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

TL;DR

This paper uses statistical-mechanical models to analyze the effectiveness of linear programming relaxations in solving combinatorial optimization problems like minimum vertex cover on random hypergraphs, identifying phase transition thresholds.

Contribution

It introduces a lattice-gas statistical-mechanical model to study LP relaxations of combinatorial problems and determines critical thresholds where relaxations become inaccurate.

Findings

LP solutions match integer programming solutions below critical thresholds

Critical average degree for LP accuracy extends known mathematical bounds

LP relaxation fails above the replica symmetry-breaking threshold

Abstract

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.