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

TL;DR

This paper presents a faster algorithm for finding minimum-cost maximum matchings in minor-free graphs, reducing the number of phases and overall runtime compared to previous algorithms, especially for planar graphs.

Contribution

It introduces a novel approach that relaxes augmenting path conditions, enabling more paths per phase and significantly improving runtime for minor-free graphs.

Findings

Achieves $ ilde{O}(n^{7/5} ext{log}(nC))$ time for minor-free graphs.

Reduces phases from $O( oot{2}n)$ to $O(n^{2/5})$.

Improves planar graph matching to $ ilde{O}(n^{6/5} ext{log}(nC))$ time.

Abstract

We give an $\tilde{O}(n^{7/5} \log (nC))$-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 $\tilde{O}(n^{10/7}\log{C})$ algorithm of Cohen et al. [SODA 2017] and the $O(n^{3/2}\log (nC))$ 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 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 $O(\sqrt{n})$ and the total execution time to $O(m\sqrt{n})$. 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 $O(\sqrt{n})$ to $O(n^{2/5})$. By using small vertex separators, the execution of each phase takes $\tilde{O}(m)$ time on average. For planar graphs, we combine our algorithm with efficient shortest path data structures to obtain a minimum-cost perfect matching in $\tilde{O}(n^{6/5} \log{(nC)})$ time. This improves upon the recent $\tilde{O}(n^{4/3}\log{(nC)})$ time algorithm by Asathulla et al. [SODA 2018].