Typical behavior of the linear programming problem (LP) is studied as a relaxation of the minimum vertex cover problem, which is a type of the integer programming problem (IP). To deal with the LP and IP by statistical mechanics, a lattice-gas model on the Erdös-Rényi random graphs is analyzed by a replica method. It is found that the LP optimal solution is typically equal to that of the IP below the critical average degree c*=e in the thermodynamic limit. The critical threshold for LP=IP is beyond a mathematical result, c=1, and coincides with the replica-symmetry-breaking threshold of the IP.