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

Authors Yingxi Li , Ellen Vitercik , Mingwei Yang

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ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online algorithm
  • Metric matching
  • Competitive analysis
  • Smoothed analysis

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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 [0, 1]^d with the Euclidean metric, 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 ≠ 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 space 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.

Cite As Get BibTex

Yingxi Li, Ellen Vitercik, and Mingwei Yang. Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 94:1-94:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.94

Author Details

Yingxi Li
  • Department of Management Science and Engineering, Stanford University, CA, USA
Ellen Vitercik
  • Department of Management Science and Engineering, Department of Computer Science, Stanford University, CA, USA
Mingwei Yang
  • Department of Management Science and Engineering, Stanford University, CA, USA

Funding

  • Li, Yingxi: Research supported in part by NSF grant CCF-2338226.
  • Vitercik, Ellen: Research supported in part by NSF grant CCF-2338226.

Acknowledgements

We would like to thank the anonymous reviewers for their many helpful comments and suggestions.

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