Parameterized Max Min Feedback Vertex Set

TL;DR

This paper advances the parameterized complexity understanding of the Max Min Feedback Vertex Set problem, providing improved algorithms based on graph parameters and correcting previous algorithmic claims.

Contribution

It introduces a generalized dynamic programming algorithm for the problem and corrects prior branching algorithms, establishing near-optimal complexity bounds.

Findings

Algorithm of time tw^{O(tw)} n^{O(1)} for graphs with bounded treewidth.

Algorithm of time vc^{O(vc)} n^{O(1)} for graphs with bounded vertex cover.

Corrected and improved the branching algorithm complexity to 9.34^k n^{O(1)}.

Abstract

Given a graph $G$ and an integer $k$, Max Min FVS asks whether there exists a minimal set of vertices of size at least $k$ whose deletion destroys all cycles. We present several results that improve upon the state of the art of the parameterized complexity of this problem with respect to both structural and natural parameters. Using standard DP techniques, we first present an algorithm of time $\textrm{tw}^{O(\textrm{tw})}n^{O(1)}$, significantly generalizing a recent algorithm of Gaikwad et al. of time $\textrm{vc}^{O(\textrm{vc})}n^{O(1)}$, where $\textrm{tw}, \textrm{vc}$ denote the input graph's treewidth and vertex cover respectively. Subsequently, we show that both of these algorithms are essentially optimal, since a $\textrm{vc}^{o(\textrm{vc})}n^{O(1)}$ algorithm would refute the ETH. With respect to the natural parameter $k$, the aforementioned recent work by Gaikwad et al. claimed an FPT branching algorithm with complexity $10^k n^{O(1)}$. We point out that this algorithm is incorrect and present a branching algorithm of complexity $9.34^k n^{O(1)}$.