Subset feedback vertex set is fixed-parameter tractable

  • Authors:

  • Marek Cygan;Marcin Pilipczuk;Michał Pilipczuk;Jakub Onufry Wojtaszczyk

  • Affiliations:

  • Institute of Informatics, University of Warsaw, Poland;Institute of Informatics, University of Warsaw, Poland;Institute of Informatics, University of Warsaw, Poland;Google Inc., Cracow, Poland and Institute of Mathematics, University of Warsaw, Poland

  • Venue:
  • ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
  • Year:
  • 2011

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Abstract

The classical FEEDBACK VERTEX SET problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. FEEDBACK VERTEX SET has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixedparameter algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named SUBSET FEEDBACK VERTEX SET (SUBSET-FVS in short) where an instance comes additionally with a set S ⊆ V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00]. The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2knO(1) instances with the size of S bounded by O(k3), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2O(klog k)nO(1) time algorithm solving the SUBSET FEEDBACK VERTEX SET problem, proving that it is indeed fixed-parameter tractable.