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arXiv · 1911.10610

On Maximum-Sum Matchings of Points

Sergey Bereg·Oscar Chacón-Rivera·David Flores-Peñaloza·Clemens Huemer·Pablo Pérez-Lantero·Carlos Seara
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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.

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Sergey Bereg, Oscar Chacón-Rivera, David Flores-Peñaloza, Clemens Huemer, Pablo Pérez-Lantero, Carlos Seara. 2019-11-24. On Maximum-Sum Matchings of Points. https://arxiv.org/abs/1911.10610

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