Polynomial-time algorithms for multimarginal optimal transport problems with structure

Abstract Multimarginal Optimal Transport (MOT) has attracted significant interest due to applications in machine learning, statistics, and the sciences. However, most applications, success of MOT is severely limited by a lack efficient algorithms. Indeed, general requires exponential time number marginals k their support sizes n . This paper develops theory about what “structure” makes solvable $$\mathrm {poly}(n,k)$$ poly ( n , k ) time. We develop unified algorithmic framework for solving characterizing structure that different algorithms require terms simple variants dual feasibility oracle. several benefits. First, it enables us show Sinkhorn algorithm, which currently popular strictly more than other do solve Second, our much simpler given problem. In particular, necessary sufficient (approximately) oracle—which amenable standard techniques. illustrate this ease-of-use developing -time three classes cost structures: (1) graphical structure; (2) set-optimization (3) low-rank plus sparse structure. For (1), we recover known result runtime; moreover, provide first computing solutions are exact sparse. structures (2)-(3), give algorithms, even approximate computation. Together, these encompass many—if not most—current MOT.