Long-range frustration in a spin-glass model of the vertex-cover problem

In a spin-glass system on a random graph, some vertices have their spins changing among different configurations of a ground-state domain. Long-range frustrations may exist among these unfrozen vertices in the sense that certain combinations of spin values for these vertices may never appear in any configuration of this domain. We present a mean field theory to tackle such long-range frustrations and apply it to the NP-hard minimum vertex-cover (hard-core gas condensation) problem. Our analytical results on the ground-state energy density and on the fraction of frozen vertices are in good agreement with known numerical and mathematical results.