Improvement of Nemhauser-Trotter Theorem and its applications in parametrized complexity

Abstract

We improve on the classical Nemhauser-Trotter Theorem, which is a key tool for the MINIMUM (WEIGHTED) VERTEX COVER problem in the design of both, approximation algorithms and exact fixedparameter algorithms. Namely, we provide in polynomial time for a graph G with vertex weights w : V -→ (0, ∞) a partition of V into three subsets V0, V1, V1/2, with no edges between V0 and V1/2 or within V0, such that the size of a minimum vertex cover for the graph induced by V1/2 is at least 1/2w(V1/2), and every minimum vertex cover C for (G,w) satisfies V1 ⊆ C ⊆ V1 UV1/2. We also demonstrate one of possible applications of this strengthening of NT-Theorem for fixed parameter tractable problems related to MIN-VC: for an integer parameter k to find all minimum vertex covers of size at most k, or to find a minimum vertex cover of size at most k under some additional constraints. © Springer-Verlag Berlin Heidelberg 2004.