Hardness results for Multimarginal Optimal Transport problems
This work addresses the fundamental limitations of solving MOT problems for researchers and practitioners in optimization and machine learning by providing a framework to understand their intractability.
This paper investigates the computational complexity of Multimarginal Optimal Transport (MOT) problems, which involve linear programming over joint probability distributions with fixed marginals. The authors develop a toolkit to prove NP-hardness and inapproximability results for various MOT problems, demonstrating that several previously resistant problems, particularly those with repulsive costs, are NP-hard to solve, even approximately.
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.