A Parameterized Algorithm for Bounded-Degree Vertex Deletion
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-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 \(O^*(3.24^k)\).
KeywordsParameterized algorithms Graph algorithms Bounded-degree vertex deletion Hitting set