Random Euclidean matching problems in one dimension

TL;DR

This paper analytically studies the optimal matching problem in one dimension with convex cost functions, deriving the average costs and scaling behaviors for large numbers of random points, and validating results with simulations.

Contribution

It provides the first analytical expressions for average optimal costs in one-dimensional matching problems with convex costs, including asymptotic scaling laws.

Findings

Average optimal cost scales as N^{-p/2} for assignment when p>1.

Average optimal cost scales as N^{-p} for matching when p>1.

Predictions match numerical simulation results.

Abstract

We discuss the optimal matching solution for both the assignment problem and the matching problem in one dimension for a large class of convex cost functions. We consider the problem in a compact set with the topology both of the interval and of the circumference. Afterwards, we assume the points' positions to be random variables identically and independently distributed on the considered domain. We analytically obtain the average optimal cost in the asymptotic regime of very large number of points $N$ and some correlation functions for a power-law type cost function in the form $c(z)=z^p$, both in the $p>1$ case and in the $p<0$ case. The scaling of the optimal mean cost with the number of points is $N^{-\frac{p}{2}}$ for the assignment and $N^{-p}$ for the matching when $p>1$, whereas in both cases it is a constant when $p<0$. Finally, our predictions are compared with the results of numerical simulations.