Polynomial-time algorithms for Multimarginal Optimal Transport problems with structure

Multimarginal Optimal Transport (MOT) has recently attracted significant interest due to its many applications. However, in most applications, the success of MOT is severely hindered by a lack of sub-exponential time algorithms. This paper develops a general theory about "structural properties" that make MOT tractable. We identify two such properties: decomposability of the cost into either (i) local interactions and simple global interactions; or (ii) low-rank interactions and sparse interactions. We also provide strong evidence that (iii) repulsive costs make MOT intractable by showing that several such problems of interest are NP-hard to solve--even approximately. These three structures are quite general, and collectively they encompass many (if not most) current MOT applications. We demonstrate our results on a variety of applications in machine learning, statistics, physics, and computational geometry.