平面多对多点匹配的精确和近似算法
Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane
摘要：
给定平面中的点的两个集合$S$和$T$，总大小为$n$，$S$和$T$之间的{多对多}匹配是对$(p,q)$的集合，使得$p\in S$、$q\in T$以及对于每个$r\in S\cup T$，$r$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（欧几里德）距离。在{minimum-cost many-to-many matching}问题中，目标是计算多对多匹配，使得对的成本之和最小化。这个问题是二分图中最小权边覆盖的限制版本，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，可以在$O(n\log n)$时间内解决问题[Colannino，Damian，Hurtado，Langerman，Meijer，Ramaswami，Souvaine，Toussaint; Graphs Comb.，2007年]。然而，没有取得进展，在一般平面的情况下，在提高立方时间界。本文给出了平面情形下的$O(n^2\cdot poly(\log n))$时间精确算法和$O( n^{3/2}\cdot poly(\log n))$时间$(1+ε)$ -近似.我们的结果肯定地解决了[Colannino et al.，图表组合，2007年]。
英文摘要：
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 $(1+ε)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].
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