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DOI: 10.4230/LIPIcs.STACS.2016.7
URN: urn:nbn:de:0030-drops-57084
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Agrawal, Akanksha ; Lokshtanov, Daniel ; Mouawad, Amer E. ; Saurabh, Saket

Simultaneous Feedback Vertex Set: A Parameterized Perspective

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For a family of graphs F, a graph G, and a positive integer k, the F-DELETION problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-DELETION generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph G = (V, cup_{i=1}^{alpha} E_{i}), where the edge set of G is partitioned into alpha color classes, is called an alpha-edge-colored graph. A natural extension of the F-DELETION problem to edge-colored graphs is the alpha-SIMULTANEOUS F-DELETION problem. In the latter problem, we are given an alpha-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G_i\S, where G_i = (V, E_i) and 1 <= i <= alpha, is in F. In this work, we study alpha-SIMULTANEOUS F-DELETION for F being the family of forests. In other words, we focus on the alpha-SIMULTANEOUS FEEDBACK VERTEX SET (alpha-SIMFVS) problem. Algorithmically, we show that, like its classical counterpart, alpha-SIMFVS parameterized by k is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant alpha. In particular, we give an algorithm running in 2^{O(alpha * k)} * n^{O(1)} time and a kernel with O(alpha * k^{3(alpha + 1)}) vertices. The running time of our algorithm implies that alpha-SIMFVS is FPT even when alpha in o(log(n)). We complement this positive result by showing that for alpha in O(log(n)), where n is the number of vertices in the input graph, alpha-SIMFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).

BibTeX - Entry

  author =	{Akanksha Agrawal and Daniel Lokshtanov and Amer E. Mouawad and Saket Saurabh},
  title =	{{Simultaneous Feedback Vertex Set: A Parameterized Perspective}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Nicolas Ollinger and Heribert Vollmer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-57084},
  doi =		{10.4230/LIPIcs.STACS.2016.7},
  annote =	{Keywords: parameterized complexity ,feedback vertex set, kernel, edge-colored graphs}

Keywords: parameterized complexity ,feedback vertex set, kernel, edge-colored graphs
Collection: 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Issue Date: 2016
Date of publication: 16.02.2016

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