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

J. Salez, D. Shah
Mathematics of Operations Research, Volume 32, No.2, pp. 468-480

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 [Aldous, D. 2001. The ζ(2) limit in the random assignment problem. RSA 18 381–418], the optimal cost was shown to converge to ζ(2) as n → ∞, as conjectured by Mézard and Parisi [Mézard, M., G. Parisi. 1987. On the solution of the random link matching problem. J. Phys. 481451–1459] through the so-called cavity method. The latter also suggested a nonrigorous decentralized strategy for finding the optimum, which turned out to be an instance of the belief propagation (BP) heuristic discussed by Pearl [Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco]. 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 𝒦nnconverges 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 et al. [Bayati, M., D. Shah, M. Sharma. 2008. Max-product for maximum weight matching: Convergence, correctness, and LP duality. IEEE Trans. Inform. Theory 54(3) 1241–1251.] and the best-known computational cost of Θ(n3) achieved by Edmonds and Karp’s algorithm [Edmonds, J., R. Karp. 1972. Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19248–264].