2009/02/03 by Justin Salez, Devavrat Shah, Salez, Justin +1
Computer Science · Mathematics · #Bayesian Modeling and Causal Inference #Complexity and Algorithms in Graphs #FOS: Mathematics #Optimization and Search Problems #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.0902.0585
Mathematics of Operations Research (2009)
arxiv created 2009/02/03 · openalex publication_date 2009/02/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The random assignment problem asks for the minimum-cost perfect matching in the complete n× 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 ζ(2) as n→∞, as conjectured by Mézard 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 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(n2) 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 Θ(n3) achieved by Edmonds and Karp's algorithm (1972).