Iterative Message Passing Algorithm for Vertex-Disjoint Shortest Paths

As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the performance of a message-passing algorithm called min-sum belief propagation (BP) for the vertex-disjoint shortest k -path problem ( k -VDSP) on weighted directed graphs, and derive the iterative message-passing update rules. As the main result of this paper, we prove that for a weighted directed graph G of order n , BP algorithm converges to the unique optimal solution of k -VDSP on G within O(n^{2}w_{max}) iterations, provided that the weight w_{e} is nonnegative integral for each arc e\in E(G) , where w_{max}=\max \{w_{e}: e\in E(G)\} . To the best of our knowledge, this is the first instance where BP algorithm is proved correct for NP-hard problems. Additionally, we establish the extensions of k -VDSP to the case of multiple sources or sinks.