FPT Algorithms for Connected Feedback Vertex Set

TL;DR

This paper introduces fixed-parameter tractable algorithms for the Connected Feedback Vertex Set problem, providing efficient solutions for general and minor-excluded graphs, and develops a new parameterized algorithm for Group Steiner Tree.

Contribution

It presents the first fixed-parameter algorithms for CFVS on general and minor-free graphs and introduces a novel parameterized algorithm for Group Steiner Tree.

Findings

CFVS can be solved in $O(2^{O(k)}n^{O(1)})$ time on general graphs.

CFVS can be solved in $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ time on H-minor-free graphs.

A new parameterized algorithm for Group Steiner Tree is developed, of independent interest.

Abstract

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists a subset F of V, of size at most k, such that G[V F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time $O(2^{O(k)}n^{O(1)})$ on general graphs and in time $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses as subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it will be useful for obtaining parameterized algorithms for other connectivity problems.