Subset Feedback Vertex Set Is Fixed-Parameter Tractable

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 fixed-parameter 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 VER-TEX 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 2 k n O(1) instances with the size of S bounded by O(k 3 ), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2 O(k log k) n O(1) time algorithm solving the SUBSET FEEDBACK VERTEX SET problem, proving that it is indeed fixed-parameter tractable.

Fixed-parameter tractability for the subset feedback set problem and the S-cycle packing problem

“…Note that, in 2010, a FPT algorithm for the subset feedback set problem is also given in [6] independently.…”

Subset Feedback Vertex Set in Chordal and Split Graphs

“…Other examples include ODD CYCLE TRANSVERSAL, DIRECTED FEEDBACK VERTEX SET and VERTEX COVER (VC). These problems have been particularly well studied in parameterized complexity [6,4,19,21,24,22].…”

“…This leads to the subset variant of classic covering problems; typical examples include SUBSET FEEDBACK VERTEX SET (SUBSET-FVS), SUBSET DIRECTED FEEDBACK VERTEX SET and SUBSET ODD CYCLE TRANSVERSAL. These three problems have received considerable attention and they have all been shown to be fixed-parameter tractable (FPT) with k as the parameter [6,4,19].…”

“…The question whether the SUBSET-FVS problem is fixed parameter tractable (FPT) when parameterized by the solution size was posed independently by Kawarabayashi and the third author in 2009. Cygan et al [6] and Kawarabayashi and Kobayashi [19] independently answered this question positively in 2011. Wahlström [24] gave the first parameterized algorithm where the dependence on k is 2 O(k) .…”

“…Our algorithm, in fact, solves SUBSET-FVS on the more general class of chordal graphs, also in O * (2 k ) time. 6 The O () notation hides polynomial factors.…”