  We consider the maximum bipartite matching problem in stochastic settings,
namely the query-commit and price-of-information models. In the query-commit
model, an edge e independently exists with probability $p_e$. We can query
whether an edge exists or not, but if it does exist, then we have to take it
into our solution. In the unweighted case, one can query edges in the order
given by the classical online algorithm of Karp, Vazirani, and Vazirani to get
a $(1-1/e)$-approximation. In contrast, the previously best known algorithm in
the weighted case is the $(1/2)$-approximation achieved by the greedy algorithm
that sorts the edges according to their weights and queries in that order.
  Improving upon the basic greedy, we give a $(1-1/e)$-approximation algorithm
in the weighted query-commit model. We use a linear program (LP) to upper bound
the optimum achieved by any strategy. The proposed LP admits several structural
properties that play a crucial role in the design and analysis of our
algorithm. We also extend these techniques to get a $(1-1/e)$-approximation
algorithm for maximum bipartite matching in the price-of-information model
introduced by Singla, who also used the basic greedy algorithm to give a
$(1/2)$-approximation.
