图中路径填充问题的消息传递算法
A Message Passing Algorithm for the Problem of Path Packing in Graphs
摘要：
研究了有向图中结点不相交有向路的装箱问题。我们考虑这个问题的一个变种，每个路径开始在一个固定的子集的根结点，路径的长度上的一个给定的范围。这个问题的动机是所谓的肾脏交换问题，但有潜在的其他应用，并在其本身的权利是有趣的。我们提出了一个新的算法，这个问题的消息传递/信念传播技术的基础上。先验这个问题没有一个相关的图形模型，所以为了应用一个信念传播算法，我们提供了一个新的表示的问题作为一个图形模型。在这个模型上的标准信念传播具有较差的缩放行为，因此我们提供了一个有效的实现，显着降低了复杂性。我们提供的数值结果比较我们的算法在人工创建的图形和现实世界的网络的性能，几种替代算法，包括基于整数规划（IP）技术的算法。这些比较表明，我们的算法比基于IP的算法更好地扩展到大型实例，并且通常比从每个根结点开始选择最长路径的简单算法找到更好的解决方案。在某些情况下，它也找到了比基于IP的算法更好的解决方案，即使后者被允许运行的时间比我们的算法长得多。
英文摘要：
We consider the problem of packing node-disjoint directed paths in a directed graph. We consider a variant of this problem where each path starts within a fixed subset of root nodes, subject to a given bound on the length of paths. This problem is motivated by the so-called kidney exchange problem, but has potential other applications and is interesting in its own right. We propose a new algorithm for this problem based on the message passing/belief propagation technique. A priori this problem does not have an associated graphical model, so in order to apply a belief propagation algorithm we provide a novel representation of the problem as a graphical model. Standard belief propagation on this model has poor scaling behavior, so we provide an efficient implementation that significantly decreases the complexity. We provide numerical results comparing the performance of our algorithm on both artificially created graphs and real world networks to several alternative algorithms, including algorithms based on integer programming (IP) techniques. These comparisons show that our algorithm scales better to large instances than IP-based algorithms and often finds better solutions than a simple algorithm that greedily selects the longest path from each root node. In some cases it also finds better solutions than the ones found by IP-based algorithms even when the latter are allowed to run significantly longer than our algorithm.
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