A simpler max-product Maximum Weight Matching algorithm and the auction algorithm

The max-product "belief propagation" algorithm has received a lot of attention recently due to its spectacular success in many application areas such as iterative decoding, computer vision and combinatorial optimization. There is a lot of ongoing work investigating the theoretical properties of the algorithm. In our previous work (2005) we showed that the max-product algorithm can be used to solve the problem of finding the Maximum Weight Matching (MWM) in a weighted complete bipartite graph. However, for a graph with n nodes the max-product algorithm requires O(n(4)) operations to find the MWM compared to O(n(3)) for best known algorithms such as those proposed by Edmonds and Karp (1972) and Bertsekas (1988). In this paper, we simplify the max-product algorithm to reduce the number of operations required to O(n(3)). The simplified algorithm has very similar dynamics to the well-known auction algorithm of Bertsekas (1988). To make this connection precise, we show that the max-product and auction algorithms, when slightly modified, are equivalent. We study the correctness of this modified algorithm. There is a tantalizing similarity between this connection and a recently observed connection between the max-product and LP-based algorithms for iterative decoding by Vontobel and Koetter.