Adaptive Greedy Algorithms for Stochastic Set Cover Problems

We study adaptive greedy algorithms for the problems of stochastic set cover with perfect and imperfect coverages. In stochastic set cover with perfect coverage, we are given a set of items and a ground set B. Evaluating an item reveals its state which is a random subset of B drawn from the state distribution of the item. Every element in B is assumed to be present in the state of some item with probability 1. For this problem, we show that the adaptive greedy algorithm has an approximation ratio of H(|B|), the |B|th Harmonic number. In stochastic set cover with imperfect coverage, an element in the ground set need not be present in the state of any item. We show a reduction from this problem to the former problem; the adaptive greedy algorithm for the reduced instance has an approxiation ratio of H(|E|), where E is the set of pairs (F, e) such that the state of item F contains e with positive probability.