Improvement of Nemhauser-Trotter Theorem and Its Applications in Parametrized Complexity

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 fixed-parameter algorithms. Namely, we provide in polynomial time for a graph G with vertex weights w : V →〈0, ∞ 〉 a partition of V into three subsets V 0, V 1, \(V_{\frac{1}{2}}\), with no edges between V 0 and \(V_{\frac{1}{2}}\) or within V 0, such that the size of a minimum vertex cover for the graph induced by \(V_{\frac{1}{2}}\) is at least \(\frac{1}{2}w(V_{\frac{1}{2}})\), and every minimum vertex cover C for (G, w) satisfies \(V_1 \subseteq C \subseteq V_1 \cup V_{\frac{1}{2}}\).