Max-Product for Maximum Weight Matching: Convergence, Correctness, and LP Duality

Abstract--Max-product "belief propagation" (BP) is an iterative, message-passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization, which involve graphs with many cycles, theoretical results about both the correctness and convergence of the algorithm are known in only a few cases (see Section I for references). In this paper, we will prove the correctness and convergence of max-product for finding the maximum weight matching (MWM) in bipartite graphs. Even though the underlying graph of the MWM problem has many cycles, somewhat surprisingly we show that the max-product algorithm converges to the correct MWM as long as the MWM is unique. We provide a bound on the number of iterations required and show that for a graph of size n, the computational cost of the algo...

• Type: Journal
• Year: 2008
• Where: TIT
• Authors: Mohsen Bayati, Devavrat Shah, Mayank Sharma
• Keywords: Algorithm, Discrete Probability Distribution, Message-passing Algorithm, TIT 2008