On a Generalization of Nemhauser and Trotter’s Local Optimization Theorem

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 \(d\ge 0\).