  Within the context of stochastic probing with commitment, we consider the
online stochastic matching problem; that is, the one-sided online bipartite
matching problem where edges adjacent to an online node must be probed to
determine if they exist based on edge probabilities that become known when an
online vertex arrives. If a probed edge exists, it must be used in the matching
(if possible). We consider the competitiveness of online algorithms in both the
adversarial order model (AOM) and the random order model (ROM). More
specifically, we consider a bipartite stochastic graph $G = (U,V,E)$ where $U$
is the set of offline vertices, $V$ is the set of online vertices and $G$ has
edge probabilities $(p_{e})_{e \in E}$ and edge weights $(w_{e})_{e \in E}$.
Additionally, $G$ has probing constraints $(\scr{C}_{v})_{v \in V}$, where
$\scr{C}_v$ indicates which sequences of edges adjacent to an online vertex $v$
can be probed. We assume that $U$ is known in advance, and that $\scr{C}_v$,
together with the edge probabilities and weights adjacent to an online vertex
are only revealed when the online vertex arrives. This model generalizes the
various settings of the classical bipartite matching problem, and so our main
contribution is in making progress towards understanding which classical
results extend to the stochastic probing model.
