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

TL;DR

This paper establishes a correspondence between couplings in $ ext{W}_$-transport with discrete targets and bipartite graph matchings, providing conditions for optimal plan existence and a numerical approximation method.

Contribution

It introduces a novel link between $ ext{W}_$-transport couplings and bipartite graph matchings, enabling new theoretical and computational tools.

Findings

Couplings with discrete targets correspond to bipartite graph perfect matchings.

Conditions are provided for the existence of optimal transport maps.

A numerical method for approximating optimal plans is proposed.

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