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 $\text{G}{}_{\mathrm{N,cN}}$ having $\text{N}$ vertices and $\text{cN}$ edges. Exact numerical results are obtained by a branch-and-bound algorithm. It is found that a transition in the coverability at a $\text{c}$-dependent threshold $\text{x=x}{}_{\mathrm{c}}\text{(c)}$ appears, where $\text{xN}$ 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 $\text{xN}$ vertices or not. For small edge concentrations $\text{c⪡0.5}$, 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 $\text{x}{}_{\mathrm{c}}\text{(c)}$ which was known previously. The main part of the paper contains an application of the replica method. Within the replica symmetric ansatz the threshold $\text{x}{}_{\mathrm{c}}\text{(c)}$ and various statistical properties of minimal vertex covers can be calculated. For $\text{c<e/2}$ the results show an excellent agreement with the numerical findings. At average vertex degree $\text{2c=e}$, an instability of the simple replica symmetric solution occurs.