On group feedback vertex set parameterized by the size of the cutset

TL;DR

This paper investigates the parameterized complexity of the Group Feedback Vertex Set problem, a generalization of classical problems, and provides a fixed-parameter algorithm based on the size of the cutset, even with an oracle group.

Contribution

It introduces a fixed-parameter algorithm for the Group Feedback Vertex Set problem parameterized by cutset size, extending previous results and handling groups via an oracle.

Findings

Provides a fixed-parameter algorithm for the problem

Works with groups given as polynomial-time oracle

Generalizes several classical problems

Abstract

We study the parameterized complexity of a robust 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 [Discr. Opt. 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 polynomial-time oracle.