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We study the vertex cover problem on finite connectivity random graphs byzero-temperature cavity method. The minimum vertex cover corresponds to theground state(s) of a proposed Ising spin model. When the connectivityc>e=2.718282, there is no state for this system as the reweighting parameter y,which takes a similar role as the inverse temperature \beta in conventionalstatistical physics, approaches infinity; consequently the ground state energyis obtained at a finite value of y when the free energy function attains itsmaximum value. The minimum vertex cover size at given c is estimated usingpopulation dynamics and compared with known rigorous bounds and numericalresults. The backbone size is also calculated.Comment: 7 pages (including 3 figures and 1 table), REVTeX4 forma

Vertex cover problem studied by cavity method: Analytics and population dynamics