The SetCover problem has been extensively studied in many different models of computation, including parallel and distributed settings. From an approximation point of view, there are two standard guarantees: an $O(\log \Delta)$-approximation (where $\Delta$ is the maximum set size) and an $O(f)$-approximation (where $f$ is the maximum number of sets containing any given element). In this paper, we introduce a new, surprisingly simple, model-independent approach to solving SetCover in unweighted graphs. We obtain multiple improved algorithms in the MPC and CRCW PRAM models. First, in the MPC model with sublinear space per machine, our algorithms can compute an $O(f)$ approximation to SetCover in $\hat{O}(\sqrt{\log \Delta} + \log f)$ rounds, where we use the $\hat{O}(x)$ notation to suppress $\mathrm{poly} \log x$ and $\mathrm{poly} \log \log n$ terms, and a $O(\log \Delta)$ approximation in $O(\log^{3/2} n)$ rounds. Moreover, in the PRAM model, we give a $O(f)$ approximate algorithm using linear work and $O(\log n)$ depth. All these bounds improve the existing round complexity/depth bounds by a $\log^{\Omega(1)} n$ factor. Moreover, our approach leads to many other new algorithms, including improved algorithms for the HypergraphMatching problem in the MPC model, as well as simpler SetCover algorithms that match the existing bounds.

翻译：集合覆盖问题已在多种计算模型中得到广泛研究，包括并行与分布式环境。从近似算法的视角，存在两种标准保证：$O(\log \Delta)$-近似（其中 $\Delta$ 表示最大集合大小）与 $O(f)$-近似（其中 $f$ 表示包含任意给定元素的最大集合数）。本文提出了一种新颖且异常简洁的、模型无关的方法来解决无权图上的集合覆盖问题。我们在MPC与CRCW PRAM模型中获得了多项改进算法。首先，在每台机器具有亚线性存储空间的MPC模型中，我们的算法可在 $\hat{O}(\sqrt{\log \Delta} + \log f)$ 轮内计算出 $O(f)$ 近似的集合覆盖解（其中 $\hat{O}(x)$ 记号用于隐藏 $\mathrm{poly} \log x$ 与 $\mathrm{poly} \log \log n$ 项），并在 $O(\log^{3/2} n)$ 轮内实现 $O(\log \Delta)$ 近似。此外，在PRAM模型中，我们提出了一个使用线性工作量与 $O(\log n)$ 深度的 $O(f)$ 近似算法。所有这些边界均将现有轮复杂度/深度边界改进了 $\log^{\Omega(1)} n$ 因子。更重要的是，我们的方法催生了诸多新算法，包括MPC模型中超图匹配问题的改进算法，以及能与现有边界匹配的更简洁的集合覆盖算法。