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：本文提出了一种在拥塞派系模型中解决$2$-控制集问题的常数轮算法。作为直接推论，我们在主存计算模型中获得了一种常数轮算法，该算法每台机器使用线性空间且总空间最优。我们的结果改进了[HPS, DISC'14]提出的$O(\log \log \log n)$轮算法和[GGKMR, PODC'18]提出的$O(\log \log \Delta)$轮算法。我们的技术同样适用于半流模型，可获得$O(1)$轮算法。本文的主要技术贡献在于一种新颖的采样过程，该过程能返回一个小子图，使得输入图中几乎所有节点都与该采样子图相邻。在采样子图上计算极大独立集可为输入图的大部分节点提供$2$-控制集。技术难点在于处理图中可能仍相对较大的剩余部分。我们通过揭示剩余图的有用结构性质克服了这一挑战，并证明运行两次该过程能以高概率得到原始输入图的$2$-控制集。