We study the Euclidean minimum weight perfect matching problem for $n$ points in the plane. It is known that any deterministic approximation algorithm whose approximation ratio depends only on $n$ requires at least $Ω(n \log n)$ time. We propose such an algorithm for the Euclidean minimum weight perfect matching problem with runtime $O(n\log n)$ and show that it has approximation ratio $O(n^{0.206})$. This improves the so far best known approximation ratio of $n/2$. We also develop an $O(n \log n)$ algorithm for the Euclidean minimum weight perfect matching problem in higher dimensions and show it has approximation ratio $O(n^{0.412})$ in all fixed dimensions.

翻译：本文研究平面上 $n$ 个点的欧几里得最小权重完美匹配问题。已知任何近似比仅依赖于 $n$ 的确定性近似算法至少需要 $Ω(n \log n)$ 时间。我们针对欧几里得最小权重完美匹配问题提出了一种运行时间为 $O(n\log n)$ 的此类算法，并证明其近似比为 $O(n^{0.206})$。这改进了目前已知的最佳近似比 $n/2$。我们还为高维空间中的欧几里得最小权重完美匹配问题开发了一种 $O(n \log n)$ 算法，并证明其在所有固定维度下具有 $O(n^{0.412})$ 的近似比。