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arXiv · 0901.2635

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

Pan Zhang·Ying Zeng·Haijun Zhou
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Abstract

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.

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BibTeXRIS

Pan Zhang, Ying Zeng, Haijun Zhou. 2009-01-17. Stability analysis on the finite-temperature replica-symmetric and first-step replica-symmetry-broken cavity solutions of the random vertex cover problem. https://doi.org/10.1103/physreve.80.021122

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