Fast Approximation Algorithms for Euclidean Minimum Weight Perfect Matching

TL;DR

This paper introduces fast approximation algorithms for the Euclidean minimum weight perfect matching problem, achieving near-linear runtime with improved approximation ratios in two and higher dimensions.

Contribution

It presents the first near-linear time algorithms with provable approximation guarantees for Euclidean minimum weight perfect matching in 2D and higher dimensions.

Findings

Achieves $O(n \, ext{log} \, n)$ runtime with $O(n^{0.206})$ approximation in 2D.

Develops an $O(n \, ext{log} \, n)$ algorithm with $O(n^{0.412})$ approximation in fixed higher dimensions.

Improves previous approximation ratio from $n/2$ to $O(n^{0.206})$ in 2D.

Abstract

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 $\Omega(n \log n)$ 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 $n/2$. 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.