On a generalization of Nemhauser and trotter’s local optimization theorem

Abstract

The Nemhauser and Trotter’s theorem applies to the famous Vertex Cover problem and can obtain a 2-approximation solution and a problem kernel of 2k vertices. This theorem is a famous theorem in combinatorial optimization and has been extensively studied. One way to generalize this theorem is to extend the result to the Bounded-Degree Vertex Deletion problem. For a fixed integer d ≥ 0, Bounded- Degree Vertex Deletion asks to delete at most k vertices of the input graph to make the maximum degree of the remaining graph at most d. Vertex Cover is a special case that d = 0. Fellows, Guo, Moser and Niedermeier proved a generalized theorem that implies an O(k)-vertex kernel for Bounded-Degree Vertex Deletion for d = 0 and 1, and for any ε > 0, an O(k1+ε)-vertex kernel for each d ≥ 2. In fact, it is still left as an open problem whether Bounded-Degree Vertex Deletion parameterized by k admits a linear-vertex kernel for each d ≥ 3. In this paper, we refine the generalized Nemhauser and Trotter’s theorem. Our result implies a linear-vertex kernel for Bounded-Degree Vertex Deletion parameterized by k for each d ≥ 0.