On the quadratic random matching problem in two-dimensional domains

TL;DR

This paper analyzes the asymptotic behavior of the minimum bipartite matching cost in 2D domains, confirming a conjecture and extending results to Riemannian manifolds and TSP applications.

Contribution

It proves the conjecture relating matching cost to domain volume and logarithmic growth, using PDE tools and domain decomposition, extending to Riemannian manifolds and TSP.

Findings

Asymptotic cost is proportional to log(n) times domain volume.

Confirmed the conjecture by Benedetto and Caglioti.

Extended results to Riemannian manifolds and TSP applications.

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 H{\"o}lder 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.