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A Robust Exact Algorithm for the Euclidean Bipartite Matching Problem

Algorithms for the minimum-cost bipartite matching can be used to estimate Wasserstein distance between two distributions. Given two sets $A$ and $B$ of $n$ points in a $2$-dimensional Euclidean space, one can use a fast implementation of the Hungarian method to compute a minimum-cost bipartite matching of $A$ and $B$ in $\tilde{O}(n^2)$ time. Let $\Delta$ be the spread, i.e., the ratio of the distance of the farthest to the closest pair of points in $A\cup B$. In this paper, we present a new algorithm to compute a minimum-cost bipartite matching of $A$ and $B$ with a similar worst-case execution time of $\tilde{O}(n^2 \log \Delta)$. However, when $A$ and $B$ are drawn independently and identically from a fixed distribution that is not known to the algorithm, the execution time of our algorithm is, in expectation, $\tilde{O}(n^{7/4}\log \Delta)$. To the best of our knowledge, our algorithm is the first one to achieve a sub-quadratic execution time even for stochastic point sets with real-valued coordinates. Our algorithm extends to any dimension $d$, where it runs in $\tilde{O}(n^{2-\frac{1}{2d}}\Phi(n))$ time for stochastic point sets $A$ and $B$; here $\Phi(n)$ is the query/update time of a dynamic weighted nearest neighbor data structure. Our algorithm can be seen as a careful adaptation of the Hungarian method in the geometric divide-and-conquer framework.

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