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DOI: 10.4230/LIPIcs.ESA.2025.115

Smoothed Analysis of Online Metric Problems

Authors: Christian Coester and Jack Umenberger

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

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)

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@InProceedings{coester_et_al:LIPIcs.ESA.2025.115,
  author =	{Coester, Christian and Umenberger, Jack},
  title =	{{Smoothed Analysis of Online Metric Problems}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{115:1--115:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.115},
  URN =		{urn:nbn:de:0030-drops-245847},
  doi =		{10.4230/LIPIcs.ESA.2025.115},
  annote =	{Keywords: Online Algorithms, Competitive Analysis, Smoothed Analysis, k-server, k-taxi, Metrical Service Systems}
}

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Smoothed Analysis of Online Metric Problems

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