13 Search Results for "Král', Daniel"


Volume

LIPIcs, Volume 170

45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

MFCS 2020, August 24-28, 2020, Prague, Czech Republic

Editors: Javier Esparza and Daniel Král'

Document
Track A: Algorithms, Complexity and Games
Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming

Authors: Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)


Abstract
An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the 𝓁₁-norm of the Graver basis is bounded by a function of the maximum 𝓁₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the 𝓁₁-norm of the Graver basis of the constraint matrix, when parameterized by the 𝓁₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.

Cite as

Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder. Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 29:1-29:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{brianski_et_al:LIPIcs.ICALP.2022.29,
  author =	{Bria\'{n}ski, Marcin and Kouteck\'{y}, Martin and Kr\'{a}l', Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na and Schr\"{o}der, Felix},
  title =	{{Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{29:1--29:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.29},
  URN =		{urn:nbn:de:0030-drops-163702},
  doi =		{10.4230/LIPIcs.ICALP.2022.29},
  annote =	{Keywords: Integer programming, width parameters, matroids, Graver basis, tree-depth, fixed parameter tractability}
}
Document
Sparsity in Algorithms, Combinatorics and Logic (Dagstuhl Seminar 21391)

Authors: Daniel Král’, Michał Pilipczuk, Sebastian Siebertz, and Blair D. Sullivan

Published in: Dagstuhl Reports, Volume 11, Issue 8 (2022)


Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 21391 "Sparsity in Algorithms, Combinatorics and Logic". The seminar took place in a hybrid format from September 26 - October 1, 2021 and brought together 61 researchers. This report includes a discussion of the motivation of the seminar, presentation of the overall organization, abstracts of talks, and a report from each of the working groups.

Cite as

Daniel Král’, Michał Pilipczuk, Sebastian Siebertz, and Blair D. Sullivan. Sparsity in Algorithms, Combinatorics and Logic (Dagstuhl Seminar 21391). In Dagstuhl Reports, Volume 11, Issue 8, pp. 115-128, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@Article{kral'_et_al:DagRep.11.8.115,
  author =	{Kr\'{a}l’, Daniel and Pilipczuk, Micha{\l} and Siebertz, Sebastian and Sullivan, Blair D.},
  title =	{{Sparsity in Algorithms, Combinatorics and Logic (Dagstuhl Seminar 21391)}},
  pages =	{115--128},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2022},
  volume =	{11},
  number =	{8},
  editor =	{Kr\'{a}l’, Daniel and Pilipczuk, Micha{\l} and Siebertz, Sebastian and Sullivan, Blair D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.11.8.115},
  URN =		{urn:nbn:de:0030-drops-157718},
  doi =		{10.4230/DagRep.11.8.115},
  annote =	{Keywords: Algorithm design, Parameterised complexity, Sparse graphs, Graph decompositions, Model theory}
}
Document
Complete Volume
LIPIcs, Volume 170, MFCS 2020, Complete Volume

Authors: Javier Esparza and Daniel Král'

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)


Abstract
LIPIcs, Volume 170, MFCS 2020, Complete Volume

Cite as

Javier Esparza and Daniel Král'. LIPIcs, Volume 170, MFCS 2020, Complete Volume. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 1-1216, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@Proceedings{esparza_et_al:LIPIcs.MFCS.2020,
  title =	{{LIPIcs, Volume 170, MFCS 2020, Complete Volume}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{1--1216},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020},
  URN =		{urn:nbn:de:0030-drops-126703},
  doi =		{10.4230/LIPIcs.MFCS.2020},
  annote =	{Keywords: LIPIcs, Volume 170, MFCS 2020, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Javier Esparza and Daniel Král'

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)


Abstract
Front Matter, Table of Contents, Preface, Conference Organization

Cite as

Javier Esparza and Daniel Král'. Front Matter, Table of Contents, Preface, Conference Organization. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 0:i-0:xviii, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{esparza_et_al:LIPIcs.MFCS.2020.0,
  author =	{Esparza, Javier and Kr\'{a}l', Daniel},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{0:i--0:xviii},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.0},
  URN =		{urn:nbn:de:0030-drops-126714},
  doi =		{10.4230/LIPIcs.MFCS.2020.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Track A: Algorithms, Complexity and Games
Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming

Authors: Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)


Abstract
A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry Δ are solvable in time g(d,Δ) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and Δ. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth. We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width.

Cite as

Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková. Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 26:1-26:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{chan_et_al:LIPIcs.ICALP.2020.26,
  author =	{Chan, Timothy F. N. and Cooper, Jacob W. and Kouteck\'{y}, Martin and Kr\'{a}l', Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na},
  title =	{{Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{26:1--26:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.26},
  URN =		{urn:nbn:de:0030-drops-124339},
  doi =		{10.4230/LIPIcs.ICALP.2020.26},
  annote =	{Keywords: Matroid algorithms, width parameters, integer programming, fixed parameter tractability, branch-width, branch-depth}
}
Document
Domination Above r-Independence: Does Sparseness Help?

Authors: Carl Einarson and Felix Reidl

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


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
Approximate Counting CSP Seen from the Other Side

Authors: Andrei A. Bulatov and Stanislav Živný

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


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
Colouring H-Free Graphs of Bounded Diameter

Authors: Barnaby Martin, Daniël Paulusma, and Siani Smith

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


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
Recovering Sparse Graphs

Authors: Jakub Gajarský and Daniel Král'

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)


Abstract
We construct a fixed parameter algorithm parameterized by d and k that takes as an input a graph G' obtained from a d-degenerate graph G by complementing on at most k arbitrary subsets of the vertex set of G and outputs a graph H such that G and H agree on all but f(d,k) vertices. Our work is motivated by the first order model checking in graph classes that are first order interpretable in classes of sparse graphs. We derive as a corollary that if G is a graph class with bounded expansion, then the first order model checking is fixed parameter tractable in the class of all graphs that can obtained from a graph G in G by complementing on at most k arbitrary subsets of the vertex set of G; this implies an earlier result that the first order model checking is fixed parameter tractable in graph classes interpretable in classes of graphs with bounded maximum degree.

Cite as

Jakub Gajarský and Daniel Král'. Recovering Sparse Graphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 29:1-29:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{gajarsky_et_al:LIPIcs.MFCS.2018.29,
  author =	{Gajarsk\'{y}, Jakub and Kr\'{a}l', Daniel},
  title =	{{Recovering Sparse Graphs}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.29},
  URN =		{urn:nbn:de:0030-drops-96111},
  doi =		{10.4230/LIPIcs.MFCS.2018.29},
  annote =	{Keywords: model checking, degenerate graphs, interpretations, bounded expansion}
}
Document
Graphic TSP in Cubic Graphs

Authors: Zdenek Dvorák, Daniel Král, and Bojan Mohar

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)


Abstract
We present a polynomial-time 9/7-approximation algorithm for the graphic TSP for cubic graphs, which improves the previously best approximation factor of 1.3 for 2-connected cubic graphs and drops the requirement of 2-connectivity at the same time. To design our algorithm, we prove that every simple 2-connected cubic n-vertex graph contains a spanning closed walk of length at most 9n/7-1, and that such a walk can be found in polynomial time.

Cite as

Zdenek Dvorák, Daniel Král, and Bojan Mohar. Graphic TSP in Cubic Graphs. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 27:1-27:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{dvorak_et_al:LIPIcs.STACS.2017.27,
  author =	{Dvor\'{a}k, Zdenek and Kr\'{a}l, Daniel and Mohar, Bojan},
  title =	{{Graphic TSP in Cubic Graphs}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{27:1--27:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.27},
  URN =		{urn:nbn:de:0030-drops-70068},
  doi =		{10.4230/LIPIcs.STACS.2017.27},
  annote =	{Keywords: Graphic TSP, approximation algorithms, cubic graphs}
}
Document
Towards an Isomorphism Dichotomy for Hereditary Graph Classes

Authors: Pascal Schweitzer

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)


Abstract
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color class size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.

Cite as

Pascal Schweitzer. Towards an Isomorphism Dichotomy for Hereditary Graph Classes. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 689-702, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{schweitzer:LIPIcs.STACS.2015.689,
  author =	{Schweitzer, Pascal},
  title =	{{Towards an Isomorphism Dichotomy for Hereditary Graph Classes}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{689--702},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.689},
  URN =		{urn:nbn:de:0030-drops-49513},
  doi =		{10.4230/LIPIcs.STACS.2015.689},
  annote =	{Keywords: graph isomorphism, modular decomposition, bounded color valence, reductions, forbidden induced subgraphs}
}
Document
Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication

Authors: Hu Fu and Robert Kleinberg

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions f1, f2 and f3, mapping {0,1}^k to {0,1}, are said to be triangle free if there is no x, y in {0,1}^k such that f1(x) = f2(y) = f3(x + y) = 1. This property is known to be strongly testable (Green 2005), but the number of queries needed is upper-bounded only by a tower of twos whose height is polynomial in 1 / epsislon, where epsislon is the distance between the tested function triple and triangle-freeness, i.e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of (1 / epsilon)^2.423 for any one-sided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to (1 / epsilon)^6.619. Interestingly, we prove this by way of a combinatorial construction called uniquely solvable puzzles that was at the heart of Coppersmith and Winograd's renowned matrix multiplication algorithm.

Cite as

Hu Fu and Robert Kleinberg. Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 669-676, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


Copy BibTex To Clipboard

@InProceedings{fu_et_al:LIPIcs.APPROX-RANDOM.2014.669,
  author =	{Fu, Hu and Kleinberg, Robert},
  title =	{{Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{669--676},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.669},
  URN =		{urn:nbn:de:0030-drops-47304},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.669},
  annote =	{Keywords: Property testing, linear invariance, fast matrix multiplication, uniquely solvable puzzles}
}
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