In this work, we present a constant-round algorithm for the $2$-ruling set problem in the Congested Clique model. As a direct consequence, we obtain a constant round algorithm in the MPC model with linear space-per-machine and optimal total space. Our results improve on the $O(\log \log \log n)$-round algorithm by [HPS, DISC'14] and the $O(\log \log \Delta)$-round algorithm by [GGKMR, PODC'18]. Our techniques can also be applied to the semi-streaming model to obtain an $O(1)$-pass algorithm. Our main technical contribution is a novel sampling procedure that returns a small subgraph such that almost all nodes in the input graph are adjacent to the sampled subgraph. An MIS on the sampled subgraph provides a $2$-ruling set for a large fraction of the input graph. As a technical challenge, we must handle the remaining part of the graph, which might still be relatively large. We overcome this challenge by showing useful structural properties of the remaining graph and show that running our process twice yields a $2$-ruling set of the original input graph with high probability.

翻译：在这项工作中，我们提出了一个在拥塞团簇模型下解决$2$-支配集问题的常数轮算法。作为直接结果，我们在MPC模型中获得了使用每台机器线性空间和最优总空间的常数轮算法。我们的结果改进了[HPS, DISC'14]的$O(\log \log \log n)$轮算法和[GGKMR, PODC'18]的$O(\log \log \Delta)$轮算法。我们的技术也可应用于半流模型，得到一个$O(1)$遍算法。我们的主要技术贡献是一种新颖的采样过程，该过程返回一个小子图，使得输入图中几乎所有节点都与该采样子图相邻。在采样子图上计算的一个MIS能够为输入图的大部分节点提供$2$-支配集。作为技术挑战，我们必须处理图中可能仍然较大的剩余部分。我们通过展示剩余图的有用结构性质来克服这一挑战，并证明运行两次该过程能够以高概率得到原始输入图的$2$-支配集。