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 ϵ of optimal in time $\tilde{O}(n^2/ϵ)$; one algorithm is straightforward to parallelize and implementable in depth $\tilde{O}(1/ϵ)$. Further, we show that additional improvements on our results must be coupled with breakthroughs in algorithmic graph theory.