Belief propagation : an asymptotically optimal algorithm for the random assignment problem

TL;DR

This paper proves that belief propagation is asymptotically optimal for solving the random assignment problem, converging efficiently to the minimum-cost perfect matching as the graph size grows large.

Contribution

It rigorously analyzes the asymptotic behavior of belief propagation on large random bipartite graphs and establishes its optimality and efficiency for the assignment problem.

Findings

BP converges in distribution to a dynamic on the Poisson Weighted Infinite Tree

BP finds an asymptotically correct assignment in O(n^2) time

Correlation decay is established for the limiting dynamic

Abstract

The random assignment problem asks for the minimum-cost perfect matching in the complete $n\times n$ bipartite graph $\Knn$ with i.i.d. edge weights, say uniform on $[0,1]$. In a remarkable work by Aldous (2001), the optimal cost was shown to converge to $\zeta(2)$ as $n\to\infty$, as conjectured by M\'ezard and Parisi (1987) through the so-called cavity method. The latter also suggested a non-rigorous decentralized strategy for finding the optimum, which turned out to be an instance of the Belief Propagation (BP) heuristic discussed by Pearl (1987). In this paper we use the objective method to analyze the performance of BP as the size of the underlying graph becomes large. Specifically, we establish that the dynamic of BP on $\Knn$ converges in distribution as $n\to\infty$ to an appropriately defined dynamic on the Poisson Weighted Infinite Tree, and we then prove correlation decay for this limiting dynamic. As a consequence, we obtain that BP finds an asymptotically correct assignment in $O(n^2)$ time only. This contrasts with both the worst-case upper bound for convergence of BP derived by Bayati, Shah and Sharma (2005) and the best-known computational cost of $\Theta(n^3)$ achieved by Edmonds and Karp's algorithm (1972).