Center of maximum-sum matchings of bichromatic points

TL;DR

This paper proves the existence of a central point in the plane that approximates the maximum-sum matching of bichromatic points within a factor of 2 of the matched pairs' distances, revealing geometric properties of optimal matchings.

Contribution

It establishes the existence of a 'center' point with bounded sum distances to matched pairs in maximum-sum matchings of bichromatic points.

Findings

Existence of a point o with 2 approximation for all matched pairs.

Bounded ratio between 2 and the maximum sum of matched pair distances.

Geometric insight into the structure of maximum-sum matchings.

Abstract

Let $R$ and $B$ be two disjoint point sets in the plane with $|R|=|B|=n$. Let $\mathcal{M}=\{(r_i,b_i),i=1,2,\ldots,n\}$ be a perfect matching that matches points of $R$ with points of $B$ and maximizes $\sum_{i=1}^n\|r_i-b_i\|$, the total Euclidean distance of the matched pairs. In this paper, we prove that there exists a point $o$ of the plane (the center of $\mathcal{M}$) such that $\|r_i-o\|+\|b_i-o\|\le \sqrt{2}~\|r_i-b_i\|$ for all $i\in\{1,2,\ldots,n\}$.