  The problem of online matching with stochastic rewards is a generalization of
the online bipartite matching problem where each edge has a probability of
success. When a match is made it succeeds with the probability of the
corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012)
focused on the special case of identical edge probabilities. Comparing against
a deterministic offline LP, they showed that the Ranking algorithm of Karp et
al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with
an improved guarantee of $0.567$ for vanishingly small probabilities. For the
case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA
2015), gave a 0.534 competitive algorithm against the same LP benchmark. For
the more general vertex-weighted version of the problem, to the best of our
knowledge, no results being $1/2$ were previously known even for identical
probabilities.
  We focus on the vertex-weighted version and give two improvements. First, we
show that a natural generalization of the Perturbed-Greedy algorithm of
Aggarwal et al. (SODA 2011), is $(1-1/e)$ competitive when probabilities
decompose as a product of two factors, one corresponding to each vertex of the
edge. This is the best achievable guarantee as it includes the case of
identical probabilities and in particular, the classical online bipartite
matching problem. Second, we give a deterministic $0.596$ competitive algorithm
for the previously well studied case of fully heterogeneous but vanishingly
small edge probabilities. A key contribution of our approach is the use of
novel path-based analysis. This allows us to compare against the natural
benchmarks of adaptive offline algorithms that know the sequence of arrivals
and the edge probabilities in advance, but not the outcomes of potential
matches.
