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 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, V 1 2 , with no edges between V0 and V 1 2 or within V0, such that the size of a minimum vertex cover for the graph induced by V 1 2 is at least 1 2 w(V 1 2 ), and every minimum vertex cover C for (G, w) satisfies V1 ⊆ C ⊆ V1 ∪ V 1 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.