Keywords: contention resolution schemes, online algorithms, matroids, prophet inequalities
Abstract: Matching based markets, like ad auctions, ride-sharing, and eBay, are inherently online and combinatorial, and therefore have been extensively studied under the lens of online stochastic combinatorial optimization models. The general framework that has emerged uses Contention Resolution Schemes (CRSs) introduced by Chekuri, Vondrák, and Zenklusen for combinatorial problems, where one first obtains a fractional solution to a (continuous) relaxation of the objective, and then proceeds to round it. When the order of rounding is controlled by an adversary, it is called an Online Contention Resolution Scheme (OCRSs), which has been successfully applied in online settings such as posted-price mechanisms, prophet inequalities and stochastic probing. The study of greedy OCRSs against an almighty adversary has emerged as one of the most interesting problems since it gives a simple-to-implement scheme against the worst possible scenario. Intuitively, a greedy OCRS has to make all its decisions before the online process starts. We present simple $1/e$ - selectable greedy OCRSs for the single-item setting, partition matroids, and transversal matroids. This improves upon the previous state-of-the-art greedy OCRSs of [FSZ16] that achieves $1/4$ for these constraints. Finally, we show that no better competitive ratio than $1/e$ is possible, making our greedy OCRSs the best possible.
Supplementary Material: zip
TL;DR: We give the first optimal greedy OCRS for the single-item setting, improving both upper and lower bounds to 1/e.