Iterative Message Passing Algorithm for Vertex-Disjoint Shortest Paths

Abstract

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 <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-path problem (<inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> of order <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, BP algorithm converges to the unique optimal solution of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-VDSP on <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> within <inline-formula> <tex-math notation="LaTeX">$O(n^{2}w_{max})$ </tex-math></inline-formula> iterations, provided that the weight <inline-formula> <tex-math notation="LaTeX">$w_{e}$ </tex-math></inline-formula> is nonnegative integral for each arc <inline-formula> <tex-math notation="LaTeX">$e\in E(G)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$w_{max}=\max \{w_{e}: e\in E(G)\}$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-VDSP to the case of multiple sources or sinks.