Stability analysis on the finite-temperature replica-symmetric and first-step replica-symmetry-broken cavity solutions of the random vertex cover problem

The vertex cover problem is a prototypical hard combinatorial optimization problem. It was studied in recent years by physicists using the cavity method of statistical mechanics. In this paper, the stability of the finite-temperature replica-symmetric (RS) and the first-step replica-symmetry-broken (1RSB) cavity solutions of the vertex cover problem on random regular graphs of finite vertex degree K are analyzed by population dynamics simulations. We found that (1) the lowest temperature for the RS solution to be stable, T(RS)(K), is not a monotonic function of K; (2) at relatively large connectivity K and temperature T slightly below the dynamic transition temperature T(d)(K), the 1RSB solutions with small but non-negative complexity values are stable, and (3) the dynamical transition temperature T(d) and Kauzmann temperature T(K) is equal to each other. Similar results are obtained on random Poissonian graphs.