Stochastic Covering and Adaptivity

We introduce a class of “stochastic covering” problems where the target set X to be covered is fixed, while the “items” used in the covering are characterized by probability distributions over subsets of X. This is a natural counterpart to the stochastic packing problems introduced in [5]. In analogy to [5], we study both adaptive and non-adaptive strategies to find a feasible solution, and in particular the adaptivity gap, introduced in [4]. It turns out that in contrast to deterministic covering problems, it makes a substantial difference whether items can be used repeatedly or not. In the model of Stochastic Covering with item multiplicity, we show that the worst case adaptivity gap is Θ(log d), where d is the size of the target set to be covered, and this is also the best approximation factor we can achieve algorithmically. In the model without item multiplicity, we show that the adaptivity gap for Stochastic Set Cover can be Ω(d). On the other hand, we show that the adaptivity gap is bounded by O(d), by exhibiting an O(d)-approximation non-adaptive algorithm.