On Maximum-Sum Matchings of Points

TL;DR

This paper investigates maximum-sum Euclidean distance matchings of point sets, showing they do not always have the common intersection property but do for uncolored points, improving a 1995 conjecture.

Contribution

It proves that maximum Euclidean distance matchings lack the common intersection property in general but always have it for uncolored points, advancing understanding of geometric matchings.

Findings

Maximum-sum Euclidean distance matchings may lack the common intersection property.

For uncolored points, such matchings always have a common intersection point.

The results improve a longstanding conjecture by Andy Fingerhut from 1995.

Abstract

Huemer et al. (Discrete Mathematics, 2019) proved that for any two point sets $R$ and $B$ with $|R|=|B|$, the perfect matching that matches points of $R$ with points of $B$, and maximizes the total \emph{squared} Euclidean distance of the matched pairs, verifies that all the disks induced by the matching have a common point. Each pair of matched points $p\in R$ and $q\in B$ induces the disk of smallest diameter that covers $p$ and $q$. Following this research line, in this paper we consider the perfect matching that maximizes the total Euclidean distance. First, we prove that this new matching for $R$ and $B$ does not always ensure the common intersection property of the disks. Second, we extend the study of this new matching for sets of $2n$ uncolored points in the plane, where a matching is just a partition of the points into $n$ pairs. As the main result, we prove that in this case all disks of the matching do have a common point. This implies a big improvement on a conjecture of Andy Fingerhut in 1995, about a maximum matching of $2n$ points in the plane.