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DOI: 10.4230/LIPIcs.ITCS.2024.39

Simple and Optimal Online Contention Resolution Schemes for k-Uniform Matroids

Authors: Atanas Dinev and S. Matthew Weinberg

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract

We provide a simple (1-O(1/(√{k)}))-selectable Online Contention Resolution Scheme for k-uniform matroids against a fixed-order adversary. If A_i and G_i denote the set of selected elements and the set of realized active elements among the first i (respectively), our algorithm selects with probability 1-1/(√{k)} any active element i such that |A_{i-1}| + 1 ≤ (1-1/(√{k)})⋅ 𝔼[|G_i|]+√k. This implies a (1-O(1/(√{k)})) prophet inequality against fixed-order adversaries for k-uniform matroids that is considerably simpler than previous algorithms [Alaei, 2014; Azar et al., 2014; Jiang et al., 2022]. We also prove that no OCRS can be (1-Ω(√{(log k)/k}))-selectable for k-uniform matroids against an almighty adversary. This guarantee is matched by the (known) simple greedy algorithm that selects every active element with probability 1-Θ(√{(log k)/k}) [Hajiaghayi et al., 2007].

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Atanas Dinev and S. Matthew Weinberg. Simple and Optimal Online Contention Resolution Schemes for k-Uniform Matroids. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dinev_et_al:LIPIcs.ITCS.2024.39,
  author =	{Dinev, Atanas and Weinberg, S. Matthew},
  title =	{{Simple and Optimal Online Contention Resolution Schemes for k-Uniform Matroids}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{39:1--39:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.39},
  URN =		{urn:nbn:de:0030-drops-195677},
  doi =		{10.4230/LIPIcs.ITCS.2024.39},
  annote =	{Keywords: online contention resolutions schemes, prophet inequalities, online algorithms, approximation algorithms}
}

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Simple and Optimal Online Contention Resolution Schemes for k-Uniform Matroids

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