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

Authors Atanas Dinev, S. Matthew Weinberg

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Atanas Dinev
  • Massachusetts Institute of Technology, Cambridge, MA, USA
S. Matthew Weinberg
  • Princeton University, NJ, USA

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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) https://doi.org/10.4230/LIPIcs.ITCS.2024.39

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].

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • online contention resolutions schemes
  • prophet inequalities
  • online algorithms
  • approximation algorithms

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