Online Metric Matching: Beyond the Worst Case

TL;DR

This paper advances online metric matching by providing improved algorithms with constant competitive ratios in high-dimensional spaces, addressing both stochastic and adversarial scenarios, and incorporating prediction-based enhancements.

Contribution

It introduces new algorithms with optimal competitive ratios for high-dimensional metric spaces and develops a framework to leverage predictions in adversarial settings.

Findings

Achieved $O(1)$-competitive algorithms for $d \,\geq\, 3$ in balanced and unbalanced markets.

Improved competitive ratio from $O((\log \log \log n)^2)$ to constant in certain regimes.

Provided a prediction-based framework that degrades gracefully with prediction accuracy.

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 an available 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 tractable setting where servers and requests are all independently drawn from the same distribution. Applying our reduction, for $[0, 1]^d$ with various choices of distributions, we achieve improved competitive ratios and nearly optimal regret 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 servers and requests are all adversarial, and a prediction of request locations is provided, we present a general framework for transforming an arbitrary algorithm that does not use predictions into an algorithm that leverages predictions. The transformation applies the given algorithm in a black-box manner, and the performance of the resulting algorithm degrades smoothly as the prediction accuracy deteriorates while preserving the worst-case guarantee.