Hardness results for Multimarginal Optimal Transport problems

TL;DR

This paper investigates the computational complexity of Multimarginal Optimal Transport (MOT), establishing NP-hardness and inapproximability results for various cost functions, including repulsive costs, thus highlighting fundamental algorithmic limitations.

Contribution

It develops a toolkit for proving NP-hardness and inapproximability in MOT and applies it to show certain MOT problems are computationally intractable.

Findings

MOT is NP-hard for several cost functions including repulsive costs.

The toolkit can prove intractability of previously resistant MOT problems.

Some MOT problems are hard to approximate within any reasonable factor.

Abstract

Multimarginal Optimal Transport (MOT) is the problem of linear programming over joint probability distributions with fixed marginals. A key issue in many applications is the complexity of solving MOT: the linear program has exponential size in the number of marginals k and their support sizes n. A recent line of work has shown that MOT is poly(n,k)-time solvable for certain families of costs that have poly(n,k)-size implicit representations. However, it is unclear what further families of costs this line of algorithmic research can encompass. In order to understand these fundamental limitations, this paper initiates the study of intractability results for MOT. Our main technical contribution is developing a toolkit for proving NP-hardness and inapproximability results for MOT problems. We demonstrate this toolkit by using it to establish the intractability of a number of MOT problems studied in the literature that have resisted previous algorithmic efforts. For instance, we provide evidence that repulsive costs make MOT intractable by showing that several such problems of interest are NP-hard to solve--even approximately.