Discrete Optimal Transport with Independent Marginals is #P-Hard

TL;DR

This paper proves that computing the Wasserstein distance between certain high-dimensional discrete distributions with independent components is #P-hard, but also offers approximation algorithms and special cases solvable efficiently.

Contribution

It establishes the computational hardness of optimal transport with independent marginals and introduces approximation algorithms and exact solutions for specific cases.

Findings

Optimal transport with independent marginals is #P-hard even for simple distributions.

A dynamic programming algorithm approximates Wasserstein distance in pseudo-polynomial time.

Certain problem instances can be solved exactly in strongly polynomial time.

Abstract

We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in polynomial time in the maximum of the number of atoms of the two distributions. However, if the components of either random vector are independent, then this number can be exponential in K even though the size of the problem description scales linearly with K. We prove that the described optimal transport problem is #P-hard even if all components of the first random vector are independent uniform Bernoulli random variables, while the second random vector has merely two atoms, and even if only approximate solutions are sought. We also develop a dynamic programming-type algorithm that approximates the Wasserstein distance in pseudo-polynomial time when the components of the first random vector follow arbitrary independent discrete distributions, and we identify special problem instances that can be solved exactly in strongly polynomial time.