  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 maps a subset of items and
their states to a non-negative real number. We aim to sequentially select a
group of items to achieve a ``target value'' while minimizing the maximum cost
across realizations (a.k.a. worst-case cost). To facilitate our study, we
assume that the utility function is \emph{worst-case submodular}, a property
that is commonly found in many machine learning applications. With this
assumption, we develop a tight $(\log (Q/\eta)+1)$-approximation policy, where
$Q$ is the ``target value'' and $\eta$ is the smallest difference between $Q$
and any achievable utility value $\hat{Q}<Q$. We also study a worst-case
maximum-coverage problem, a dual problem of the minimum-cost-cover problem,
whose goal is to select a group of items to maximize its worst-case utility
subject to a budget constraint. To solve this problem, we develop a
$(1-1/e)/2$-approximation solution.
