Statistical-mechanical Analysis of Linear Programming Relaxation for Combinatorial Optimization Problems

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erdös-Rényi random graphs of $α$-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statistical-mechanical analyses reveal for $α=2$ that the LP optimal solution is typically equal to that given by the IP below the critical average degree $c=e$ in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result $c=1$, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with $α\geq 3$, minimum vertex covers on $α$-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree $c=e/(α-1)$ where the replica symmetry is broken.

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