An O(n^3) time algorithm for the maximum weight b-matching problem on bipartite graphs

Abstract

A matching between two sets A and B assigns some elements of A to some elements of B. Finding the similarity between two sets of elements by advantage of the matching is widely used in computational biology. Frequently, the capacities of the elements are limited. That is, the number of the elements that can be matched to each element should not exceed a given number. We describe the first O(n 3 ) time algorithm for matching two sets of elements with limited capacities.We use bipartite graphs to model relationships between pairs of objects. Let G = (A ∪ B, E) denote an undirected bipartite graph with real edge weights and node capacities b(v). The b-matching of G matches each vertex v in A (B) to at least 1 and at most b(v) vertices in B (A). In this paper, we present an O(n 3 ) time algorithm for finding the maximum weight b-matching of G, where |A| + |B| = O(n). Our algorithm improves the previous best time complexity of O(n 3 log n) for this problem.