一般图中加权匹配的LP松弛与最大积的等价性
Equivalence of LP Relaxation and Max-Product for Weighted Matching in General Graphs
摘要：
最大乘积置信传播是一种局部迭代算法，用于找到概率分布的模式/MAP估计。虽然它已被成功地应用于各种各样的应用，有相对较少的理论保证收敛性和正确性的一般圈图，可能有许多短周期。其中，提供确切的"必要和顺从“特征的更少。在本文中，我们研究的问题，使用最大积找到最大的重量匹配的任意图的边重量。这是通过首先构建一个概率分布，其模式对应于最佳匹配，然后运行最大乘积来完成的。加权匹配也可以作为一个整数规划，其中有一个LP松弛。这种放松并不总是紧张的。在本文中，我们表明，\begin{enumerate} \item If the LP relaxation is tight, then max-product always converges, and that too to the correct answer. \item If the LP relaxation is loose, then max-product does not converge. \end{enumerate}这提供了一个精确的，数据依赖的表征最大产品的性能，和精确的连接到LP松弛，这是一个很好的研究优化技术。此外，由于LP松弛是紧的二部图，我们的结果推广了其他最近的结果使用最大产品，以找到加权匹配的二部图。
英文摘要：
Max-product belief propagation is a local, iterative algorithm to find the mode/MAP estimate of a probability distribution. While it has been successfully employed in a wide variety of applications, there are relatively few theoretical guarantees of convergence and correctness for general loopy graphs that may have many short cycles. Of these, even fewer provide exact ``necessary and sufficient'' characterizations. In this paper we investigate the problem of using max-product to find the maximum weight matching in an arbitrary graph with edge weights. This is done by first constructing a probability distribution whose mode corresponds to the optimal matching, and then running max-product. Weighted matching can also be posed as an integer program, for which there is an LP relaxation. This relaxation is not always tight. In this paper we show that \begin{enumerate} \item If the LP relaxation is tight, then max-product always converges, and that too to the correct answer. \item If the LP relaxation is loose, then max-product does not converge. \end{enumerate} This provides an exact, data-dependent characterization of max-product performance, and a precise connection to LP relaxation, which is a well-studied optimization technique. Also, since LP relaxation is known to be tight for bipartite graphs, our results generalize other recent results on using max-product to find weighted matchings in bipartite graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
