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On maximum-sum matchings of bichromatic points

Oscar Chacón-Rivera, Pablo Pérez-Lantero

TL;DR

This work addresses max-sum matchings between bichromatic planar point sets under continuous (semi-)metrics, focusing on Euclidean distance and its square (quadrance). It introduces a precise 3-point characterization, linking max-sum optimality to the nonemptiness of five specific intersections of sets derived from the metric, with distinct boundary geometries for $d=\|\cdot\|$ and $d=\|\cdot\|^2$. Using this framework, it offers a new, elementary proof of the common-intersection property for the disks induced by a max-sum matching under squared Euclidean distance, and extends the argument to Euclidean quadrance via Helly's theorem. The results deepen the understanding of Tverberg-graph-type properties in geometric matchings and provide a unifying perspective across metrics.

Abstract

Huemer et al. (Discrete Math, 2019) proved that for any two finite point sets $R$ and $B$ in the plane with $|R| = |B|$, 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 nonempty common intersection. A pair of matched points induces the disk that has the segment connecting the points as diameter. In this note, we characterize these maximum-sum matchings for some family of continuous (semi-)metrics, focusing on both the Euclidean distance and squared Euclidean distance. Using this characterization, we give a different but simpler proof for the common intersection property proved by Huemer et al..

On maximum-sum matchings of bichromatic points

TL;DR

This work addresses max-sum matchings between bichromatic planar point sets under continuous (semi-)metrics, focusing on Euclidean distance and its square (quadrance). It introduces a precise 3-point characterization, linking max-sum optimality to the nonemptiness of five specific intersections of sets derived from the metric, with distinct boundary geometries for and . Using this framework, it offers a new, elementary proof of the common-intersection property for the disks induced by a max-sum matching under squared Euclidean distance, and extends the argument to Euclidean quadrance via Helly's theorem. The results deepen the understanding of Tverberg-graph-type properties in geometric matchings and provide a unifying perspective across metrics.

Abstract

Huemer et al. (Discrete Math, 2019) proved that for any two finite point sets and in the plane with , the perfect matching that matches points of with points of , and maximizes the total squared Euclidean distance of the matched pairs, has the property that all the disks induced by the matching have a nonempty common intersection. A pair of matched points induces the disk that has the segment connecting the points as diameter. In this note, we characterize these maximum-sum matchings for some family of continuous (semi-)metrics, focusing on both the Euclidean distance and squared Euclidean distance. Using this characterization, we give a different but simpler proof for the common intersection property proved by Huemer et al..
Paper Structure (4 sections, 7 theorems, 13 equations, 3 figures)

This paper contains 4 sections, 7 theorems, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Characterization of max-sum matchings of 6 points
  4. Common intersection for the Euclidean quadrance

Key Result

Lemma 2.1

If the five intersections $H(a,b) \cap H(b,c) \cap H(c,a)$, $h(a,b) \cap h(b,c) \cap h(c,a)$, $H(a,b)\cap h(a,b)$, $H(b,c)\cap h(b,c)$, and $H(c,a)\cap h(c,a)$ are all nonempty, then $\{ (a,a'),(b,b'),(c,c') \}$ is a max-sum matching of $\{a,b,c\}$ and $\{a',b',c'\}$.

Figures (3)

  • Figure 1: Definition of the sets $H(a,b)$ and $h(a,b)$ for (a) $\| \cdot \|$, and (b) $\| \cdot \|^2$ (the labels $b'$ and $a'$ are used to indicate that they are in the boundaries of $H(a,b)$ and $h(a,b)$, respectively).
  • Figure 2: (a) Strips $S(a,b)$, $S(b,c)$, and $S(c,a)$ with their common intersection. (b-c) Proof of Proposition \ref{['prop:1']}.
  • Figure 3: (a) Proof of Proposition \ref{['prop:2']}. (b-c) Proof of Case 2. (d) Proof of Case 3.

Theorems & Definitions (11)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Theorem 3.1: Helly helly1923
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 1 more