  We study single-sample prophet inequalities (SSPIs), i.e., prophet
inequalities where only a single sample from each prior distribution is
available. Besides a direct, and optimal, SSPI for the basic single choice
problem [Rubinstein et al., 2020], most existing SSPI results were obtained via
an elegant, but inherently lossy, reduction to order-oblivious secretary (OOS)
policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an
intuitive and versatile greedy-based technique that yields SSPIs directly
rather than through the reduction to OOSs. Our results can be seen as
generalizing and unifying a number of existing results in the area of prophet
and secretary problems. Our algorithms significantly improve on the competitive
guarantees for a number of interesting scenarios (including general matching
with edge arrivals, bipartite matching with vertex arrivals, and certain
matroids), and capture new settings (such as budget additive combinatorial
auctions). Complementing our algorithmic results, we also consider mechanism
design variants. Finally, we analyze the power and limitations of different
SSPI approaches by providing a partial converse to the reduction from SSPI to
OOS given by Azar et al.
