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Optimal Transport is a popular distance metric for measuring similaritybetween distributions. Exact algorithms for computing Optimal Transport can beslow, which has motivated the development of approximate numerical solvers(e.g. Sinkhorn method). We introduce a new and very simple combinatorialapproach to find an $\varepsilon$-approximation of the OT distance. Ouralgorithm achieves a near-optimal execution time of $O(n^2/\varepsilon^2)$ forcomputing OT distance and, for the special case of the assignment problem, theexecution time improves to $O(n^2/\varepsilon)$. Our algorithm is based on thepush-relabel framework for min-cost flow problems.  Unlike the other combinatorial approach (Lahn, Mulchandani and Raghvendra,NeurIPS 2019) which does not have a fast parallel implementation, our algorithmhas a parallel execution time of $O(\log n/\varepsilon^2)$. Interestingly,unlike the Sinkhorn algorithm, our method also readily provides a compacttransport plan as well as a solution to an approximate version of the dualformulation of the OT problem, both of which have numerous applications inMachine Learning. For the assignment problem, we provide both a CPUimplementation as well as an implementation that exploits GPU parallelism.Experiments suggest that our algorithm is faster than the Sinkhorn algorithm,both in terms of CPU and GPU implementations, especially while computingmatchings with a high accuracy.

A Push-Relabel Based Additive Approximation for Optimal Transport