Let $P$ be a set of at most $n$ points and let $R$ be a set of at most $n$ geometric ranges, such as for example disks or rectangles, where each $p \in P$ has an associated supply $s_{p} > 0$, and each $r \in R$ has an associated demand $d_{r} > 0$. A (many-to-many) matching is a set $\mathcal{A}$ of ordered triples $(p,r,a_{pr}) \in P \times R \times \mathbb{R}_{>0}$ such that $p \in r$ and the $a_{pr}$'s satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing $\sum_{(p,r,a_{pr}) \in \mathcal{A}} a_{pr}$. Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of $n$ red points $P$ and a set of $n$ blue points $Q$ that minimizes the length of the longest edge. For the $L_\infty$-metric, we can do this in time $O(n^{1+\varepsilon})$ in any fixed dimension, for the $L_2$-metric in the plane in time $O(n^{4/3 + \varepsilon})$, for any $\varepsilon > 0$.

翻译：设 $P$ 为最多包含 $n$ 个点的集合，$R$ 为最多包含 $n$ 个几何范围（例如圆盘或矩形）的集合，其中每个 $p \in P$ 关联一个供给量 $s_{p} > 0$，每个 $r \in R$ 关联一个需求量 $d_{r} > 0$。（多对多）匹配是一个有序三元组集合 $\mathcal{A} \subset P \times R \times \mathbb{R}_{>0}$，满足 $p \in r$ 且 $a_{pr}$ 满足供给与需求约束。本文展示了如何计算最大匹配，即最大化 $\sum_{(p,r,a_{pr}) \in \mathcal{A}} a_{pr}$ 的匹配。利用我们的技术，还可解决最小瓶颈问题，例如计算将 $n$ 个红点 $P$ 与 $n$ 个蓝点 $Q$ 进行完全匹配并最小化最长边长度的问题。对于 $L_\infty$ 度量，可在任意固定维度中以 $O(n^{1+\varepsilon})$ 时间完成；对于平面中的 $L_2$ 度量，可在 $O(n^{4/3 + \varepsilon})$ 时间内完成，其中 $\varepsilon > 0$ 为任意常数。