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

Abstract

Max-product "belief propagation" is an iterative, local, 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, computer vision and combinatorial optimization which involve graphs with many cycles, theoretical results about both correctness and convergence of the algorithm are known in few cases [21], [18], [23], [16].In this paper we consider the problem of finding the Maximum Weight Matching (MWM) in a weighted complete bipartite graph. We define a probability distribution on the bipartite graph whose MAP assignment corresponds to the MWM. We use the max-product algorithm for finding the MAP of this distribution or equivalently, the MWM on the bipartite graph. Even though the underlying bipartite graph has many short cycles, we find that surprisingly, the max-product algorithm always converges to the correct MAP assignment as long as the MAP assignment is unique. We provide a bound on the number of iterations required by the algorithm and evaluate the computational cost of the algorithm. We find that for a graph of size n, the computational cost of the algorithm scales as O(n 3 ), which is the same as the computational cost of the best known algorithm. Finally, we establish the precise relation between the max-product algorithm and the celebrated auction algorithm proposed by Bertsekas. This suggests possible connections between dual algorithm and max-product algorithm for discrete optimization problems.

Belief Propagation for Min-Cost Network Flow: Convergence and Correctness

“…This is in contrast with the asymptotic convergence established for many iterative algorithms in the theory of continuous optimization. We note that this result is similar in flavor to those established in the context of BP's convergence for combinatorial optimization [5,4,29]. However, it differs from the convergence results in [18,19] where the estimates converge to the optimal solution with an exponential rate, but are not established to reach exact optimal in finitely many steps.…”

“…However they do not provide any guarantee on the convergence of max-product. Bayati, Shah and Sharma [5] considered the performance of BP for finding the maximum weight matching in a bipartite graph. They established that BP converges in pseudo-polynomial time to the optimal solution when the optimal solution is unique [5].…”

“…Bayati, Shah and Sharma [5] considered the performance of BP for finding the maximum weight matching in a bipartite graph. They established that BP converges in pseudo-polynomial time to the optimal solution when the optimal solution is unique [5]. Bayati et al [4] as well as Sanghavi et al [28] generalized this result by establishing correctness and convergence of the BP algorithm for b-matching problem when the linear programming relaxation corresponding to the node constraints has a unique integral optimal solution.…”

“…This immediately implies that BP finds the correct optimal solution for MCF for large enough N leading to Theorem 4.1. We note that this strategy is similar to that of [5]. However, the technical details are quite different.…”

Optimizing Social Welfare for Network Bargaining Games in the Face of Unstability, Greed and Spite

“…Moreover, as described in [15,3] a damping factor κ ∈ (0, 1) is used in the update; we can think of κ = 1 2 . Although later on we will also consider the LP relaxation of b-matching, unlike previous works [21,6,15], we do not require the assumption that the LP relaxation has a unique integral optimum.…”