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Documents authored by Toufar, Tomáš

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Toufar, Tomáš

Document
DOI: 10.4230/LIPIcs.MFCS.2019.33
Parameterized Complexity of Fair Vertex Evaluation Problems

Authors: Dušan Knop, Tomáš Masařík, and Tomáš Toufar

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

Abstract
A prototypical graph problem is centered around a graph-theoretic property for a set of vertices and a solution to it is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of a certain quality, where we use size as a quality measure. In this work, we are changing the measure to the fair measure (cf. Lin and Sahni [Li-Shin Lin and Sartaj Sahni, 1989]). The fair measure of a set of vertices S is (at most) k if the number of neighbors in the set S of any vertex (in the input graph) does not exceed k. One possible way to study graph problems is by defining the property in a certain logic. For a given objective, an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. More formally, we study the {MSO} Fair Vertex Evaluation, where the graph-theoretic property is described by an {MSO} formula. In the presented paper we show that there is an FPT algorithm for the {MSO} Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph and the size of the formula. One may define an extended variant of {MSO} Fair Vertex Evaluation for formulas with l free variables; here we measure a maximum number of neighbors in each of the l sets. However, such variant is {W[1]}-hard for parameter l even on graphs with twin cover one. Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is {W[1]}-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width.
Cite as

Dušan Knop, Tomáš Masařík, and Tomáš Toufar. Parameterized Complexity of Fair Vertex Evaluation Problems. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{knop_et_al:LIPIcs.MFCS.2019.33,
  author =	{Knop, Du\v{s}an and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s}},
  title =	{{Parameterized Complexity of Fair Vertex Evaluation Problems}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{33:1--33:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.33},
  URN =		{urn:nbn:de:0030-drops-109775},
  doi =		{10.4230/LIPIcs.MFCS.2019.33},
  annote =	{Keywords: Fair objective, metatheorem, fair vertex cover, twin cover, modular width}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.18
Target Set Selection in Dense Graph Classes

Authors: Pavel Dvorák, Dusan Knop, and Tomás Toufar

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
In this paper we study the Target Set Selection problem from a parameterized complexity perspective. Here for a given graph and a threshold for each vertex the task is to find a set of vertices (called a target set) to activate at the beginning which activates the whole graph during the following iterative process. A vertex outside the active set becomes active if the number of so far activated vertices in its neighborhood is at least its threshold. We give two parameterized algorithms for a special case where each vertex has the threshold set to the half of its neighbors (the so called Majority Target Set Selection problem) for parameterizations by the neighborhood diversity and the twin cover number of the input graph. We complement these results from the negative side. We give a hardness proof for the Majority Target Set Selection problem when parameterized by (a restriction of) the modular-width - a natural generalization of both previous structural parameters. We show that the Target Set Selection problem parameterized by the neighborhood diversity when there is no restriction on the thresholds is W[1]-hard.
Cite as

Pavel Dvorák, Dusan Knop, and Tomás Toufar. Target Set Selection in Dense Graph Classes. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dvorak_et_al:LIPIcs.ISAAC.2018.18,
  author =	{Dvor\'{a}k, Pavel and Knop, Dusan and Toufar, Tom\'{a}s},
  title =	{{Target Set Selection in Dense Graph Classes}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{18:1--18:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.18},
  URN =		{urn:nbn:de:0030-drops-99666},
  doi =		{10.4230/LIPIcs.ISAAC.2018.18},
  annote =	{Keywords: parameterized complexity, target set selection, dense graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.26
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Authors: Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.
Cite as

Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý. Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2018.26,
  author =	{Dvor\'{a}k, Pavel and Feldmann, Andreas Emil and Knop, Du\v{s}an and Masar{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s} and Vesel\'{y}, Pavel},
  title =	{{Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf 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.2018.26},
  URN =		{urn:nbn:de:0030-drops-85158},
  doi =		{10.4230/LIPIcs.STACS.2018.26},
  annote =	{Keywords: Steiner Tree, Steiner Forest, Approximation Algorithms, Parameterized Algorithms, Lossy Kernelization}
}

Toufar, Tomás

Document
DOI: 10.4230/LIPIcs.MFCS.2019.33
Parameterized Complexity of Fair Vertex Evaluation Problems

Authors: Dušan Knop, Tomáš Masařík, and Tomáš Toufar

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

Abstract
A prototypical graph problem is centered around a graph-theoretic property for a set of vertices and a solution to it is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of a certain quality, where we use size as a quality measure. In this work, we are changing the measure to the fair measure (cf. Lin and Sahni [Li-Shin Lin and Sartaj Sahni, 1989]). The fair measure of a set of vertices S is (at most) k if the number of neighbors in the set S of any vertex (in the input graph) does not exceed k. One possible way to study graph problems is by defining the property in a certain logic. For a given objective, an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. More formally, we study the {MSO} Fair Vertex Evaluation, where the graph-theoretic property is described by an {MSO} formula. In the presented paper we show that there is an FPT algorithm for the {MSO} Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph and the size of the formula. One may define an extended variant of {MSO} Fair Vertex Evaluation for formulas with l free variables; here we measure a maximum number of neighbors in each of the l sets. However, such variant is {W[1]}-hard for parameter l even on graphs with twin cover one. Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is {W[1]}-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width.
Cite as

Dušan Knop, Tomáš Masařík, and Tomáš Toufar. Parameterized Complexity of Fair Vertex Evaluation Problems. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{knop_et_al:LIPIcs.MFCS.2019.33,
  author =	{Knop, Du\v{s}an and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s}},
  title =	{{Parameterized Complexity of Fair Vertex Evaluation Problems}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{33:1--33:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.33},
  URN =		{urn:nbn:de:0030-drops-109775},
  doi =		{10.4230/LIPIcs.MFCS.2019.33},
  annote =	{Keywords: Fair objective, metatheorem, fair vertex cover, twin cover, modular width}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.18
Target Set Selection in Dense Graph Classes

Authors: Pavel Dvorák, Dusan Knop, and Tomás Toufar

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
In this paper we study the Target Set Selection problem from a parameterized complexity perspective. Here for a given graph and a threshold for each vertex the task is to find a set of vertices (called a target set) to activate at the beginning which activates the whole graph during the following iterative process. A vertex outside the active set becomes active if the number of so far activated vertices in its neighborhood is at least its threshold. We give two parameterized algorithms for a special case where each vertex has the threshold set to the half of its neighbors (the so called Majority Target Set Selection problem) for parameterizations by the neighborhood diversity and the twin cover number of the input graph. We complement these results from the negative side. We give a hardness proof for the Majority Target Set Selection problem when parameterized by (a restriction of) the modular-width - a natural generalization of both previous structural parameters. We show that the Target Set Selection problem parameterized by the neighborhood diversity when there is no restriction on the thresholds is W[1]-hard.
Cite as

Pavel Dvorák, Dusan Knop, and Tomás Toufar. Target Set Selection in Dense Graph Classes. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{dvorak_et_al:LIPIcs.ISAAC.2018.18,
  author =	{Dvor\'{a}k, Pavel and Knop, Dusan and Toufar, Tom\'{a}s},
  title =	{{Target Set Selection in Dense Graph Classes}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{18:1--18:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.18},
  URN =		{urn:nbn:de:0030-drops-99666},
  doi =		{10.4230/LIPIcs.ISAAC.2018.18},
  annote =	{Keywords: parameterized complexity, target set selection, dense graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.26
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Authors: Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.
Cite as

Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý. Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{dvorak_et_al:LIPIcs.STACS.2018.26,
  author =	{Dvor\'{a}k, Pavel and Feldmann, Andreas Emil and Knop, Du\v{s}an and Masar{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s} and Vesel\'{y}, Pavel},
  title =	{{Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf 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.2018.26},
  URN =		{urn:nbn:de:0030-drops-85158},
  doi =		{10.4230/LIPIcs.STACS.2018.26},
  annote =	{Keywords: Steiner Tree, Steiner Forest, Approximation Algorithms, Parameterized Algorithms, Lossy Kernelization}
}
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