An improved FPT algorithm for Independent Feedback Vertex Set

TL;DR

This paper introduces an improved fixed-parameter tractable (FPT) algorithm for the Independent Feedback Vertex Set problem, significantly reducing the exponential time complexity compared to previous algorithms.

Contribution

The paper presents a new FPT algorithm with a faster exponential time bound for the Independent Feedback Vertex Set problem, using a refined measure in the branching process.

Findings

Achieved an $O^*((1+{ ext{golden ratio}^2})^{k})$-time algorithm

Improved the exponential factor from 4.1481 to approximately 1.619^k

Simplified analysis through a refined measure in the branching process

Abstract

We study the Independent Feedback Vertex Set problem - a variant of the classic Feedback Vertex Set problem where, given a graph $G$ and an integer $k$, the problem is to decide whether there exists a vertex set $S\subseteq V(G)$ such that $G\setminus S$ is a forest and $S$ is an independent set of size at most $k$. We present an $O^\ast((1+\varphi^{2})^{k})$-time FPT algorithm for this problem, where $\varphi<1.619$ is the golden ratio, improving the previous fastest $O^\ast(4.1481^{k})$-time algorithm given by Agrawal et al [IPEC 2016]. The exponential factor in our time complexity bound matches the fastest deterministic FPT algorithm for the classic Feedback Vertex Set problem. On the technical side, the main novelty is a refined measure of an input instance in a branching process, that allows for a simpler and more concise description and analysis of the algorithm.