An Extension of the Nemhauser--Trotter Theorem to Generalized Vertex Cover with Applications

The Nemhauser–Trotter theorem provides an algorithm which is frequently used as a subroutine in approximation algorithms for the classical Vertex Cover problem. In this paper we present an extension of this theorem so it fits a more general variant of Vertex Cover, namely, the Generalized Vertex Cover problem, where edges are allowed not to be covered at a certain predetermined penalty. We show that many applications of the original Nemhauser–Trotter theorem can be applied using our extension to Generalized Vertex Cover. These applications include a (2−2/d)-approximation algorithm for graphs of bounded degree d, a polynomial-time approximation scheme (PTAS) for planar graphs, a (2− lg lgn/2 lgn)-approximation algorithm for general graphs, and a 2k kernel for the parameterized Generalized Vertex Cover problem.