In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets $R,B \subset [Δ]^2$ with $|R|+|B|=n$, the goal is to select a set of edges between $R$ and $B$ so that every point is incident to at least one edge and the total Euclidean length is minimized. In the general case that $R$ and $B$ are point sets in the plane, the best-known algorithm for the many-to-many matching problem takes $\tilde{O}(n^2)$ time. We present an exact $\tilde{O}(n^{1.5} \log Δ)$ time algorithm for point sets in $[Δ]^2$. To the best of our knowledge, this is the first subquadratic exact algorithm for planar many-to-many matching under bounded integer coordinates.

翻译：本文研究整数坐标平面点集上的多对多匹配问题：给定两个不相交集合$R,B \subset [Δ]^2$，且$|R|+|B|=n$，目标是选择$R$与$B$之间的一组边，使得每个点至少与一条边关联，且总欧氏长度最小化。在$R$和$B$为平面点集的一般情况下，多对多匹配问题的最优算法需$\tilde{O}(n^2)$时间。我们针对$[Δ]^2$中的点集提出了一种精确的$\tilde{O}(n^{1.5} \log Δ)$时间算法。据我们所知，这是整数有界坐标下平面多对多匹配的首个次二次精确算法。