Hardness results for Multimarginal Optimal Transport problems

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