Worst-Case Adaptive Submodular Cover

In this paper, we study the adaptive submodular cover problem under the worst-case setting. This problem generalizes many previously studied problems, namely, the pool-based active learning and the stochastic submodular set cover. The input of our problem is a set of items (e.g., medical tests) and each item has a random state (e.g., the outcome of a medical test), whose realization is initially unknown. One must select an item at a fixed cost in order to observe its realization. There is an utility function which is defined over items and their states. Our goal is to sequentially select a group of items to achieve a ``goal value'' while minimizing the maximum cost across realizations (a.k.a. worst-case cost). To facilitate our study, we introduce a broad class of stochastic functions, called \emph{worst-case submodular function}. Assume the utility function is worst-case submodular, we develop a tight $(\log (Q/\eta)+1)$-approximation policy, where $Q$ is the ``goal value'' and $\eta$ is the minimum gap between $Q$ and any attainable utility value $\hat{Q}<Q$. We also study a worst-case maximum-coverage problem, whose goal is to select a group of items to maximize its worst-case utility subject to a budget constraint. This is a flipped problem of the minimum-cost-cover problem, and to solve this problem, we develop a $(1-1/e)/2$-approximation solution.