置信传播：随机指派问题的一个渐近最优算法
Belief propagation : an asymptotically optimal algorithm for the random assignment problem
摘要：
随机分配问题要求在具有独立同分布的完全$n\times n$二部图$\Knn$中实现最小费用完美匹配。边权重，比如在$[0,1]$上均匀。在Aldous（2001）的一项杰出的工作中，最优成本被证明为以$n\to\infty$的形式收敛到$ζ(2)$，正如Mézard和Parisi（1987）通过所谓的空腔方法所证明的那样。后者还提出了一种非严格的分散策略来寻找最优解，这是Pearl（1987）讨论的信念传播（BP）启发式的一个例子。在本文中，我们使用客观的方法来分析BP的性能作为基础图的大小变得很大。具体来说，我们建立的动态BP $\Knn$收敛分布为$n\to\infty$适当定义的动态泊松加权无限树，然后我们证明相关衰减为这个限制动态。因此，我们得到BP发现一个渐近正确的分配在$O(n^2)$时间。这与Bayati，Shah和Sharma（2005）推导的BP收敛的最坏情况上界以及Edmonds和Karp算法（1972）实现的$Θ(n^3)$的最著名计算成本形成对比。
英文摘要：
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 $ζ(2)$ 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 $Θ(n^3)$ achieved by Edmonds and Karp's algorithm (1972).
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