Fast Algorithms for Continuous Optimal Transport Between Histograms

We give the first (relative) (1+ε)-approximation algorithm for the continuous optimal transport (OT) problem between two axis-aligned histograms in the plane, each of which is represented as a piecewise-constant function over a rectangular subdivision. Our algorithm runs in nearly quadratic time in the complexity of the histograms in the worst case. In contrast to the discrete and semi-discrete optimal transport problems, which always admit linear-size OT plans, we demonstrate that there is a quadratic lower bound on the complexity of even a (1+ε)-approximate OT plan in the continuous setting in the worst case. This suggests that the runtime of our approximation algorithm nearly matches the worst-case lower-bound complexity of an explicit (1+ε)-approximate OT plan between two histograms. We additionally provide near-linear time algorithms with either weaker approximation guarantees or restrictions on the input histograms.