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