  Real-world problems such as ad allocation and matching have been extensively
studied under the lens of combinatorial optimization. In several applications,
uncertainty in the input appears naturally and this has led to the study of
online stochastic optimization models for such problems. For the offline case,
these constrained combinatorial optimization problems have been extensively
studied, and Contention Resolution Schemes (CRSs), introduced by Chekuri,
Vondr\'{a}k, and Zenklusen, have emerged in recent years as a general framework
to obtaining a solution. The idea behind a CRS is to first obtain a fractional
solution to a (continuous) relaxation of the objective and then round the
fractional solution to an integral one. When the order of rounding is
controlled by an adversary, Online Contention Resolution Schemes (OCRSs) can be
used instead, and have been successfully applied in settings such as prophet
inequalities and stochastic probing.
  In this work, we focus on greedy OCRSs, which provide guarantees against the
strongest possible adversary, an almighty adversary. 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, which improve upon the previous
state-of-the-art greedy OCRSs for these constraints. We also show that our
greedy OCRSs are optimal, even for the simple single-item case.
