Beyond Pointwise Submodularity: Non-Monotone Adaptive Submodular Maximization subject to a Knapsack Constraint

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.