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We consider the problem of packing node-disjoint directed paths in a directedgraph. We consider a variant of this problem where each path starts within afixed 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 haspotential other applications and is interesting in its own right.  We propose a new algorithm for this problem based on the messagepassing/belief propagation technique. A priori this problem does not have anassociated graphical model, so in order to apply a belief propagation algorithmwe provide a novel representation of the problem as a graphical model. Standardbelief propagation on this model has poor scaling behavior, so we provide anefficient implementation that significantly decreases the complexity. Weprovide numerical results comparing the performance of our algorithm on bothartificially created graphs and real world networks to several alternativealgorithms, including algorithms based on integer programming (IP) techniques.These comparisons show that our algorithm scales better to large instances thanIP-based algorithms and often finds better solutions than a simple algorithmthat greedily selects the longest path from each root node. In some cases italso finds better solutions than the ones found by IP-based algorithms evenwhen the latter are allowed to run significantly longer than our algorithm.

A Message Passing Algorithm for the Problem of Path Packing in Graphs