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2022 On the quadratic random matching problem in two-dimensional domains
Luigi Ambrosio, Michael Goldman, Dario Trevisan
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Electron. J. Probab. 27: 1-35 (2022). DOI: 10.1214/22-EJP784

Abstract

We investigate the average minimum cost of a bipartite matching, with respect to the squared Euclidean distance, between two samples of n i.i.d. random points on a bounded Lipschitz domain in the Euclidean plane, whose common law is absolutely continuous with strictly positive Hölder continuous density. We confirm in particular the validity of a conjecture by D. Benedetto and E. Caglioti stating that the asymptotic cost as n grows is given by the logarithm of n multiplied by an explicit constant times the volume of the domain. Our proof relies on a reduction to the optimal transport problem between the associated empirical measures and a Whitney-type decomposition of the domain, together with suitable upper and lower bounds for local and global contributions, both ultimately based on PDE tools. We further show how to extend our results to more general settings, including Riemannian manifolds, and also give an application to the asymptotic cost of the random quadratic bipartite travelling salesperson problem.

Funding Statement

L.A. thanks the support of the PRIN 2017 project “Gradient Flows, Optimal Transport and Metric Measure Structures”. M.G. was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR). D.T. was partially supported by Gnampa project 2020 “Problemi di ottimizzazione con vincoli via trasporto ottimo e incertezza”.

Citation

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Luigi Ambrosio. Michael Goldman. Dario Trevisan. "On the quadratic random matching problem in two-dimensional domains." Electron. J. Probab. 27 1 - 35, 2022. https://doi.org/10.1214/22-EJP784

Information

Received: 1 November 2021; Accepted: 19 April 2022; Published: 2022
First available in Project Euclid: 3 May 2022

Digital Object Identifier: 10.1214/22-EJP784

Subjects:
Primary: 35J05 , ‎39B62 , 60D05 , 60F25 , 90C05

Keywords: geometric probability , Matching problem , Optimal transport

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