Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion

2025-10-23

AI Summary

This paper studies the online metric matching problem in the Euclidean metric space $[0,1]^d$: $n$ servers are placed adversarially in advance, and $n$ requests arrive sequentially; each request must be irrevocably matched to an idle server upon arrival, incurring cost equal to the Euclidean distance. Requests are independent but not identically distributed, satisfying only a mild smoothness condition. We present the first $o(log n)$-competitive algorithm for non-i.i.d. inputs in a nontrivial metric space, breaking the $Omega(log n)$ lower bound inherent to classical metric embedding approaches. Key technical innovations include a single-sample learning framework, deterministic metric embedding, smoothness analysis based on the cost of suboptimal solutions, and offline optimal cost lower bounds derived via stochastic majorization theory. For any dimension $d eq 2$, we design an $O(1)$-competitive algorithm requiring only one sample per distribution—without prior knowledge of the distributions—significantly improving over all existing methods.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the online metric matching problem, $n$ servers and $n$ requests lie in a metric space. Servers are available upfront, and requests arrive sequentially. An arriving request must be matched immediately and irrevocably to an available server, incurring a cost equal to their distance. The goal is to minimize the total matching cost. We study this problem in the Euclidean metric $[0, 1]^d$, when servers are adversarial and requests are independently drawn from distinct distributions that satisfy a mild smoothness condition. Our main result is an $O(1)$-competitive algorithm for $d eq 2$ that requires no distributional knowledge, relying only on a single sample from each request distribution. To our knowledge, this is the first algorithm to achieve an $o(log n)$ competitive ratio for non-trivial metrics beyond the i.i.d. setting. Our approach bypasses the $Ω(log n)$ barrier introduced by probabilistic metric embeddings: instead of analyzing the embedding distortion and the algorithm separately, we directly bound the cost of the algorithm on the target metric of a simple deterministic embedding. We then combine this analysis with lower bounds on the offline optimum for Euclidean metrics, derived via majorization arguments, to obtain our guarantees.

Problem

• Analyzing online metric matching with adversarial servers
• Using single samples from smooth request distributions
• Achieving constant competitive ratio in Euclidean spaces

Innovation

• Uses single sample per request distribution
• Employs deterministic embedding bypassing probabilistic barriers
• Combines majorization arguments for Euclidean optimum bounds