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

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.