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 $d\ge 0$, 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 $\varepsilon >0$, an $O(k^{1+\varepsilon })$-vertex kernel for each $d\ge 2$. 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 $d\ge 3$. 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 $d\ge 0$.

M. Xiao—Supported by NFSC of China under the Grant 61370071.