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Typical behavior of the linear programming method for combinatorial optimization problems: From a statistical-mechanical perspective (1309.6925v1)
Published 26 Sep 2013 in cond-mat.dis-nn, cond-mat.stat-mech, cs.IT, and math.IT
Abstract: 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 Erd\"os-R\'enyi 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.