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 \(\Vert c \Vert _{L^\infty (\pi )}\). 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.

在这个简短的说明中，我们表明，给定一个成本函数c，两个概率度量的任何耦合\(\pi \)，其中第二个是离散度量，可以与包含完美匹配的某个二部图相关联，基于值的无限运输成本\(\Vert c \Vert _{L^\infty (\pi )}\)。耦合和二部图之间的这种对应关系是明确构造的。当目标度量是离散的时，我们将此结果应用于\({\mathcal {W}}_\infty \)最优传输问题，第一个是确保存在由映射引起的最优计划的条件，第二个是近似最优计划的数值方法。