Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Jaffke, Lars; Kwon, O-joung; Telle, Jan Arne http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-85348
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A Unified Polynomial-Time Algorithm for Feedback Vertex Set on Graphs of Bounded Mim-Width

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Abstract

We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an n^{O(w)}-time algorithm that solves Feedback Vertex Set. This provides a unified algorithm for many well-known classes, such as Interval graphs and Permutation graphs, and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular k-Trapezoid graphs for fixed k. In all these classes the decomposition is computable in polynomial time, as shown by Belmonte and Vatshelle [Theor. Comput. Sci. 2013]. We show that powers of graphs of tree-width w-1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results extensively provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1. Given a tree decomposition of width w-1, a path decomposition of width w, or a clique-width w-expression of a graph G, one can for any value of k find a mim-width decomposition of its k-power in polynomial time, and apply our algorithm to solve Feedback Vertex Set on the k-power in time n^{O(w)}. In contrast to Feedback Vertex Set, we show that Hamiltonian Cycle is NP-complete even on graphs of linear mim-width 1, which further hints at the expressive power of the mim-width parameter.

BibTeX - Entry

@InProceedings{jaffke_et_al:LIPIcs:2018:8534,
  author =	{Lars Jaffke and O-joung Kwon and Jan Arne Telle},
  title =	{{A Unified Polynomial-Time Algorithm for Feedback Vertex Set on Graphs of Bounded Mim-Width}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{42:1--42:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Rolf Niedermeier and Brigitte Vall{\'e}e},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8534},
  URN =		{urn:nbn:de:0030-drops-85348},
  doi =		{10.4230/LIPIcs.STACS.2018.42},
  annote =	{Keywords: graph width parameters, graph classes, feedback vertex set, leaf powers}
}

Keywords: graph width parameters, graph classes, feedback vertex set, leaf powers
Seminar: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
Issue date: 2018
Date of publication: 2018


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