Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
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The categories of this paper are: eess
4
The categories of this paper are: cond-mat