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arXiv · 1309.6925

Typical behavior of the linear programming method for combinatorial optimization problems: From a statistical-mechanical perspective

Satoshi Takabe·Koji Hukushima
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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 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.

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Satoshi Takabe, Koji Hukushima. 2013-09-26. Typical behavior of the linear programming method for combinatorial optimization problems: From a statistical-mechanical perspective. https://doi.org/10.7566/jpsj.83.043801

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