${\mathcal{W}}_{\infty }$-transport with discrete target as a combinatorial matching problem

Abstract

In this short note, we show that given a cost function $c$, any coupling $\pi$ of two probability measures where the second is a discrete measure can be associated to a certain bipartite graph containing a perfect matching, based on the value of the infinity transport cost $\norm{c}_{L^\infty(π)}$. This correspondence between couplings and bipartite graphs is explicitly constructed. We give two applications of this result to the ${\mathcal{W}}_{\infty }$ optimal transport problem when the target measure is discrete, the first is a condition to ensure existence of an optimal plan induced by a mapping, and the second is a numerical approach to approximating optimal plans.

12 pages, comments welcome!