Fast Approximation Algorithms for Euclidean Minimum Weight Perfect Matching

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 Ω(n log n) runtime. We propose such an algorithm for the Euclidean minimum weight perfect matching problem with runtime O(nlog n) and show that it has approximation ratio O(n0.2995). 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(n0.599) in all fixed dimensions.