Fast Approximation Algorithms for Euclidean Minimum Weight Perfect Matching

2024-07-10

Workshop on Approximation and Online Algorithms

AI Summary

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.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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 $O(nlog n)$ 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.

Problem

• Euclidean minimum weight perfect matching for planar points
• Deterministic approximation algorithm with O(n log n) runtime
• Improved approximation ratio in higher dimensions

Innovation

• Euclidean matching algorithm in O(n log n) time
• Achieves O(n^0.206) approximation ratio
• Extends to higher dimensions efficiently