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 $K$, and (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. Similar results are obtained on random Poissonian graphs.