Clustering analysis of the ground-state structure of the vertex-cover problem

Abstract

Vertex cover is one of the classical NP-complete problems in theoretical computer science. A vertex cover of a graph is a subset of vertices such that for each edge at least one of the two endpoints is contained in the subset. When studied on Erdos-Renyi random graphs (with connectivity c) one observes a threshold behavior: In the thermodynamic limit the size of the minimal vertex cover is independent of the specific graph. Recent analytical studies show that on the phase boundary, for small connectivities c