A Parameterized Algorithm for Bounded-Degree Vertex Deletion

TL;DR

This paper presents a deterministic parameterized algorithm for the d-bounded-degree vertex deletion problem, achieving optimal or improved running times for fixed d, thus solving an open problem in the field.

Contribution

It introduces a uniform deterministic algorithm with optimal running time for all fixed d, including a new bound for d=2, advancing the understanding of the problem's complexity.

Findings

Deterministic algorithm for all fixed d with O^*((d+1)^k) time.

Improved running time for d=2 case to O^*(3.0645^k).

Answers an open problem on hitting set problem complexity for fixed d.

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 $d\geq 0$. In terms of parameterized complexity, the problem parameterized by $k$ is W[2]-hard for unbounded $d$ and fixed-parameter tractable for each fixed $d\geq 0$. Previously, (randomized) parameterized algorithms for this problem with running time bound $O^*((d+1)^k)$ are only known for $d\leq2$. In this paper, we give a uniform parameterized algorithm deterministically solving this problem in $O^*((d+1)^k)$ time for each $d\geq 3$. Note that it is an open problem whether the $d'$-hitting set problem can be solved in $O^*((d'-1)^k)$ time for $d'\geq 3$. 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 $O^*(3.24^k)$.