Center of maximum-sum matchings of bichromatic points

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 $i\in\{1,2,\ldots,n\}$.