In the stochastic set cover problem (Grandoni et al., FOCS '08), we are given a collection $\mathcal{S}$ of $m$ sets over a universe $\mathcal{U}$ of size $N$, and a distribution $D$ over elements of $\mathcal{U}$. 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 $u \in \mathcal{U}$ to a set $S(u)$ such that if $U \subseteq \mathcal{U}$ 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 $D_1, \ldots, D_n$. 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.

翻译：在随机集合覆盖问题（Grandoni等人，FOCS '08）中，我们给定一个由$m$个集合组成的集合族$\mathcal{S}$，其定义在大小为$N$的全集$\mathcal{U}$上，以及一个关于$\mathcal{U}$中元素的分布$D$。算法从$D$中依次抽取$n$个元素，并且必须在每个元素到达时购买一个集合来覆盖它；目标是最小化此过程中所购买集合的总成本。一种通用算法事先将每个元素$u \in \mathcal{U}$映射到一个集合$S(u)$，使得如果通过从分布$D$中抽取$n$次得到$U \subseteq \mathcal{U}$，则该算法承诺输出$S(U)$。Grandoni等人针对此随机集合覆盖问题给出了一种$O(\log mN)$竞争的通用算法。我们在更一般的预言机版本（其中$U$由从$n$个不同分布$D_1, \ldots, D_n$中抽取得到）上，通过给出一种简单的、多项式时间的$O(\log mn)$竞争通用算法，全面改进了这一结果。此外，我们证明我们不需要完全预知这些分布：实际上，每个分布的一个样本就足够了。我们针对两阶段预言机设置和带样本的在线设置展示了类似的结果。我们通过一种从单样本预言机设置到随机顺序设置的通用归约得到了我们的结果；这种归约适用于包括所有覆盖问题在内的一类广泛的极小化问题。我们利用这一框架给出了非度量设施选址和集合多重覆盖的随机顺序算法；利用我们的框架，这些算法自动转化为通用预言机算法。