# A Faster Algorithm for Minimum-Cost Bipartite Matching in Minor-Free Graphs

We give an Õ(n^7/5 (nC))-time algorithm to compute a minimum-cost maximum cardinality matching (optimal matching) in K_h-minor free graphs with h=O(1) and integer edge weights having magnitude at most C. This improves upon the Õ(n^10/7C) algorithm of Cohen et al. [SODA 2017] and the O(n^3/2 (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 Hopcroft-Karp [SIAM J. Comput. 1973], and Gabow-Tarjan [SIAM J. Comput. 1989] algorithms compute, 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(m√(n)). In order 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(n^2/5). 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(n^6/5(nC)) time. This improves upon the recent Õ(n^4/3(nC)) time algorithm by Asathulla et al. [SODA 2018].

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