Fast Approximation Algorithms for Euclidean Minimum Weight Perfect Matching
This paper studies the minimum-weight perfect matching problem for $n$ points in Euclidean space. For the two-dimensional case, we present the first deterministic approximation algorithm with $O(n log n)$ runtime—breaking the long-standing belief that the $Omega(n log n)$ lower bound was inherently unimprovable—and achieve an approximation ratio of $O(n^{0.206})$, substantially improving upon the prior best ratio of $n/2$. We further generalize this divide-and-conquer framework to any fixed dimension $d$, attaining a unified $O(n^{0.412})$ approximation ratio in $O(n log n)$ time. Our algorithm relies on geometric partitioning, local matching pruning, and recursive decomposition—requiring neither random sampling nor linear programming solvers. It thus bridges theoretical optimality and practical implementability.