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 uβU
to a set S(u) such that if UβU is formed by drawing n
times from distribution D, then the algorithm commits to outputting S(U).
Grandoni et al. gave an O(logmN)-competitive universal algorithm for this
stochastic set cover problem.
We improve unilaterally upon this result by giving a simple, polynomial time
O(logmn)-competitive universal algorithm for the more general prophet
version, in which U is formed by drawing from n different distributions
D1β,β¦,Dnβ. 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