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.