Online Metric Matching: Beyond the Worst Case

TL;DR

This work studies online metric matching with $m$ servers and $n$ requests in a metric space, seeking to minimize total distance. It introduces a generic, black-box framework to leverage predictions, converting prediction-oblivious online algorithms into prediction-aware ones with performance that degrades gracefully with prediction error. The authors establish strong results in semi-stochastic and prediction settings for Euclidean metrics, achieving $O(1)$-competitive guarantees in dimensions $d\ge 3$ with smooth distributions and near-optimal regret in several regimes, while also extending to unbalanced markets. They further show that greedy online algorithms with local, non-increasing tie-breaking preserve their competitive ratios in unbalanced adversarial settings and provide a unified approach bridging adversarial, stochastic, and predictive paradigms with practical implications for real-time matching markets. The work opens multiple avenues, including constant-factor results for broader metrics, robustness under imperfect predictions, and extensions to more general cost structures.

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.

Theorems & Definitions (52)

• Definition 1: Smoothness
• Lemma 1: talagrand1992ajtaidobric1995asymptoticsbobkov2019one
• Lemma 2
• Lemma 3
• Lemma 4: DBLP:conf/sigecom/ChenKKZ23
• Theorem 1
• proof
• Theorem 2
• Lemma 5
• proof