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

09/18/2023

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by   Atanas Dinev, et al.

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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 [Ala14, AKW14, JMZ22]. 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 accepts every active element with probability 1-Θ(√(log k/k)) [HKS07].