Belief propagation for optimal edge cover in the random complete graph

TL;DR

This paper uses the objective method to analyze the minimum-cost edge cover in large random complete graphs, proving known asymptotic results and demonstrating belief propagation's effectiveness in approximating the optimal solution.

Contribution

It provides a new proof of the asymptotic minimum cost using the objective method and shows belief propagation converges to the optimal edge cover in this setting.

Findings

Proof of the known asymptotic minimum cost using the objective method

Belief propagation converges asymptotically to the optimal solution

Near-optimal solutions with lower complexity than existing algorithms

Abstract

We apply the objective method of Aldous to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and W\"{a}stlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the $\zeta(2)$ limit of the random assignment problem. A proof via the objective method is useful because it provides us with more information on the nature of the edge's incident on a typical root in the minimum-cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This can be applied in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.