A Parameterized Algorithm for Bounded-Degree Vertex Deletion

Abstract

The d-bounded-degree vertex deletion problem, to delete at most k vertices in a given graph to make the maximum degree of the remaining graph at most d, finds applications in computational biology, social network analysis and some others. It can be regarded as a special case of the $(d+2)$-hitting set problem and generates the famous vertex cover problem. The d-bounded-degree vertex deletion problem is NP-hard for each fixed $d\ge 0$. In terms of parameterized complexity, the problem parameterized by k is W[2]-hard for unbounded d and fixed-parameter tractable for each fixed $d\ge 0$. Previously, (randomized) parameterized algorithms for this problem with running time bound $O^*((d+1)^k)$ are only known for $d\le 2$. In this paper, we give a uniform parameterized algorithm deterministically solving this problem in $O^*((d+1)^k)$ time for each $d\ge 3$. Note that it is an open problem whether the $d'$-hitting set problem can be solved in $O^*((d'-1)^k)$ time for $d'\ge 3$. Our result answers this challenging open problem affirmatively for a special case. Furthermore, our algorithm also gets a running time bound of $O^*(3.0645^k)$ for the case that $d=2$, improving the previous deterministic bound of $O^*(3.24^k)$.

M. Xiao—Supported by NFSC of China under the Grant 61370071 and Fundamental Research Funds for the Central Universities under the Grant ZYGX2015J057.