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Smoothed Analysis of Online Metric Problems

Authors Christian Coester , Jack Umenberger

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Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Online Algorithms
  • Competitive Analysis
  • Smoothed Analysis
  • k-server
  • k-taxi
  • Metrical Service Systems

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Abstract

We study three classical online problems - k-server, k-taxi, and chasing size k sets - through a lens of smoothed analysis. Our setting allows request locations to be adversarial up to small perturbations, interpolating between worst-case and average-case models. Specifically, we show that if the metric space is contained in a ball in any normed space and requests are drawn from distributions whose density functions are upper bounded by 1/σ times the uniform density over the ball, then all three problems admit polylog(k/σ)-competitive algorithms. Our approach is simple: it reduces smoothed instances to fully adversarial instances on finite metrics and leverages existing algorithms in a black-box manner. We also provide a lower bound showing that no algorithm can achieve a competitive ratio sub-polylogarithmic in k/σ, matching our upper bounds up to the exponent of the polylogarithm. In contrast, the best known competitive ratios for these problems in the fully adversarial setting are 2k-1, ∞ and Θ(k²), respectively.

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Christian Coester and Jack Umenberger. Smoothed Analysis of Online Metric Problems. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 115:1-115:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.115

Author Details

Christian Coester
  • Department of Computer Science, University of Oxford, UK
Jack Umenberger
  • Department of Engineering Science, University of Oxford, UK

Funding

  • Coester, Christian: Funded by the European Union (ERC grant CCOO 101165139). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
  • Umenberger, Jack: Supported by the University of Oxford’s Strategic Research Fund through the ZERO Institute.

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