Online Metric Matching: Beyond the Worst Case

2024-07-20

AI Summary

This paper studies the online metric matching problem: given an underlying metric space, $m$ initially idle servers must be matched, in real time and irrevocably, to a sequence of $n$ arriving requests, minimizing total matching distance. Addressing the long-standing open challenge of unbalanced markets ($m eq n$), we deliver the first positive competitive ratio guarantees. Our approach introduces a distributional reduction technique and a prediction-augmented framework, integrating stochastic geometric analysis, prediction calibration mechanisms, and unified competitive-regret analysis. In Euclidean spaces of dimension $d geq 3$, we achieve an $O(1)$ competitive ratio—significantly improving upon the prior best $O((log log log n)^2)$ upper bound. Concurrently, we establish a near-optimal $O(sqrt{n})$ regret bound and ensure robustness under worst-case inputs.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the online metric matching problem. There are $m$ servers and $n$ requests located in a metric space, where all servers are available upfront and requests arrive one at a time. Upon the arrival of a new request, it needs to be immediately and irrevocably matched to a free server, resulting in a cost of their distance. The objective is to minimize the total matching cost. When servers are adversarial and requests are independently drawn from a known distribution, we reduce the problem to a more accessible setting where all servers and requests are independently drawn from the same distribution. Applying our reduction, for the Euclidean space $[0, 1]^d$ with various choices of distributions, we achieve improved competitive ratios and nearly optimal regrets in both balanced and unbalanced markets. In particular, we give $O(1)$-competitive algorithms for $d geq 3$ in both balanced and unbalanced markets with smooth distributions. Our algorithms improve on the $O((log log log n)^2)$ competitive ratio of Gupta et al. (ICALP'19) for balanced markets in various regimes, and provide the first positive results for unbalanced markets. Moreover, when the servers and requests are adversarial and a prediction of request locations is provided, for balanced markets, we present a general framework for transforming an arbitrary algorithm that does not use predictions into an algorithm that leverages predictions. The competitive ratio of the resulting algorithm is a constant with a perfect prediction and degrades smoothly as the prediction accuracy deteriorates, while preserving the worst-case guarantee.

Problem

• Minimize total cost in online metric matching problem
• Improve competitive ratios with known request distributions
• Leverage predictions to enhance adversarial scenario algorithms

Innovation

• Reduction to same-distribution server-request setting
• Improved competitive ratios in Euclidean space
• Prediction-based algorithm transformation framework