Hardness results for Multimarginal Optimal Transport problems

Multimarginal Optimal Transport ($\mathrm{MOT}$) is the problem of linear programming over joint probability distributions with fixed marginals. A key issue in many applications is the complexity of solving $\mathrm{MOT}$: 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 $\mathrm{MOT}$ is $\mathrm{poly}(n,k)$-time solvable for certain families of costs that have $\mathrm{poly}(n,k)$-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 $\mathrm{MOT}$.

Our main technical contribution is developing a toolkit for proving $\mathrm{NP}$-hardness and inapproximability results for $\mathrm{MOT}$ problems. This toolkit reduces proving intractability of $\mathrm{MOT}$ problems to proving intractability of more amenable discrete optimization problems. We demonstrate this toolkit by using it to establish the intractability of a number of $\mathrm{MOT}$ problems studied in the literature that have resisted previous algorithmic efforts. For instance, we provide evidence that repulsive costs make $\mathrm{MOT}$ intractable by showing that several such problems of interest are $\mathrm{NP}$-hard to solve—even approximately.