A generalization of Nemhauser and Trotterʼs local optimization theorem

Abstract

The Nemhauser–Trotter local optimization theorem applies to the NP-hard Vertex Cover problem and has applications in approximation as well as parameterized algorithmics. We generalize Nemhauser and Trotterʼs result to vertex deletion problems, introducing a novel algorithmic strategy based on purely combinatorial arguments (not referring to linear programming as the Nemhauser–Trotter result originally did). The essence of our strategy can be understood as a doubly iterative process of cutting away “easy parts” of the input instance, finally leaving a “hard core” whose size is (almost) linearly related to the cardinality of the solution set. We exhibit our approach using a generalization of Vertex Cover, called Bounded-Degree Vertex Deletion. For some fixed d ⩾ 0 , Bounded-Degree Vertex Deletion asks to delete at most k vertices from a graph in order to transform it into a graph with maximum vertex degree at most d. Vertex Cover is the special case of d = 0 . Our generalization of the Nemhauser–Trotter-Theorem implies that Bounded-Degree Vertex Deletion, parameterized by k, admits an O ( k ) -vertex problem kernel for d ⩽ 1 and, for any ϵ > 0 , an O ( k 1 + ϵ ) -vertex problem kernel for d ⩾ 2 . Finally, we provide a W[2]-completeness result for Bounded-Degree Vertex Deletion in case of unbounded d-values.