the 1D average energy minimisation problem.
Cite as
Duncan Adamson, Argyrios Deligkas, Vladimir V. Gusev, and Igor Potapov. The Complexity of Periodic Energy Minimisation. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{adamson_et_al:LIPIcs.MFCS.2022.8,
author = {Adamson, Duncan and Deligkas, Argyrios and Gusev, Vladimir V. and Potapov, Igor},
title = {{The Complexity of Periodic Energy Minimisation}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {8:1--8:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.8},
URN = {urn:nbn:de:0030-drops-168065},
doi = {10.4230/LIPIcs.MFCS.2022.8},
annote = {Keywords: Optimisation of periodic structures, tiling, undecidability, NP-hardness}
}
Document
Weighted Counting of Matchings in Unbounded-Treewidth Graph Families
Abstract
We consider a weighted counting problem on matchings, denoted PrMatching(𝒢), on an arbitrary fixed graph family 𝒢. The input consists of a graph G ∈ 𝒢 and of rational probabilities of existence on every edge of G, assuming independence. The output is the probability of obtaining a matching of G in the resulting distribution, i.e., a set of edges that are pairwise disjoint. It is known that, if 𝒢 has bounded treewidth, then PrMatching(𝒢) can be solved in polynomial time. In this paper we show that, under some assumptions, bounded treewidth in fact characterizes the tractable graph families for this problem. More precisely, we show intractability for all graph families 𝒢 satisfying the following treewidth-constructibility requirement: given an integer k in unary, we can construct in polynomial time a graph G ∈ 𝒢 with treewidth at least k. Our hardness result is then the following: for any treewidth-constructible graph family 𝒢, the problem PrMatching(𝒢) is intractable. This generalizes known hardness results for weighted matching counting under some restrictions that do not bound treewidth, e.g., being planar, 3-regular, or bipartite; it also answers a question left open in [Amarilli et al., 2016]. We also obtain a similar lower bound for the weighted counting of edge covers.
Cite as
Antoine Amarilli and Mikaël Monet. Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{amarilli_et_al:LIPIcs.MFCS.2022.9,
author = {Amarilli, Antoine and Monet, Mika\"{e}l},
title = {{Weighted Counting of Matchings in Unbounded-Treewidth Graph Families}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {9:1--9:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.9},
URN = {urn:nbn:de:0030-drops-168078},
doi = {10.4230/LIPIcs.MFCS.2022.9},
annote = {Keywords: Treewidth, counting complexity, matchings, Fibonacci sequence}
}
Document
On Upward-Planar L-Drawings of Graphs
Abstract
In an upward-planar L-drawing of a directed acyclic graph (DAG) each edge e is represented as a polyline composed of a vertical segment with its lowest endpoint at the tail of e and of a horizontal segment ending at the head of e. Distinct edges may overlap, but not cross. Recently, upward-planar L-drawings have been studied for st-graphs, i.e., planar DAGs with a single source s and a single sink t containing an edge directed from s to t. It is known that a plane st-graph, i.e., an embedded st-graph in which the edge (s,t) is incident to the outer face, admits an upward-planar L-drawing if and only if it admits a bitonic st-ordering, which can be tested in linear time.
We study upward-planar L-drawings of DAGs that are not necessarily st-graphs. On the combinatorial side, we show that a plane DAG admits an upward-planar L-drawing if and only if it is a subgraph of a plane st-graph admitting a bitonic st-ordering. This allows us to show that not every tree with a fixed bimodal embedding admits an upward-planar L-drawing. Moreover, we prove that any acyclic cactus with a single source (or a single sink) admits an upward-planar L-drawing, which respects a given outerplanar embedding if there are no transitive edges. On the algorithmic side, we consider DAGs with a single source (or a single sink). We give linear-time testing algorithms for these DAGs in two cases: (i) when the drawing must respect a prescribed embedding and (ii) when no restriction is given on the embedding, but the DAG is biconnected and series-parallel.
Cite as
Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, and Giordano Da Lozzo. On Upward-Planar L-Drawings of Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{angelini_et_al:LIPIcs.MFCS.2022.10,
author = {Angelini, Patrizio and Chaplick, Steven and Cornelsen, Sabine and Da Lozzo, Giordano},
title = {{On Upward-Planar L-Drawings of Graphs}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {10:1--10:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.10},
URN = {urn:nbn:de:0030-drops-168085},
doi = {10.4230/LIPIcs.MFCS.2022.10},
annote = {Keywords: graph drawing, planar L-drawings, directed graphs, bitonic st-ordering, upward planarity, series-parallel graphs}
}
Document
RAC Drawings of Graphs with Low Degree
Abstract
Motivated by cognitive experiments providing evidence that large crossing-angles do not impair the readability of a graph drawing, RAC (Right Angle Crossing) drawings were introduced to address the problem of producing readable representations of non-planar graphs by supporting the optimal case in which all crossings form 90° angles.
In this work, we make progress on the problem of finding RAC drawings of graphs of low degree. In this context, a long-standing open question asks whether all degree-3 graphs admit straight-line RAC drawings. This question has been positively answered for the Hamiltonian degree-3 graphs. We improve on this result by extending to the class of 3-edge-colorable degree-3 graphs. When each edge is allowed to have one bend, we prove that degree-4 graphs admit such RAC drawings, a result which was previously known only for degree-3 graphs. Finally, we show that 7-edge-colorable degree-7 graphs admit RAC drawings with two bends per edge. This improves over the previous result on degree-6 graphs.
Cite as
Patrizio Angelini, Michael A. Bekos, Julia Katheder, Michael Kaufmann, and Maximilian Pfister. RAC Drawings of Graphs with Low Degree. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{angelini_et_al:LIPIcs.MFCS.2022.11,
author = {Angelini, Patrizio and Bekos, Michael A. and Katheder, Julia and Kaufmann, Michael and Pfister, Maximilian},
title = {{RAC Drawings of Graphs with Low Degree}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {11:1--11:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.11},
URN = {urn:nbn:de:0030-drops-168090},
doi = {10.4230/LIPIcs.MFCS.2022.11},
annote = {Keywords: Graph Drawing, RAC graphs, Straight-line and bent drawings}
}
Document
Polynomial Time Algorithm for ARRIVAL on Tree-Like Multigraphs
Abstract
A rotor walk in a directed graph can be thought of as a deterministic version of a Markov Chain, where a pebble moves from vertex to vertex following a simple rule until a terminal vertex, or sink, has been reached. The ARRIVAL problem, as defined by Dohrau et al. [Dohrau et al., 2017], consists in determining which sink will be reached. While the walk itself can take an exponential number of steps, this problem belongs to the complexity class NP ∩ co-NP without being known to be in P. In this work, we define a class of directed graphs, namely tree-like multigraphs, which are multigraphs having the global shape of an undirected tree. We prove that in this class, ARRIVAL can be solved in almost linear time, while the number of steps of a rotor walk can still be exponential. Then, we give an application of this result to solve some deterministic analogs of stochastic models (e.g., Markovian decision processes, Stochastic Games).
Cite as
David Auger, Pierre Coucheney, and Loric Duhazé. Polynomial Time Algorithm for ARRIVAL on Tree-Like Multigraphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{auger_et_al:LIPIcs.MFCS.2022.12,
author = {Auger, David and Coucheney, Pierre and Duhaz\'{e}, Loric},
title = {{Polynomial Time Algorithm for ARRIVAL on Tree-Like Multigraphs}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {12:1--12:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.12},
URN = {urn:nbn:de:0030-drops-168103},
doi = {10.4230/LIPIcs.MFCS.2022.12},
annote = {Keywords: Rotor-routing, Rotor Walk, Reachability Problem, Game Theory, Tree-like Multigraph}
}
Document
Graph Realization of Distance Sets
Abstract
The Distance Realization problem is defined as follows. Given an n × n matrix D of nonnegative integers, interpreted as inter-vertex distances, find an n-vertex weighted or unweighted graph G realizing D, i.e., whose inter-vertex distances satisfy dist_G(i,j) = D_{i,j} for every 1 ≤ i < j ≤ n, or decide that no such realizing graph exists. The problem was studied for general weighted and unweighted graphs, as well as for cases where the realizing graph is restricted to a specific family of graphs (e.g., trees or bipartite graphs). An extension of Distance Realization that was studied in the past is where each entry in the matrix D may contain a range of consecutive permissible values. We refer to this extension as Range Distance Realization (or Range-DR). Restricting each range to at most k values yields the problem k-Range Distance Realization (or k-Range-DR). The current paper introduces a new extension of Distance Realization, in which each entry D_{i,j} of the matrix may contain an arbitrary set of acceptable values for the distance between i and j, for every 1 ≤ i < j ≤ n. We refer to this extension as Set Distance Realization (Set-DR), and to the restricted problem where each entry may contain at most k values as k-Set Distance Realization (or k-Set-DR).
We first show that 2-Range-DR is NP-hard for unweighted graphs (implying the same for 2-Set-DR). Next we prove that 2-Set-DR is NP-hard for unweighted and weighted trees. We then explore Set-DR where the realization is restricted to the families of stars, paths, or cycles. For the weighted case, our positive results are that for each of these families there exists a polynomial time algorithm for 2-Set-DR. On the hardness side, we prove that 6-Set-DR is NP-hard for stars and 5-Set-DR is NP-hard for paths and cycles. For the unweighted case, our results are the same, except for the case of unweighted stars, for which k-Set-DR is polynomially solvable for any k.
Cite as
Amotz Bar-Noy, David Peleg, Mor Perry, and Dror Rawitz. Graph Realization of Distance Sets. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{barnoy_et_al:LIPIcs.MFCS.2022.13,
author = {Bar-Noy, Amotz and Peleg, David and Perry, Mor and Rawitz, Dror},
title = {{Graph Realization of Distance Sets}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {13:1--13:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.13},
URN = {urn:nbn:de:0030-drops-168119},
doi = {10.4230/LIPIcs.MFCS.2022.13},
annote = {Keywords: Graph Realization, distance realization, network design}
}
Document
On the Role of the High-Low Partition in Realizing a Degree Sequence by a Bipartite Graph
Abstract
We consider the problem of characterizing degree sequences that can be realized by a bipartite graph. If a partition of the sequence into the two sides of the bipartite graph is given as part of the input, then a complete characterization has been established over 60 years ago. However, the general question, in which a partition and a realizing graph need to be determined, is still open. We investigate the role of an important class of special partitions, called High-Low partitions, which separate the degrees of a sequence into two groups, the high degrees and the low degrees. We show that when the High-Low partition exists and satisfies some natural properties, analysing the High-Low partition resolves the bigraphic realization problem. For sequences that are known to be not realizable by a bipartite graph or that are undecided, we provide approximate realizations based on the High-Low partition.
Cite as
Amotz Bar-Noy, Toni Böhnlein, David Peleg, and Dror Rawitz. On the Role of the High-Low Partition in Realizing a Degree Sequence by a Bipartite Graph. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{barnoy_et_al:LIPIcs.MFCS.2022.14,
author = {Bar-Noy, Amotz and B\"{o}hnlein, Toni and Peleg, David and Rawitz, Dror},
title = {{On the Role of the High-Low Partition in Realizing a Degree Sequence by a Bipartite Graph}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {14:1--14:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.14},
URN = {urn:nbn:de:0030-drops-168121},
doi = {10.4230/LIPIcs.MFCS.2022.14},
annote = {Keywords: Graph Realization, Bipartite Graphs, Degree Sequences, Graphic Sequences, Bigraphic Sequences, Approximate Realization, Multigraph Realization}
}
Document
Towards a Model Theory of Ordered Logics: Expressivity and Interpolation
Abstract
We consider the family of guarded and unguarded ordered logics, that constitute a recently rediscovered family of decidable fragments of first-order logic (FO), in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. While the complexities of their satisfiability problems are now well-established, their model theory, however, is poorly understood. Our paper aims to provide some insight into it.
We start by providing suitable notions of bisimulation for ordered logics. We next employ bisimulations to compare the relative expressive power of ordered logics, and to characterise our logics as bisimulation-invariant fragments of FO à la van Benthem.
Afterwards, we study the Craig Interpolation Property (CIP). We refute yet another claim from the infamous work by Purdy, by showing that the fluted and forward fragments do not enjoy CIP. We complement this result by showing that the ordered fragment and the guarded ordered logics enjoy CIP. These positive results rely on novel and quite intricate model constructions, which take full advantage of the "forwardness" of our logics.
Cite as
Bartosz Bednarczyk and Reijo Jaakkola. Towards a Model Theory of Ordered Logics: Expressivity and Interpolation. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bednarczyk_et_al:LIPIcs.MFCS.2022.15,
author = {Bednarczyk, Bartosz and Jaakkola, Reijo},
title = {{Towards a Model Theory of Ordered Logics: Expressivity and Interpolation}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {15:1--15:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.15},
URN = {urn:nbn:de:0030-drops-168132},
doi = {10.4230/LIPIcs.MFCS.2022.15},
annote = {Keywords: ordered fragments, fluted fragment, guarded fragment, model theory, Craig Interpolation Property, expressive power, model checking}
}
Document
Algebraic Representations of Unique Bipartite Perfect Matching
Abstract
We obtain complete characterizations of the Unique Bipartite Perfect Matching function, and of its Boolean dual, using multilinear polynomials over the reals. Building on previous results [Beniamini, 2020; Beniamini and Nisan, 2021], we show that, surprisingly, the dual description is sparse and has low 𝓁₁-norm - only exponential in Θ(n log n), and this result extends even to other families of matching-related functions. Our approach relies on the Möbius numbers in the matching-covered lattice, and a key ingredient in our proof is Möbius' inversion formula.
These polynomial representations yield complexity-theoretic results. For instance, we show that unique bipartite matching is evasive for classical decision trees, and nearly evasive even for generalized query models. We also obtain a tight Θ(n log n) bound on the log-rank of the associated two-party communication task.
Cite as
Gal Beniamini. Algebraic Representations of Unique Bipartite Perfect Matching. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{beniamini:LIPIcs.MFCS.2022.16,
author = {Beniamini, Gal},
title = {{Algebraic Representations of Unique Bipartite Perfect Matching}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {16:1--16:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.16},
URN = {urn:nbn:de:0030-drops-168140},
doi = {10.4230/LIPIcs.MFCS.2022.16},
annote = {Keywords: Bipartite Perfect Matching, Boolean Functions, Partially Ordered Sets}
}
Document
Fixed-Point Cycles and Approximate EFX Allocations
Abstract
We study edge-labelings of the complete bidirected graph K^↔_n with functions that map the set [d] = {1, … , d} to itself. We call a directed cycle in K^↔_n a fixed-point cycle if composing the labels of its edges in order results in a map that has a fixed point, and we say that a labeling is fixed-point-free if no fixed-point cycle exists. For a given d, we ask for the largest value of n, denoted R_f(d), for which there exists a fixed-point-free labeling of K^↔_n. Determining R_f(d) for all d > 0 is a natural Ramsey-type question, generalizing some well-studied zero-sum problems in extremal combinatorics. The problem was recently introduced by Chaudhury, Garg, Mehlhorn, Mehta, and Misra [EC 2021], who proved that d ≤ R_f(d) ≤ d⁴+d and showed that the problem has close connections to EFX allocations, a central problem of fair allocation in social choice theory.
In this paper we show the improved bound R_f(d) ≤ d^{2 + o(1)}, yielding an efficient (1-ε)-EFX allocation with n agents and O((n/ε)^{0.67}) unallocated goods; this improves the bound of O((n/ε)^{0.8}) of Chaudhury, Garg, Mehlhorn, Mehta, and Misra.
{Additionally, we prove the stronger upper bound 2d-2, in the case where all edge-labels are permutations. A very special case of this problem, that of finding zero-sum cycles in digraphs whose edges are labeled with elements of ℤ_d, was recently considered by Alon and Krivelevich [JGT 2021] and by Mészáros and Steiner [EJC 2021]. Our result improves the bounds obtained by these authors and extends them to labelings with elements of an arbitrary (not necessarily commutative) group, while also simplifying the proof.}
Cite as
Benjamin Aram Berendsohn, Simona Boyadzhiyska, and László Kozma. Fixed-Point Cycles and Approximate EFX Allocations. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{berendsohn_et_al:LIPIcs.MFCS.2022.17,
author = {Berendsohn, Benjamin Aram and Boyadzhiyska, Simona and Kozma, L\'{a}szl\'{o}},
title = {{Fixed-Point Cycles and Approximate EFX Allocations}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {17:1--17:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.17},
URN = {urn:nbn:de:0030-drops-168153},
doi = {10.4230/LIPIcs.MFCS.2022.17},
annote = {Keywords: fixed-point, zero-sum cycle, Ramsey theory, fair allocation, EFX}
}
Document
Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic Circuits
Abstract
We show that any product-depth Δ algebraic circuit for the Iterated Matrix Multiplication Polynomial IMM_{n,d} (when d = O(log n/log log n)) must be of size at least n^Ω(d^{1/(φ²)^Δ}) where φ = 1.618… is the golden ratio. This improves the recent breakthrough result of Limaye, Srinivasan and Tavenas (FOCS'21) who showed a super polynomial lower bound of the form n^Ω(d^{1/4^Δ}) for constant-depth circuits.
One crucial idea of the (LST21) result was to use set-multilinear polynomials where each of the sets in the underlying partition of the variables could be of different sizes. By picking the set sizes more carefully (depending on the depth we are working with), we first show that any product-depth Δ set-multilinear circuit for IMM_{n,d} (when d = O(log n)) needs size at least n^Ω(d^{1/φ^Δ}). This improves the n^Ω(d^{1/2^Δ}) lower bound of (LST21). We then use their Hardness Escalation technique to lift this to general circuits.
We also show that our lower bound cannot be improved significantly using these same techniques. For the specific two set sizes used in (LST21), they showed that their lower bound cannot be improved. We show that for any d^o(1) set sizes (out of maximum possible d), the scope for improving our lower bound is minuscule: there exists a set-multilinear circuit that has product-depth Δ and size almost matching our lower bound such that the value of the measure used to prove the lower bound is maximum for this circuit. This results in a barrier to further improvement using the same measure.
Cite as
C. S. Bhargav, Sagnik Dutta, and Nitin Saxena. Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic Circuits. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bhargav_et_al:LIPIcs.MFCS.2022.18,
author = {Bhargav, C. S. and Dutta, Sagnik and Saxena, Nitin},
title = {{Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic Circuits}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {18:1--18:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.18},
URN = {urn:nbn:de:0030-drops-168161},
doi = {10.4230/LIPIcs.MFCS.2022.18},
annote = {Keywords: polynomials, lower bounds, algebraic circuits, proof barrier, fibonacci numbers}
}
Document
Conflict-Free Coloring on Claw-Free Graphs and Interval Graphs
Abstract
A Conflict-Free Open Neighborhood coloring, abbreviated CFON^* coloring, of a graph G = (V,E) using k colors is an assignment of colors from a set of k colors to a subset of vertices of V(G) such that every vertex sees some color exactly once in its open neighborhood. The minimum k for which G has a CFON^* coloring using k colors is called the CFON^* chromatic number of G, denoted by χ_{ON}^*(G). The analogous notion for closed neighborhood is called CFCN^* coloring and the analogous parameter is denoted by χ_{CN}^*(G). The problem of deciding whether a given graph admits a CFON^* (or CFCN^*) coloring that uses k colors is NP-complete. Below, we describe briefly the main results of this paper.
- For k ≥ 3, we show that if G is a K_{1,k}-free graph then χ_{ON}^*(G) = O(k²log Δ), where Δ denotes the maximum degree of G. Dębski and Przybyło in [J. Graph Theory, 2021] had shown that if G is a line graph, then χ_{CN}^*(G) = O(log Δ). As an open question, they had asked if their result could be extended to claw-free (K_{1,3}-free) graphs, which are a superclass of line graphs. Since it is known that the CFCN^* chromatic number of a graph is at most twice its CFON^* chromatic number, our result positively answers the open question posed by Dębski and Przybyło.
- We show that if the minimum degree of any vertex in G is Ω(Δ/{log^ε Δ}) for some ε ≥ 0, then χ_{ON}^*(G) = O(log^{1+ε}Δ). This is a generalization of the result given by Dębski and Przybyło in the same paper where they showed that if the minimum degree of any vertex in G is Ω(Δ), then χ_{ON}^*(G)= O(logΔ).
- We give a polynomial time algorithm to compute χ_{ON}^*(G) for interval graphs G. This answers in positive the open question posed by Reddy [Theoretical Comp. Science, 2018] to determine whether the CFON^* chromatic number can be computed in polynomial time on interval graphs.
- We explore biconvex graphs, a subclass of bipartite graphs and give a polynomial time algorithm to compute their CFON^* chromatic number. This is interesting as Abel et al. [SIDMA, 2018] had shown that it is NP-complete to decide whether a planar bipartite graph G has χ_{ON}^*(G) = k where k ∈ {1, 2, 3}.
Cite as
Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, and Rogers Mathew. Conflict-Free Coloring on Claw-Free Graphs and Interval Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bhyravarapu_et_al:LIPIcs.MFCS.2022.19,
author = {Bhyravarapu, Sriram and Kalyanasundaram, Subrahmanyam and Mathew, Rogers},
title = {{Conflict-Free Coloring on Claw-Free Graphs and Interval Graphs}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {19:1--19:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.19},
URN = {urn:nbn:de:0030-drops-168173},
doi = {10.4230/LIPIcs.MFCS.2022.19},
annote = {Keywords: Conflict-free coloring, Interval graphs, Bipartite graphs, Claw-free graphs}
}
Document
Skolem Meets Schanuel
Abstract
The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this paper is an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct - a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool Skolem point to the practical applicability of this method.
Cite as
Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, David Purser, and James Worrell. Skolem Meets Schanuel. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bilu_et_al:LIPIcs.MFCS.2022.20,
author = {Bilu, Yuri and Luca, Florian and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Purser, David and Worrell, James},
title = {{Skolem Meets Schanuel}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {20:1--20:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.20},
URN = {urn:nbn:de:0030-drops-168180},
doi = {10.4230/LIPIcs.MFCS.2022.20},
annote = {Keywords: Skolem Problem, Skolem Conjecture, Exponential Local-Global Principle, p-adic Schanuel Conjecture}
}
Document
Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems
Abstract
When computing stable matchings, it is usually assumed that the preferences of the agents in the matching market are fixed. However, in many realistic scenarios, preferences change over time. Consequently, an initially stable matching may become unstable. Then, a natural goal is to find a matching which is stable with respect to the modified preferences and as close as possible to the initial one. For Stable Marriage/Roommates, this problem was formally defined as Incremental Stable Marriage/Roommates by Bredereck et al. [AAAI '20]. As they showed that Incremental Stable Roommates and Incremental Stable Marriage with Ties are NP-hard, we focus on the parameterized complexity of these problems. We answer two open questions of Bredereck et al. [AAAI '20]: We show that Incremental Stable Roommates is W[1]-hard parameterized by the number of changes in the preferences, yet admits an intricate XP-algorithm, and we show that Incremental Stable Marriage with Ties is W[1]-hard parameterized by the number of ties. Furthermore, we analyze the influence of the degree of "similarity" between the agents' preference lists, identifying several polynomial-time solvable and fixed-parameter tractable cases, but also proving that Incremental Stable Roommates and Incremental Stable Marriage with Ties parameterized by the number of different preference lists are W[1]-hard.
Cite as
Niclas Boehmer, Klaus Heeger, and Rolf Niedermeier. Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{boehmer_et_al:LIPIcs.MFCS.2022.21,
author = {Boehmer, Niclas and Heeger, Klaus and Niedermeier, Rolf},
title = {{Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {21:1--21:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.21},
URN = {urn:nbn:de:0030-drops-168194},
doi = {10.4230/LIPIcs.MFCS.2022.21},
annote = {Keywords: Stable Marriage, Stable Roommates, adapting to changing preferences, NP-hardness, W\lbrack1\rbrack-hardness, XP, FPT, master lists, incremental algorithms}
}
Document
Tree Exploration in Dual-Memory Model
Abstract
We study the problem of online tree exploration by a deterministic mobile agent. Our main objective is to establish what features of the model of the mobile agent and the environment allow linear exploration time. We study agents that, upon entering a node, do not receive as input the edge via which they entered. In such model, deterministic memoryless exploration is infeasible, hence the agent needs to be allowed to use some memory. The memory can be located at the agent or at each node. The existing lower bounds show that if the memory is either only at the agent or only at the nodes, then the exploration needs superlinear time. We show that tree exploration in dual-memory model, with constant memory at the agent and logarithmic in the degree at each node is possible in linear time when one of the two additional features is present: fixed initial state of the memory at each node (so called clean memory) or a single movable token. We present two algorithms working in linear time for arbitrary trees in these two models. On the other hand, in our lower bound we show that if the agent has a single bit of memory and one bit is present at each node, then the exploration may require quadratic time even on paths, if the initial memory at nodes could be set arbitrarily (so called dirty memory). This shows that having clean node memory or a token allows linear time exploration of trees in the dual-memory model, but having neither of those features may lead to quadratic exploration time even on a simple path.
Cite as
Dominik Bojko, Karol Gotfryd, Dariusz R. Kowalski, and Dominik Pająk. Tree Exploration in Dual-Memory Model. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bojko_et_al:LIPIcs.MFCS.2022.22,
author = {Bojko, Dominik and Gotfryd, Karol and Kowalski, Dariusz R. and Paj\k{a}k, Dominik},
title = {{Tree Exploration in Dual-Memory Model}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.22},
URN = {urn:nbn:de:0030-drops-168207},
doi = {10.4230/LIPIcs.MFCS.2022.22},
annote = {Keywords: Graph exploration, agent, memory, tree, deterministic algorithms, lower bound}
}
Document
On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum-Of-Squares
Abstract
Vanishing sums of roots of unity can be seen as a natural generalization of knapsack from Boolean variables to variables taking values over the roots of unity. We show that these sums are hard to prove for polynomial calculus and for sum-of-squares, both in terms of degree and size.
Cite as
Ilario Bonacina, Nicola Galesi, and Massimo Lauria. On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum-Of-Squares. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bonacina_et_al:LIPIcs.MFCS.2022.23,
author = {Bonacina, Ilario and Galesi, Nicola and Lauria, Massimo},
title = {{On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum-Of-Squares}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {23:1--23:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.23},
URN = {urn:nbn:de:0030-drops-168211},
doi = {10.4230/LIPIcs.MFCS.2022.23},
annote = {Keywords: polynomial calculus, sum-of-squares, roots of unity, knapsack}
}
Document
Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions
Abstract
We introduce a family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions. These calculi recover many of the nice features of the qubit ZX-calculus which were lost in previous proposals for higher-dimensional systems. We then prove that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations. Adding a discard construction, we obtain a calculus complete for mixed state stabiliser quantum mechanics in odd prime dimensions, and this furthermore gives a complete axiomatisation for the related diagrammatic language for affine co-isotropic relations.
Cite as
Robert I. Booth and Titouan Carette. Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{booth_et_al:LIPIcs.MFCS.2022.24,
author = {Booth, Robert I. and Carette, Titouan},
title = {{Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {24:1--24:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.24},
URN = {urn:nbn:de:0030-drops-168225},
doi = {10.4230/LIPIcs.MFCS.2022.24},
annote = {Keywords: ZX-calculus, completeness, quantum, stabiliser, qudits}
}
Document
Extending Partial Representations of Circle Graphs in Near-Linear Time
Abstract
The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph H ⊆ G and a representation H of H. The question is whether G admits a representation G whose restriction to H is H. We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams.
We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in O((n + m) α(n + m)) time, where α is the inverse Ackermann function. This improves over an O(n³)-time algorithm by Chaplick, Fulek and Klavík [2019]. The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest.
Cite as
Guido Brückner, Ignaz Rutter, and Peter Stumpf. Extending Partial Representations of Circle Graphs in Near-Linear Time. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bruckner_et_al:LIPIcs.MFCS.2022.25,
author = {Br\"{u}ckner, Guido and Rutter, Ignaz and Stumpf, Peter},
title = {{Extending Partial Representations of Circle Graphs in Near-Linear Time}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {25:1--25:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.25},
URN = {urn:nbn:de:0030-drops-168233},
doi = {10.4230/LIPIcs.MFCS.2022.25},
annote = {Keywords: circle graphs, partial representation extension, split decomposition tree, recognition algorithm}
}
Document
Boundaries to Single-Agent Stability in Additively Separable Hedonic Games
Abstract
Coalition formation considers the question of how to partition a set of agents into coalitions with respect to their preferences. Additively separable hedonic games (ASHGs) are a dominant model where cardinal single-agent values are aggregated into preferences by taking sums. Output partitions are typically measured by means of stability, and we follow this approach by considering stability based on single-agent movements (to join other coalitions), where a coalition is defined as stable if there exists no beneficial single-agent deviation. Permissible deviations should always lead to an improvement for the deviator, but they may also be constrained by demanding the consent of agents involved in the deviations, i.e., by agents in the abandoned or welcoming coalition. Most of the existing research focuses on the unanimous consent of one or both of these coalitions, but more recent research relaxes this to majority-based consent. Our contribution is twofold. First, we settle the computational complexity of the existence of contractually Nash stable partitions, where deviations are constrained by the unanimous consent of the abandoned coalition. This resolves the complexity of the last classical stability notion for ASHGs. Second, we identify clear boundaries to the tractability of stable partitions under majority-based stability concepts by proving elaborate hardness results for restricted classes of ASHGs. Slight further restrictions lead to positive results.
Cite as
Martin Bullinger. Boundaries to Single-Agent Stability in Additively Separable Hedonic Games. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bullinger:LIPIcs.MFCS.2022.26,
author = {Bullinger, Martin},
title = {{Boundaries to Single-Agent Stability in Additively Separable Hedonic Games}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {26:1--26:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.26},
URN = {urn:nbn:de:0030-drops-168249},
doi = {10.4230/LIPIcs.MFCS.2022.26},
annote = {Keywords: Coalition Formation, Hedonic Games, Stability}
}
Document
Bounded Degree Nonnegative Counting CSP
Abstract
Constraint satisfaction problems (CSP) encompass an enormous variety of computational problems. In particular, all partition functions from statistical physics, such as spin systems, are special cases of counting CSP (#CSP). We prove a complete complexity classification for every counting problem in #CSP with nonnegative valued constraint functions that is valid when every variable occurs a bounded number of times in all constraints. We show that, depending on the set of constraint functions ℱ, every problem in the complexity class #CSP(ℱ) defined by ℱ is either polynomial time computable for all instances without the bounded occurrence restriction, or is #P-hard even when restricted to bounded degree input instances. The constant bound in the degree depends on ℱ. The dichotomy criterion on ℱ is decidable. As a second contribution, we prove a slightly modified but more streamlined decision procedure (from [Jin-Yi Cai et al., 2011]) for tractability. This enables us to fully classify a family of directed weighted graph homomorphism problems. This family contains both P-time tractable problems and #P-hard problems. To our best knowledge, this is the first family of such problems explicitly classified that are not acyclic, thereby the Lovász-goodness criterion of Dyer-Goldberg-Paterson [Martin E. Dyer et al., 2006] cannot be applied.
Cite as
Jin-Yi Cai and Daniel P. Szabo. Bounded Degree Nonnegative Counting CSP. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{cai_et_al:LIPIcs.MFCS.2022.27,
author = {Cai, Jin-Yi and Szabo, Daniel P.},
title = {{Bounded Degree Nonnegative Counting CSP}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {27:1--27:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.27},
URN = {urn:nbn:de:0030-drops-168250},
doi = {10.4230/LIPIcs.MFCS.2022.27},
annote = {Keywords: Computational Counting Complexity, Constraint Satisfaction Problems, Counting CSPs, Complexity Dichotomy, Nonnegative Counting CSP, Graph Homomorphisms}
}
Document
Continuous Rational Functions Are Deterministic Regular
Abstract
A word-to-word function is rational if it can be realized by a non-deterministic one-way transducer. Over finite words, it is a classical result that any rational function is regular, i.e. it can be computed by a deterministic two-way transducer, or equivalently, by a deterministic streaming string transducer (a one-way automaton which manipulates string registers).
This result no longer holds for infinite words, since a non-deterministic one-way transducer can guess, and check along its run, properties such as infinitely many occurrences of some pattern, which is impossible for a deterministic machine. In this paper, we identify the class of rational functions over infinite words which are also computable by a deterministic two-way transducer. It coincides with the class of rational functions which are continuous, and this property can thus be decided. This solves an open question raised in a previous paper of Dave et al.
Cite as
Olivier Carton and Gaëtan Douéneau-Tabot. Continuous Rational Functions Are Deterministic Regular. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{carton_et_al:LIPIcs.MFCS.2022.28,
author = {Carton, Olivier and Dou\'{e}neau-Tabot, Ga\"{e}tan},
title = {{Continuous Rational Functions Are Deterministic Regular}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {28:1--28:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.28},
URN = {urn:nbn:de:0030-drops-168268},
doi = {10.4230/LIPIcs.MFCS.2022.28},
annote = {Keywords: infinite words, rational functions, determinization, continuity, streaming string transducers, two-way transducers}
}
Document
On Kernels for d-Path Vertex Cover
Abstract
In this paper we study the kernelization of the d-Path Vertex Cover (d-PVC) problem. Given a graph G, the problem requires finding whether there exists a set of at most k vertices whose removal from G results in a graph that does not contain a path (not necessarily induced) with d vertices. It is known that d-PVC is NP-complete for d ≥ 2. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with 𝒪(dk^d) edges. We improve on this by giving better kernels. Specifically, we give kernels with 𝒪(k²) vertices and edges for the cases when d = 4 and d = 5. Further, we give a kernel with 𝒪(k⁴d^{2d+9}) vertices and edges for general d.
Cite as
Radovan Červený, Pratibha Choudhary, and Ondřej Suchý. On Kernels for d-Path Vertex Cover. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2022.29,
author = {\v{C}erven\'{y}, Radovan and Choudhary, Pratibha and Such\'{y}, Ond\v{r}ej},
title = {{On Kernels for d-Path Vertex Cover}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {29:1--29:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.29},
URN = {urn:nbn:de:0030-drops-168279},
doi = {10.4230/LIPIcs.MFCS.2022.29},
annote = {Keywords: Parameterized complexity, Kernelization, d-Hitting Set, d-Path Vertex Cover, Expansion Lemma}
}
Document
On Synthesizing Computable Skolem Functions for First Order Logic
Abstract
Skolem functions play a central role in the study of first order logic, both from theoretical and practical perspectives. While every Skolemized formula in first-order logic makes use of Skolem constants and/or functions, not all such Skolem constants and/or functions admit effectively computable interpretations. Indeed, the question of whether there exists an effectively computable interpretation of a Skolem function, and if so, how to automatically synthesize it, is fundamental to their use in several applications, such as planning, strategy synthesis, program synthesis etc.
In this paper, we investigate the computability of Skolem functions and their automated synthesis in the full generality of first order logic. We first show a strong negative result, that even under mild assumptions on the vocabulary, it is impossible to obtain computable interpretations of Skolem functions. We then show a positive result, providing a precise characterization of first-order theories that admit effective interpretations of Skolem functions, and also present algorithms to automatically synthesize such interpretations. We discuss applications of our characterization as well as complexity bounds for Skolem functions (interpreted as Turing machines).
Cite as
Supratik Chakraborty and S. Akshay. On Synthesizing Computable Skolem Functions for First Order Logic. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2022.30,
author = {Chakraborty, Supratik and Akshay, S.},
title = {{On Synthesizing Computable Skolem Functions for First Order Logic}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {30:1--30:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.30},
URN = {urn:nbn:de:0030-drops-168285},
doi = {10.4230/LIPIcs.MFCS.2022.30},
annote = {Keywords: Skolem functions, Automated, Synthesis, First order logic, Computability}
}
Document
Sample Compression Schemes for Balls in Graphs
Abstract
One of the open problems in machine learning is whether any set-family of VC-dimension d admits a sample compression scheme of size O(d). In this paper, we study this problem for balls in graphs. For balls of arbitrary radius r, we design proper sample compression schemes of size 4 for interval graphs, of size 6 for trees of cycles, and of size 22 for cube-free median graphs. We also design approximate sample compression schemes of size 2 for balls of δ-hyperbolic graphs.
Cite as
Jérémie Chalopin, Victor Chepoi, Fionn Mc Inerney, Sébastien Ratel, and Yann Vaxès. Sample Compression Schemes for Balls in Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chalopin_et_al:LIPIcs.MFCS.2022.31,
author = {Chalopin, J\'{e}r\'{e}mie and Chepoi, Victor and Mc Inerney, Fionn and Ratel, S\'{e}bastien and Vax\`{e}s, Yann},
title = {{Sample Compression Schemes for Balls in Graphs}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-256-3},
ISSN = {1868-8969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.31},
URN = {urn:nbn:de:0030-drops-168298},
doi = {10.4230/LIPIcs.MFCS.2022.31},
annote = {Keywords: Proper Sample Compression Schemes, Balls, Graphs, VC-dimension}
}
Document
On Algorithms Based on Finitely Many Homomorphism Counts
Abstract
It is well known [L. Lovász, 1967] that up to isomorphism a graph G is determined by the homomorphism counts hom(F, G), i.e., by the number of homomorphisms from F to G where F ranges over all graphs. Moreover, it suffices that F ranges over the graphs with at most as many vertices as G. Thus, in principle, we can answer any query concerning G with only accessing the hom(⋅, G)’s instead of G itself. In this paper, we deal with queries for which there is a hom algorithm, i.e., there are finitely many graphs F₁, …, F_k such that for any graph G whether it is a Yes-instance of the query is already determined by the vector hom^⟶_{F₁, …, F_k}(G): = (hom(F₁, G), …, hom(F_k, G)).
We observe that planarity of graphs and 3-colorability of graphs, properties expressible in monadic second-order logic, have no hom algorithm. On the other hand, queries expressible as a Boolean combination of universal sentences in first-order logic FO have a hom algorithm. Even though it is not easy to find FO definable queries without a hom algorithm, we succeed to show this for the non-existence of an isolated vertex, a property expressible by the FO sentence ∀ x∃ y Exy, somehow the "simplest" graph property not definable by a Boolean combination of universal sentences. These results provide a characterization of the prefix classes of first-order logic with the property that each query definable by a sentence of the prefix class has a hom algorithm.
For adaptive hom algorithms, i.e., algorithms that might access a hom(F_{i+1}, G) with F_{i+1} depending on hom(F_j, G) for 1 ≤ j ≤ i we show that three homomorphism counts hom(⋅, G) are sufficient and in general necessary to determine the (isomorphism type of) G. In particular, by three adaptive queries we can answer any question on G. Moreover, adaptively accessing two hom(⋅, G)’s is already enough to detect an isolated vertex.
In 1993 Chaudhuri and Vardi [S. Chaudhuri and M. Y. Vardi, 1993] showed the analogue of the Lovász Isomorphism Theorem for the right homomorphism vector of a graph G, i.e, the vector of values hom(G,F) where F ranges over all graphs characterizes the isomorphism type of G. We study to what extent our results carry over to the right homomorphism vector.
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Yijia Chen, Jörg Flum, Mingjun Liu, and Zhiyang Xun. On Algorithms Based on Finitely Many Homomorphism Counts. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 32:1-32:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chen_et_al:LIPIcs.MFCS.2022.32,
author = {Chen, Yijia and Flum, J\"{o}rg and Liu, Mingjun and Xun, Zhiyang},
title = {{On Algorithms Based on Finitely Many Homomorphism Counts}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {32:1--32:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
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