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

1 TREC - Theory of networks and communications
DI-ENS - Département d'informatique de l'École normale supérieure, Inria Paris-Rocquencourt
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 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 $n\to\infty$ 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).
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Journal articles
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Cited literature [20 references]

https://hal.inria.fr/inria-00358331
Contributor : Justin Salez <>
Submitted on : Tuesday, February 3, 2009 - 1:15:04 PM
Last modification on : Thursday, July 1, 2021 - 5:58:02 PM
Long-term archiving on: : Tuesday, June 8, 2010 - 7:08:07 PM

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### Identifiers

• HAL Id : inria-00358331, version 1
• ARXIV : 0902.0585

### Citation

Justin Salez, Devavrat Shah. Belief propagation : an asymptotically optimal algorithm for the random assignment problem. Mathematics of Operations Research, INFORMS, 2009. ⟨inria-00358331⟩

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