Euclidean Maximum Matchings in the Plane---Local to Global

TL;DR

This paper investigates the relationship between local and global maximum matchings in the Euclidean plane, providing improved bounds for how well local maxima approximate global maxima and exploring properties of crossing matchings.

Contribution

It improves bounds on the ratio of local to global maximum matchings in the plane and proves the uniqueness and optimality of pairwise crossing matchings.

Findings

Improved bounds for 5_k: 7bd 5_2< 0.93 and 5_3< 0.98.

Every pairwise crossing matching is unique and globally maximum.

Disks with increased radii have a common intersection if radii are scaled by 2/7bd.

Abstract

Let $M$ be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that $M$ is globally maximum if it is a maximum-length matching on all points. We say that $M$ is $k$-local maximum if for any subset $M'=\{a_1b_1,\dots,a_kb_k\}$ of $k$ edges of $M$ it holds that $M'$ is a maximum-length matching on points $\{a_1,b_1,\dots,a_k,b_k\}$. We show that local maximum matchings are good approximations of global ones. Let $\mu_k$ be the infimum ratio of the length of any $k$-local maximum matching to the length of any global maximum matching, over all finite point sets in the Euclidean plane. It is known that $\mu_k\geqslant \frac{k-1}{k}$ for any $k\geqslant 2$. We show the following improved bounds for $k\in\{2,3\}$: $\sqrt{3/7}\leqslant\mu_2< 0.93 $ and $\sqrt{3}/2\leqslant\mu_3< 0.98$. We also show that every pairwise crossing matching is unique and it is globally maximum. Towards our proof of the lower bound for $\mu_2$ we show the following result which is of independent interest: If we increase the radii of pairwise intersecting disks by factor $2/\sqrt{3}$, then the resulting disks have a common intersection.