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In this paper, we study the non-monotone adaptive submodular maximizationproblem subject to a knapsack and a $k$-system constraints. The input of ourproblem is a set of items, where each item has a particular state drawn from aknown 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 autility function which is defined over items and states. Our objective is tosequentially select a group of items to maximize the expected utility. Althoughthe cardinality-constrained non-monotone adaptive submodular maximization hasbeen well studied in the literature, whether there exists a constantapproximation solution for the knapsack-constrained or $k$-system constrainedadaptive submodular maximization problem remains an open problem. It fact, ithas only been settled given the additional assumption of pointwisesubmodularity. In this paper, we remove the common assumption on pointwisesubmodularity and propose the first constant approximation solutions for bothcases. Inspired by two recent studies on non-monotone adaptive submodularmaximization, we develop a sampling-based randomized algorithm that achieves a$\frac{1}{10}$ approximation for the case of a knapsack constraint and thatachieves a $\frac{1}{2k+4}$ approximation ratio for the case of a $k$-systemconstraint.

Beyond Pointwise Submodularity: Non-Monotone Adaptive Submodular Maximization subject to Knapsack and $k$-System Constraints