Belief propagation for optimal edge cover in the random complete graph

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.

💡 Research Summary

The paper studies the minimum‑cost edge‑cover problem on the complete graph (K_n) when each edge weight is drawn independently from a common distribution (typically exponential). An edge‑cover is a set of edges such that every vertex is incident to at least one selected edge, and the goal is to minimize the total weight of the selected edges. Earlier work by Hessler and Wästlund derived the limiting expected optimal cost as (n\to\infty) using a combinatorial argument, showing that the limit equals (\zeta(2)=\pi^2/6). However, that approach does not reveal the local structure of the optimal cover around a typical vertex.

The authors apply Aldous’s “objective method” together with local weak convergence to re‑derive the limit and to obtain a detailed probabilistic description of the optimal cover. They show that the neighbourhood of a uniformly random vertex in (K_n) converges in distribution to the Poisson Weighted Infinite Tree (PWIT), an infinite rooted tree whose edges have independent exponential weights. The edge‑cover problem on the PWIT can be expressed through a recursive distributional equation (RDE): \