Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $p\in S$, $q\in T$ and for each $r\in S\cup T$, $r$ appears in at least one such pair. The {cost of a pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In the {minimum-cost many-to-many matching} problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in $O(n^3)$ time. In a more restricted setting where all the points are on a line, the problem can be solved in $O(n\log n)$ time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an $O(n^2\cdot poly(\log n))$ time exact algorithm and an $O( n^{3/2}\cdot poly(\log n))$ time $(1+\epsilon)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].

中文翻译：

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平面内多对多点匹配的精确和近似算法

给定飞机中的两组点 $S$ 和 $T$，总大小为 $n$，$S$ 和 $T$ 之间的{多对多}匹配是一组对 $(p,q )$ 使得 $p\in S$, $q\in T$ 并且对于每个 $r\in S\cup T$，$r$ 出现在至少一个这样的对中。{cost of a pair} $(p,q)$ 是 $p$ 和 $q$ 之间的（欧几里得）距离。在 {minimum-cost many-to-many matching} 问题中，目标是计算多对多匹配，使得对成本的总和最小化。这个问题是二部图中最小权重边覆盖的受限版本，因此可以在 $O(n^3)$ 时间内解决。在所有点都在一条线上的更受限制的设置中，问题可以在 $O(n\log n)$ 时间内解决 [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; 图梳，2007]。然而，在一般平面情况下，在改进三次时间界限方面没有取得任何进展。在本文中，我们得到了一个 $O(n^2\cdot poly(\log n))$ 时间精确算法和一个 $O( n^{3/2}\cdot poly(\log n))$ time $ (1+\epsilon)$-在平面情况下的近似值。我们的结果肯定地解决了 [Colannino et al., Graphs Comb., 2007] 中提出的一个开放问题。