On linear programs with random costs
Mathematical Programming: Series A and B
Probabilistic reasoning in intelligent systems: networks of plausible inference
Probabilistic reasoning in intelligent systems: networks of plausible inference
A lower bound on the expected cost of an optimal assignment
Mathematics of Operations Research
Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems
Journal of the ACM (JACM)
Constructive bounds and exact expectation for the random assignment problem
Random Structures & Algorithms
The ζ (2) limit in the random assignment problem
Random Structures & Algorithms
Proofs of the Parisi and Coppersmith-Sorkin random assignment conjectures
Random Structures & Algorithms
Max-Product for Maximum Weight Matching: Convergence, Correctness, and LP Duality
IEEE Transactions on Information Theory
ACNS'11 Proceedings of the 9th international conference on Applied cryptography and network security
| Hi-index | 0.00 |
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. RSA18 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.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 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 et al. [Bayati, M., D. Shah, M. Sharma. 2008. Max-product for maximum weight matching: Convergence, correctness, and LP duality. IEEE Trans. Inform. Theory54(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. ACM19 248--264].