  In this paper, we study the non-monotone adaptive submodular maximization
problem subject to a knapsack and a $k$-system constraints. 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. 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. Although
the cardinality-constrained non-monotone adaptive submodular maximization has
been well studied in the literature, whether there exists a constant
approximation solution for the knapsack-constrained or $k$-system constrained
adaptive submodular maximization problem remains an open problem. It fact, it
has only been settled given the additional assumption of pointwise
submodularity. In this paper, we remove the common assumption on pointwise
submodularity and propose the first constant approximation solutions for both
cases. Inspired by two recent studies on non-monotone adaptive submodular
maximization, we develop a sampling-based randomized algorithm that achieves a
$\frac{1}{10}$ approximation for the case of a knapsack constraint and that
achieves a $\frac{1}{2k+4}$ approximation ratio for the case of a $k$-system
constraint.
