Geometric matching is an important topic in computational geometry and has been extensively studied over decades. In this paper, we study a geometric-matching problem, known as geometric many-to-many matching. In this problem, the input is a set $S$ of $n$ colored points in $\mathbb{R}^d$, which implicitly defines a graph $G = (S,E(S))$ where $E(S) = \{(p,q): p,q \in S \text{ have different colors}\}$, and the goal is to compute a minimum-cost subset $E^* \subseteq E(S)$ of edges that cover all points in $S$. Here the cost of $E^*$ is the sum of the costs of all edges in $E^*$, where the cost of a single edge $e$ is the Euclidean distance (or more generally, the $L_p$-distance) between the two endpoints of $e$. Our main result is a $(1+\varepsilon)$-approximation algorithm with an optimal running time $O_\varepsilon(n \log n)$ for geometric many-to-many matching in any fixed dimension, which works under any $L_p$-norm. This is the first near-linear approximation scheme for the problem in any $d \geq 2$. Prior to this work, only the bipartite case of geometric many-to-many matching was considered in $\mathbb{R}^1$ and $\mathbb{R}^2$, and the best known approximation scheme in $\mathbb{R}^2$ takes $O_\varepsilon(n^{1.5} \cdot \mathsf{poly}(\log n))$ time.

翻译：几何匹配是计算几何中的重要课题，历经数十年广泛研究。本文研究一类称为几何多对多匹配的几何匹配问题。该问题中，输入为$\mathbb{R}^d$空间中一组含$n$个彩色点的集合$S$，其隐式定义图$G = (S,E(S))$，其中$E(S) = \{(p,q): p,q \in S \text{具有不同颜色}\}$，目标在于计算能够覆盖$S$中所有点的最小代价边子集$E^* \subseteq E(S)$。此处$E^*$的代价定义为其中所有边的代价之和，而单条边$e$的代价为其两端点间的欧氏距离（更一般地，$L_p$距离）。我们的主要成果是在任意固定维度下，针对几何多对多匹配提出一种$(1+\varepsilon)$-近似算法，其最优运行时间为$O_\varepsilon(n \log n)$，且适用于任意$L_p$范数。这是该问题在任意$d \geq 2$维度下首个近线性近似方案。此前工作仅考虑$\mathbb{R}^1$和$\mathbb{R}^2$中几何多对多匹配的二部情形，其中$\mathbb{R}^2$中已知最优近似方案需耗时$O_\varepsilon(n^{1.5} \cdot \mathsf{poly}(\log n))$。