Belief Propagation (BP), a distributed, message-passing algorithm, has been widely used in different disciplines including information theory, artificial intelligence, statistics and optimization problems in graphical models such as Bayesian networks and Markov random fields. Despite BP algorithm has a great success in many application fields and many progress about BP algorithm has been made, the rigorous analysis about the correctness and convergence of BP algorithm are known in only a few cases for arbitrary graph. In this paper, we will investigate the correctness and convergence of BP algorithm for determining the optimal solutions of the Chinese postman problem in both undirected and directed graphs. As a main result, we prove that BP algorithm converges to the optimal solution of the undirected Chinese postman problem within O(n) iterations where n represents the number of vertices, provided that the optimal solution is unique. For the directed case, we consider the directed Chinese postman problem with capacity and show that BP algorithm also converges to its optimal solution after O(n) iterations, as long as its corresponding linear programming relaxation has the unique optimal solution.
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We are truly grateful to the reviewers’ comments and suggestions on improving the manuscript, which helped us greatly to improve the quality of our paper. The research was partially supported by the National Natural Science Foundation of China under Grant Nos. 11871280, 11471003, 11401389,11531014, 11871081, China Scholarship Council under Grant Nos. 201607910003, 201608330111, Zhejiang Provincial Natural Science Foundation of China under Grant Nos. LY17A010017, 11401389 and Qing Lan Project.
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Dai, G., Li, F., Sun, Y. et al. Convergence and correctness of belief propagation for the Chinese postman problem. J Glob Optim 75, 813–831 (2019). https://doi.org/10.1007/s10898-019-00749-2
- Belief propagation (BP)
- Message-passing algorithm
- Min-sum algorithm
- Undirected Chinese postman problem
- Directed Chinese postman problem