Abstract
The Earth Mover Distance (EMD) between two sets of points A, B subseteq R^d with A = B is the minimum total Euclidean distance of any perfect matching between A and B. One of its generalizations is asymmetric EMD, which is the minimum total Euclidean distance of any matching of size A between sets of points A,B subseteq R^d with A <= B. The problems of computing EMD and asymmetric EMD are wellstudied and have many applications in computer science, some of which also ask for the EMDoptimal matching itself. Unfortunately, all known algorithms require at least quadratic time to compute EMD exactly. Approximation algorithms with nearly linear time complexity in n are known (even for finding approximately optimal matchings), but suffer from exponential dependence on the dimension.
In this paper we show that significant improvements in exact and approximate algorithms for EMD would contradict conjectures in finegrained complexity. In particular, we prove the following results:
 Under the Orthogonal Vectors Conjecture, there is some c>0 such that EMD in Omega(c^{log^* n}) dimensions cannot be computed in truly subquadratic time.
 Under the Hitting Set Conjecture, for every delta>0, no truly subquadratic time algorithm can find a (1 + 1/n^delta)approximate EMD matching in omega(log n) dimensions.
 Under the Hitting Set Conjecture, for every eta = 1/omega(log n), no truly subquadratic time algorithm can find a (1 + eta)approximate asymmetric EMD matching in omega(log n) dimensions.
BibTeX  Entry
@InProceedings{rohatgi:LIPIcs:2019:11227,
author = {Dhruv Rohatgi},
title = {{Conditional Hardness of Earth Mover Distance}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
pages = {12:112:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771252},
ISSN = {18688969},
year = {2019},
volume = {145},
editor = {Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11227},
URN = {urn:nbn:de:0030drops112270},
doi = {10.4230/LIPIcs.APPROXRANDOM.2019.12},
annote = {Keywords: Earth Mover Distance, Hardness of Approximation, FineGrained Complexity}
}
Keywords: 

Earth Mover Distance, Hardness of Approximation, FineGrained Complexity 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019) 
Issue Date: 

2019 
Date of publication: 

17.09.2019 