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 $k\in \{2,3\}$: $\mu _2\geqslant \sqrt{3/7}$ and $\mu _3\geqslant 1/\sqrt{2}$. 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.

Supported by NSERC.