Worst-Case Adaptive Submodular Cover
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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 worst-case submodular function. Assume the utility function is worst-case submodular, we develop a tight (log (Q/η)+1)-approximation policy, where Q is the “goal value” and η is the minimum gap between Q and any attainable utility value Q̂<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.
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