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

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

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  1. https://github.com/mohit-bansil/W_infinity_2D.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, Los Angeles, USA

    Mohit Bansil

  2. Department of Mathematics, Michigan State University, East Lansing, USA

    Jun Kitagawa

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  1. Mohit Bansil
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  2. Jun Kitagawa
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Correspondence to Jun Kitagawa.

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JK’s research was supported in part by National Science Foundation Grants DMS-1700094 and DMS-2000128.

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Bansil, M., Kitagawa, J. \({\mathcal {W}}_\infty \)-transport with discrete target as a combinatorial matching problem. Arch. Math. 117, 189–202 (2021). https://doi.org/10.1007/s00013-021-01606-z

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  • DOI: https://doi.org/10.1007/s00013-021-01606-z

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