On maximum-sum matchings of points

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.

Center of maximum-sum matchings of bichromatic points

“…Huemer et al [6] proved that if M is any perfect matching of R and B that maximizes the total squared Euclidean distance of the matched points, i.e., it maximizes (p,q)∈M p − q 2 , then all disks of B M have a point in common. As proved by Bereg et al [2], the disks of our max-sum matching M of R ∪ B intersect pairwise, fact that will be used in this paper, but the common intersection is not always possible.…”

“…Bereg et al [2] obtained an approximation to this conjecture. They proved that for any point set P of 2n uncolored points in the plane and a max-sum matching M = {(a i , b i ), i = 1, .…”

Intersecting balls induced by a geometric graph II

“…In 2021, Bereg, Chacón-Rivera, Flores-Peñaloza, Huemer, Pérez-Lantero, and Seara [4,Theorem 3.14] showed that a max-sum matching of any even set in the plane is a Tverberg graph. Our second result is a new proof of a slightly stronger version of their theorem.…”

Euclidean Maximum Matchings in the Plane—Local to Global

“…To do so, we consider the set D of diametral disks of edges in M . A recent result of Bereg et al [5] combined with Helly's theorem [22,30] implies that the disks in D have a common intersection. We take a point in this intersection and connect it to endpoints of all edges of M to obtain a star S. Then we show that w(M * ) ⩽ w(S) ⩽ √ 2 • w(M ), which proves the weaker lower bound.…”

“…Maximum-weight matching is among well-studied structures in graph theory and combinatorial optimization. It has been studied from both combinatorial and computational points of view in both abstract and geometric settings, see for example [1,3,5,11,9,12,13,16,17,19,24,25,31]. Over the years, it has found applications in several areas such as scheduling, facility location, and network switching.…”