On Group Feedback Vertex Set Parameterized by the Size of the Cutset

Abstract

We study parameterized complexity of a generalization of the classical Feedback Vertex Set problem, namely the Group Feedback Vertex Set problem: we are given a graph $$G$$ with edges labeled with group elements, and the goal is to compute the smallest set of vertices that hits all cycles of $$G$$ that evaluate to a non-null element of the group. This problem generalizes not only Feedback Vertex Set, but also Subset Feedback Vertex Set, Multiway Cut and Odd Cycle Transversal . Completing the results of Guillemot (Discrete Optim 8(1):61–71, 2011), we provide a fixed-parameter algorithm for the parameterization by the size of the cutset only. Our algorithm works even if the group is given as a blackbox performing group operations.