Set Covering with Our Eyes Wide Shut
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In the stochastic set cover problem (Grandoni et al., FOCS '08), we are given a collection 𝒮 of m sets over a universe 𝒰 of size N, and a distribution D over elements of 𝒰. The algorithm draws n elements one-by-one from D and must buy a set to cover each element on arrival; the goal is to minimize the total cost of sets bought during this process. A universal algorithm a priori maps each element u ∈𝒰 to a set S(u) such that if U ⊆𝒰 is formed by drawing n times from distribution D, then the algorithm commits to outputting S(U). Grandoni et al. gave an O(log mN)-competitive universal algorithm for this stochastic set cover problem. We improve unilaterally upon this result by giving a simple, polynomial time O(log mn)-competitive universal algorithm for the more general prophet version, in which U is formed by drawing from n different distributions D_1, …, D_n. Furthermore, we show that we do not need full foreknowledge of the distributions: in fact, a single sample from each distribution suffices. We show similar results for the 2-stage prophet setting and for the online-with-a-sample setting. We obtain our results via a generic reduction from the single-sample prophet setting to the random-order setting; this reduction holds for a broad class of minimization problems that includes all covering problems. We take advantage of this framework by giving random-order algorithms for non-metric facility location and set multicover; using our framework, these automatically translate to universal prophet algorithms.
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