Statistical mechanics perspective on the phase transition in vertex covering of finite-connectivity random graphs

Abstract

The vertex-cover problem is studied for random graphs G N,cN having N vertices and cN edges. Exact numerical results are obtained by a branch-and-bound algorithm. It is found that a transition in the coverability at a c-dependent threshold x=x c(c) appears, where xN is the cardinality of the vertex cover. This transition coincides with a sharp peak of the typical numerical effort, which is needed to decide whether there exists a cover with xN vertices or not. For small edge concentrations c⪡0.5, a cluster expansion is performed, giving very accurate results in this regime. These results are extended using methods developed in statistical physics. The so-called annealed approximation reproduces a rigorous bound on x c(c) which was known previously. The main part of the paper contains an application of the replica method. Within the replica symmetric ansatz the threshold x c(c) and various statistical properties of minimal vertex covers can be calculated. For c<e/2 the results show an excellent agreement with the numerical findings. At average vertex degree 2c=e, an instability of the simple replica symmetric solution occurs.