Optimal Analysis of Greedy for Stochastic Online Euclidean Matching

2026-09-20

AI Summary

研究了贪婪算法在随机在线欧几里得匹配问题中的表现，通过分析平环上的算法并转移估计到立方体上，证明了其在不同维度下的竞争比。

Abstract

We study Greedy for online metric matching with $n$ servers and $n$ requests sampled independently and uniformly from $[0,1]^d$. Servers are available initially, and Greedy irrevocably matches each arriving request to its closest available server, incurring a cost of their distance. We prove that Greedy has competitive ratio $O(1)$ for every fixed $d\ne2$, and $Θ(\sqrt{\log n})$ for $d=2$. Previously, constant competitiveness was shown for $d = 1$ [BFP23], and no non-trivial results for this setting were known for higher dimensions. Our proof first analyzes Greedy on the flat torus and then transfers the estimates back to the cube.

Problem

• Greedy
• Stochastic Online Euclidean Matching
• Competitive Ratio
• Dimensions

Innovation

• Greedy Algorithm
• Online Euclidean Matching
• Competitive Ratio
• Flat Torus