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 Erd\"os-R\'enyi random graphs of $\alpha$-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 $\alpha=2$ 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 $\alpha\geq 3$, minimum vertex covers on $\alpha$-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 $c=e/(\alpha-1)$ where the replica symmetry is broken.