A faster algorithm for minimum-cost bipartite matching in minor-free graphs

Abstract

We give an Õ(n7/5 log(nC)) time1 algorithm to compute a minimum-cost maximum cardinality matching (optimal matching) in Kh-minor free graphs with h = O(1) and integer edge weights having magnitude at most C. This improves upon the Õ(n10/7 log C) algorithm of Cohen et al.. [SODA 2017] and the O(n3/2 log(nC)) algorithm of Gabow and Tarjan [SIAM J. Comput. 1989]. For a graph with m edges and n vertices, the well-known Hungarian Algorithm computes a shortest augmenting path in each phase in O(m) time, yielding an optimal matching in O(mn) time. The Gabow-Tarjan [SIAM J. Comput. 1989] algorithm computes, in each phase, a maximal set of vertex-disjoint shortest augmenting paths (for appropriately defined costs) in O(m) time. This reduces the number of phases from n to O(n) and the total execution time to O(mn). To obtain our speed-up, we relax the conditions on the augmenting paths and iteratively compute, in each phase, a set of carefully selected augmenting paths that are not restricted to be shortest or vertex-disjoint. As a result, our algorithm computes substantially more augmenting paths in each phase, reducing the number of phases from O(n) to O(n2/5). By using small vertex separators, the execution of each phase takes Õ(m) time on average. For planar graphs, we combine our algorithm with efficient shortest path data structures to obtain a minimum-cost perfect matching in Õ(n6/5 log (nC)) time. This improves upon the recent Õ(n4/3 log (nC)) time algorithm by Asathulla et al. [SODA 2018].