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 $d_{r} > 0$. An assignment is a set $\mathcal{A}$ of ordered triples $(p,r,a_{pr}) \in P \times R \times \mathbb{R}_{>0}$ such that $p \in r$. We show how to compute a maximum assignment that satisfies the constraints given by the supplies and demands. 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 $n$ 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$P$为至多$n$个点的集合，$R$为至多$n$个几何范围（例如圆盘或矩形）的集合，其中每个$p \in P$具有关联供给量$s_{p} > 0$，每个$r \in R$具有关联需求量$d_{r} > 0$。一个指派是一个有序三元组$(p,r,a_{pr}) \in P \times R \times \mathbb{R}_{>0}$的集合$\mathcal{A}$，满足$p \in r$。我们展示了如何计算满足供给和需求约束的最大指派。利用我们的技术，我们还可以解决最小瓶颈问题，例如计算一组$n$个红点$P$与$n$个蓝点$Q$之间的完美匹配，使得最长边的长度最小化。对于$L_\infty$度量，我们可以在任意固定维度中以$O(n^{1+\varepsilon})$的时间完成；对于平面上的$L_2$度量，可以在$O(n^{4/3 + \varepsilon})$的时间内完成，其中$\varepsilon > 0$。