Oblivious Online Contention Resolution Schemes

TL;DR

This paper introduces an optimal oblivious online contention resolution scheme that works without prior distribution knowledge, with applications in stochastic optimization, and proves limitations for schemes with limited samples.

Contribution

It presents a simple, optimal oblivious OCRS with no samples and establishes fundamental limitations for schemes with few samples in certain matroid settings.

Findings

Proposes a $rac{1}{e}$-selectable oblivious OCRS.

Proves optimality of the scheme via Ramsey theory.

Shows no $O(1)$ sample scheme can be $ ext{Omega}(1)$-balanced for certain matroids.

Abstract

Contention resolution schemes (CRSs) are powerful tools for obtaining "ex post feasible" solutions from candidates that are drawn from "ex ante feasible" distributions. Online contention resolution schemes (OCRSs), the online version, have found myriad applications in Bayesian and stochastic problems, such as prophet inequalities and stochastic probing. When the ex ante distribution is unknown, it was unknown whether good CRSs/OCRSs exist with no sample (in which case the scheme is oblivious) or few samples from the distribution. In this work, we give a simple $\frac{1}{e}$-selectable oblivious single item OCRS by mixing two simple schemes evenly, and show, via a Ramsey theory argument, that it is optimal. On the negative side, we show that no CRS or OCRS with $O(1)$ samples can be $\Omega(1)$-balanced/selectable (i.e., preserve every active candidate with a constant probability) for graphic or transversal matroids.