Minimum-weight perfect matching for non-intrinsic distances on the line

TL;DR

This paper introduces a recursive approach to solve the minimum-weight perfect matching problem on a line with arbitrary distances, connecting it to one-dimensional optimal transport problems.

Contribution

It presents a novel bottom-up recursion relation for computing weights of partial matchings in this setting.

Findings

Recursion relation simplifies solving the matching problem.

Links the matching problem to Monge-Kantorovich transport.

Provides a new perspective on non-intrinsic distances on the line.

Abstract

Consider a real line equipped with a (not necessarily intrinsic) distance. We deal with the minimum-weight perfect matching problem for a complete graph whose points are located on the line and whose edges have weights equal to distances along the line. This problem is closely related to one-dimensional Monge-Kantorovich trasnport optimization. The main result of the present note is a "bottom-up" recursion relation for weights of partial minimum-weight matchings.