A generalization of Nemhauser and Trotter's local optimization theorem
Journal of Computer and System Sciences
Parameterized reductions and algorithms for another vertex cover generalization
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Parameterized reductions and algorithms for a graph editing problem that generalizes vertex cover
Theoretical Computer Science
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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 $(2-\lg\lg n/2\lg n)$-approximation algorithm for general graphs, and a $2k$ kernel for the parameterized Generalized Vertex Cover problem.