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DOI: 10.4230/LIPIcs.ESA.2016.59
URN: urn:nbn:de:0030-drops-64066
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Kratsch, Stefan

A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter

LIPIcs-ESA-2016-59.pdf (0.5 MB)


In the Vertex Cover problem we are given a graph G=(V,E) and an integer k and have to determine whether there is a set X subseteq V of size at most k such that each edge in E has at least one endpoint in X. The problem can be easily solved in time O^*(2^k), making it fixed-parameter tractable (FPT) with respect to k. While the fastest known algorithm takes only time O^*(1.2738^k), much stronger improvements have been obtained by studying parameters that are smaller than k. Apart from treewidth-related results, the arguably best algorithm for Vertex Cover runs in time O^*(2.3146^p), where p = k - LP(G) is only the excess of the solution size k over the best fractional vertex cover [Lokshtanov et al., TALG 2014]. Since p <= k but k cannot be bounded in terms of p alone, this strictly increases the range of tractable instances.

Recently, Garg and Philip (SODA 2016) greatly contributed to understanding the parameterized complexity of the Vertex Cover problem. They prove that 2LP(G) - MM(G) is a lower bound for the vertex cover size of G, where MM(G) is the size of a largest matching of G, and proceed to study parameter l = k - (2LP(G)-MM(G)). They give an algorithm of running time O^*(3^l), proving that Vertex Cover is FPT in l. It can be easily observed that l <= p whereas p cannot be bounded in terms of l alone. We complement the work of Garg and Philip by proving that Vertex Cover admits a randomized polynomial kernelization in terms of l, i.e., an efficient preprocessing to size polynomial in l. This improves over parameter p = k - LP(G) for which this was previously known [Kratsch and Wahlström, FOCS 2012].

BibTeX - Entry

  author =	{Stefan Kratsch},
  title =	{{A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{59:1--59:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Piotr Sankowski and Christos Zaroliagis},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-64066},
  doi =		{10.4230/LIPIcs.ESA.2016.59},
  annote =	{Keywords: Vertex cover, parameterized complexity, kernelization}

Keywords: Vertex cover, parameterized complexity, kernelization
Collection: 24th Annual European Symposium on Algorithms (ESA 2016)
Issue Date: 2016
Date of publication: 18.08.2016

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