On maximum-sum matchings of points

Sergey Bereg (), Oscar P. Chacón-Rivera (), David Flores-Peñaloza (), Clemens Huemer (), Pablo Pérez-Lantero () and Carlos Seara ()
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Sergey Bereg: University of Texas at Dallas
Oscar P. Chacón-Rivera: Universidad de Santiago de Chile (USACH), Las Sophoras n° 173
David Flores-Peñaloza: Av. Universidad 3000, Circuito Exterior s/n, Alcaldía Coyoacán, CP 04510, Ciudad Universitaria
Clemens Huemer: Universitat Politècnica de Catalunya
Pablo Pérez-Lantero: Universidad de Santiago de Chile (USACH), Las Sophoras n° 173
Carlos Seara: Universitat Politècnica de Catalunya

Journal of Global Optimization, 2023, vol. 85, issue 1, No 6, 128 pages

Abstract: Abstract Huemer et al. (Discrete Mathematics, 2019) proved that for any two point sets R and B with $$|R|=|B|$$ | R | = | B | , the perfect matching that matches points of R with points of B, and maximizes the total squared Euclidean distance of the matched pairs, has the property that all the disks induced by the matching have a common point. Each pair of matched points $$p\in R$$ p ∈ R and $$q\in B$$ q ∈ 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.

Date: 2023
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DOI: 10.1007/s10898-022-01199-z

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