π²_β-transport with discrete target as a combinatorial matching problem
In this short note, we show that given a cost function c, any coupling Ο 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 c_L^β(Ο). This correspondence between couplings and bipartite graphs is explicitly constructed. We give two applications of this result to the π²_β 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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