We study the problem of finding a Euclidean minimum weight perfect matching for $n$ points in the plane. It is known that a deterministic approximation algorithm for this problems must have at least $\Omega(n \log n)$ runtime. 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：我们研究了在平面上为$n$个点寻找欧几里得最小权重完美匹配的问题。已知针对该问题的确定性近似算法至少需要$\Omega(n \log n)$的运行时间。我们为此问题提出了一种运行时间为$O(n\log n)$的算法，并证明其近似比为$O(n^{0.206})$。这改进了目前已知的最优近似比$n/2$。我们还为更高维度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。