Belief Propagation: An Asymptotically Optimal Algorithm for the Random Assignment Problem

• 成果类型: Article
• 署名作者: Salez, Justin; Shah, Devavrat
• 署名单位: Inria; Universite PSL; Ecole Normale Superieure (ENS); Massachusetts Institute of Technology (MIT)
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.1090.0380
• 发表日期: 2009
• 页码: 468-480

摘要

The random assignment problem asks for the minimum-cost perfect matching in the complete n x n bipartite graph K-nn with i.i.d. edge weights, say uniform on [0, 1]. In a remarkable work by Aldous [Aldous, D. 2001. The zeta(2) limit in the random assignment problem. RSA 18 381-418], the optimal cost was shown to converge to zeta(2) as n -> infinity, as conjectured by Mezard and Parisi [Mezard, M., G. Parisi. 1987. On the solution of the random link matching problem. J. Phys. 48 1451-1459] 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 K-nn converges in distribution as n -> infinity 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 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 Theta(n(3)) achieved by Edmonds and Karp's algorithm [Edmonds, J., R. Karp. 1972. Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19 248-264].