An improved FPT algorithm for Independent Feedback Vertex Set

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 $S\subseteq V(G)$suchthat$G\setminus S$isaforestand$S$ is an independent set of size at most $k$. We present an $O^{\ast }((1+{\phi }^2{)}^k)$-time FPT algorithm for this problem, where $\phi <1.619$ 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.