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DOI: 10.4230/LIPIcs.MFCS.2022.29

On Kernels for d-Path Vertex Cover

Authors: Radovan Červený, Pratibha Choudhary, and Ondřej Suchý

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

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.

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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}
}

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DOI: 10.4230/LIPIcs.MFCS.2019.32

Faster FPT Algorithm for 5-Path Vertex Cover

Authors: Radovan Červený and Ondřej Suchý

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

Abstract

The problem of d-Path Vertex Cover, d-PVC lies in determining a subset F of vertices of a given graph G=(V,E) such that G \ F does not contain a path on d vertices. The paths we aim to cover need not to be induced. It is known that the d-PVC problem is NP-complete for any d >= 2. When parameterized by the size of the solution k, 5-PVC has direct trivial algorithm with O(5^kn^{O(1)}) running time and, since d-PVC is a special case of d-Hitting Set, an algorithm running in O(4.0755^kn^{O(1)}) time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in O(4^kn^{O(1)}) time.

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Radovan Červený and Ondřej Suchý. Faster FPT Algorithm for 5-Path Vertex Cover. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cerveny_et_al:LIPIcs.MFCS.2019.32,
  author =	{\v{C}erven\'{y}, Radovan and Such\'{y}, Ond\v{r}ej},
  title =	{{Faster FPT Algorithm for 5-Path Vertex Cover}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{32:1--32:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.32},
  URN =		{urn:nbn:de:0030-drops-109761},
  doi =		{10.4230/LIPIcs.MFCS.2019.32},
  annote =	{Keywords: graph algorithms, Hitting Set, iterative compression, parameterized complexity, d-Path Vertex Cover}
}

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Documents authored by Červený, Radovan

On Kernels for d-Path Vertex Cover

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Faster FPT Algorithm for 5-Path Vertex Cover

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