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

In this short note, we show that given a cost function $c$, any coupling $π$ 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.

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