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 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 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(klogk) n ^O(1) time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.