Maximum Minimal Feedback Vertex Set: A Parameterized Perspective

TL;DR

This paper investigates the Max Min Feedback Vertex Set problem from a parameterized complexity perspective, providing fixed parameter tractable algorithms, complexity bounds, and approximation schemes based on graph parameters.

Contribution

It introduces new FPT algorithms, complexity results, and an approximation scheme for the Max Min FVS problem, expanding understanding of its computational complexity.

Findings

Developed an FPT algorithm with time $10^k n^{O(1)}$.

Established a parameterized complexity bound based on vertex cover number.

Designed an FPT-approximation scheme with $(1- ext{epsilon})$-approximation ratio.

Abstract

In this paper we study a maximization version of the classical Feedback Vertex Set (FVS) problem, namely, the Max Min FVS problem, in the realm of parameterized complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal feedback vertex set of size at least $k$. We obtain following results for Max Min FVS. 1) We first design a fixed parameter tractable (FPT) algorithm for Max Min FVS running in time $10^kn^{\mathcal{O}(1)}$. 2) Next, we consider the problem parameterized by the vertex cover number of the input graph (denoted by $\mathsf{vc}(G)$), and design an algorithm with running time $2^{\mathcal{O}(\mathsf{vc}(G)\log \mathsf{vc}(G))}n^{\mathcal{O}(1)}$. We complement this result by showing that the problem parameterized by $\mathsf{vc}(G)$ does not admit a polynomial compression unless coNP $\subseteq$ NP/poly. 3) Finally, we give an FPT-approximation scheme (fpt-AS) parameterized by $\mathsf{vc}(G)$. That is, we design an algorithm that for every $\epsilon >0$, runs in time $2^{\mathcal{O}\left(\frac{\mathsf{vc}(G)}{\epsilon}\right)} n^{\mathcal{O}(1)}$ and returns a minimal feedback vertex set of size at least $(1-\epsilon){\sf opt}$.