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。本文研究一种名为几何多对多匹配的几何匹配问题。该问题的输入是 $\mathbb{R}^d$ 空间中由 $n$ 个带颜色点构成的集合 $S$，其隐式定义了一个图 $G = (S,E(S))$，其中 $E(S) = \{(p,q): p,q \in S \text{ 且颜色不同}\}$，目标是最小化覆盖 $S$ 中所有点的边子集 $E^* \subseteq E(S)$ 的成本。这里 $E^*$ 的成本定义为其中所有边的成本之和，单条边 $e$ 的成本为两端点之间的欧氏距离（或更一般的 $L_p$ 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问题提出一种 $(1+\varepsilon)$ 近似算法，其最优运行时间为 $O_\varepsilon(n \log n)$，适用于任意 $L_p$ 范数。这是该问题在 $d \geq 2$ 维度中首个近线性近似方案。此前，几何多对多匹配问题仅在 $\mathbb{R}^1$ 和 $\mathbb{R}^2$ 中考虑过二部情形，且 $\mathbb{R}^2$ 中已知最佳近似方案需要 $O_\varepsilon(n^{1.5} \cdot \mathsf{poly}(\log n))$ 时间。