An $\tilde{O}(n^{5/4})$ Time $\varepsilon$-Approximation Algorithm for RMS Matching in a Plane

The 2-Wasserstein distance (or RMS distance) is a useful measure of similarity between probability distributions that has exciting applications in machine learning. For discrete distributions, the problem of computing this distance can be expressed in terms of finding a minimum-cost perfect matching on a complete bipartite graph given by two multisets of points $A,B \subset \mathbb{R}^2$, with $|A|=|B|=n$, where the ground distance between any two points is the squared Euclidean distance between them. Although there is a near-linear time relative $\varepsilon$-approximation algorithm for the case where the ground distance is Euclidean (Sharathkumar and Agarwal, JACM 2020), all existing relative $\varepsilon$-approximation algorithms for the RMS distance take $\Omega(n^{3/2})$ time. This is primarily because, unlike Euclidean distance, squared Euclidean distance is not a metric. In this paper, for the RMS distance, we present a new $\varepsilon$-approximation algorithm that runs in $O(n^{5/4}\mathrm{poly}\{\log n,1/\varepsilon\})$ time. Our algorithm is inspired by a recent approach for finding a minimum-cost perfect matching in bipartite planar graphs (Asathulla et al., TALG 2020). Their algorithm depends heavily on the existence of sub-linear sized vertex separators as well as shortest path data structures that require planarity. Surprisingly, we are able to design a similar algorithm for a complete geometric graph that is far from planar and does not have any vertex separators. Central components of our algorithm include a quadtree-based distance that approximates the squared Euclidean distance and a data structure that supports both Hungarian search and augmentation in sub-linear time.