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.

翻译：设$M$为平面上点集的一个完美匹配，其中每条边均为两点间的线段。若$M$在所有点上构成最大长度匹配，则称其为全局最大匹配。若对于$M$中任意$k$条边的子集$M'=\{a_1b_1,\dots,a_kb_k\}$，$M'$在点集$\{a_1,b_1,\dots,a_k,b_k\}$上均为最大长度匹配，则称$M$为$k$-局部最大匹配。本文证明局部最大匹配可有效逼近全局最大匹配。令$\mu_k$表示在欧几里得平面上所有有限点集中，任意$k$-局部最大匹配长度与任意全局最大匹配长度的下确界比值。已知对于任意$k\geqslant 2$有$\mu_k\geqslant \frac{k-1}{k}$。本文针对$k\in\{2,3\}$给出改进界：$\sqrt{3/7}\leqslant\mu_2< 0.93$ 且 $\sqrt{3}/2\leqslant\mu_3< 0.98$。同时证明任意两两相交的匹配具有唯一性且必为全局最大匹配。在证明$\mu_2$下界的过程中，我们得到了以下独立意义的结果：若将两两相交圆盘的半径扩大$2/\sqrt{3}$倍，则所得圆盘存在公共交点。