Towards optimal running timesfor optimal transport

We provide faster algorithms for approximating the optimal transport distance, e.g. earth mover's distance, between two discrete probability distributions on n elements. We present two algorithms which compute couplings between marginal distributions with an expected transportation cost that is within an additive is an element of of optimal in time (O) over tilde (n(2)/is an element of); one algorithm is straightforward to parallelize and implementable in depth (O) over tilde (1/is an element of). Further, we show that additional improvements on our results must be coupled with breakthroughs in algorithmic graph theory. (c) 2023 Published by Elsevier B.V.