  In this paper, we study the non-monotone adaptive submodular maximization
problem subject to a knapsack constraint. The input of our problem is a set of
items, where each item has a particular state drawn from a known prior
distribution. However, the state of an item is initially unknown, one must
select an item in order to reveal the state of that item. Moreover, each item
has a fixed cost. There is a utility function which is defined over items and
states. Our objective is to sequentially select a group of items to maximize
the expected utility subject to a knapsack constraint. Although the
cardinality-constrained, as well as the more general matroid-constrained,
adaptive submodular maximization has been well studied in the literature,
whether there exists a constant approximation solution for the
knapsack-constrained adaptive submodular maximization problem remains an open
problem. We fill this gap by proposing the first constant approximation
solution. In particular, our main contribution is to develop a sampling-based
randomized algorithm that achieves a $\frac{1}{10}$ approximation for
maximizing an adaptive submodular function subject to a knapsack constraint.
