An improved parameterized algorithm for the p-cluster vertex deletion problem

Abstract

In the p-Cluster Vertex Deletion problem, we are given a graph $G=(V,E)$ and two parameters k and p, and the goal is to determine if there exists a subset X of at most k vertices such that the removal of X results in a graph consisting of exactly p disjoint maximal cliques. Let $r=p/k$. In this paper, we design a branching algorithm with time complexity $O(\alpha ^k+|V||E|)$, where $\alpha$ depends on r and has a rough upper bound $\min \{1.618^{1+r},2\}$. With a more precise analysis, we show that $\alpha =1.28\cdot 3.57^{r}$ for $r\le 0.219$; $\alpha =(1-r)^{r-1}r^{-r}$ for $0.219< r<1/2$; and $\alpha =2$ for $r\ge 1/2$, respectively. Our algorithm also works with the same time complexity for the variant that the number of clusters is at most p. Our result improves the previous best time complexity $O^*(1.84^{p+k})$ and implies that for fixed p the problem can be solved as efficiently as Vertex Cover.