Fast Approximation Algorithms for Euclidean Minimum Weight Perfect Matching

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 $\mathrm{\Omega }(n\log n)$ 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 $n\mathrm{/}2$. 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.