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

TL;DR

This paper introduces a simple, optimal online contention resolution scheme for $k$-uniform matroids that achieves near-perfect selectability against fixed-order adversaries, improving simplicity and matching theoretical limits.

Contribution

The authors present a new simple algorithm for online contention resolution in $k$-uniform matroids that achieves near-optimal selectability and simplifies previous approaches.

Findings

Achieves $(1-O(1/\sqrt{k}))$ selectability against fixed-order adversaries.

Proves no scheme can surpass $(1-\Omega(\sqrt{\log k / k}))$ selectability against an all-powerful adversary.

Matches the simple greedy algorithm's performance bound.

Abstract

We provide a simple $(1-O(\frac{1}{\sqrt{k}}))$-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 $1-\frac{1}{\sqrt{k}}$ any active element $i$ such that $|A_{i-1}| + 1 \leq (1-\frac{1}{\sqrt{k}})\cdot \mathbb{E}[|G_i|]+\sqrt{k}$. This implies a $(1-O(\frac{1}{\sqrt{k}}))$ 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 $(1-\Omega(\sqrt{\frac{\log k}{k}}))$-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 $1-\Theta(\sqrt{\frac{\log k}{k}})$ [HKS07].