Enumerating minimal subset feedback vertex sets

Abstract

The Subset Feedback Vertex Set problem takes as input a weighted graph G and a vertex subset S of G, and the task is to find a set of vertices of total minimum weight to be removed from G such that in the remaining graph no cycle contains a vertex of S. This problem is a generalization of two classical NP-complete problems: Feedback Vertex Set and Multiway Cut. We show that it can be solved in time O(1.8638n) for input graphs on n vertices. To the best of our knowledge, no exact algorithm breaking the trivial 2 nnO(1)-time barrier has been known for Subset Feedback Vertex Set, even in the case of unweighted graphs. The mentioned running time is a consequence of the more general main result of this paper: we show that all minimal subset feedback vertex sets of a graph can be enumerated in O(1.8638n) time. © 2011 Springer-Verlag.