Beyond pointwise submodularity: Non-monotone adaptive submodular maximization in linear time

In this paper, we study the non-monotone adaptive submodular maximization problem subject to a cardinality constraint. We first revisit random greedy algorithm proposed in \citep{gotovos2015non}, where they show that achieves $1/e$ approximation ratio if objective function is and pointwise submodular. It not clear whether same guarantee holds under submodularity (without resorting submodularity) or not. Our contribution submodularity. One limitation of it requires $O(n\times k)$ value oracle queries, $n$ size ground set $k$ second develop linear-time for problem. $1/e-\epsilon$ (this bound improved $1-1/e-\epsilon$ monotone case), using only $O(n\epsilon^{-2}\log \epsilon^{-1})$ queries. Notably, independent For case, propose faster expectation with $O(n \log \frac{1}{\epsilon})$ also generalize our by considering partition matroid constraint, fully functions.