  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.
