  In this paper, we study a new stochastic submodular maximization problem with
state-dependent costs and rejections. The input of our problem is a budget
constraint $B$, and a set of items whose states (i.e., the marginal
contribution and the cost of an item) are drawn from a known probability
distribution. The only way to know the realized state of an item is to probe
that item. We allow rejections, i.e., after probing an item and knowing its
actual state, we must decide immediately and irrevocably whether to add that
item to our solution or not. Our objective is to sequentially probe/selet a
best group of items subject to a budget constraint on the total cost of the
selected items. We present a constant approximate solution to this problem. We
show that our solution can be extended to an online setting.
