In this paper, we present efficient distributed algorithms for classical symmetry breaking problems, maximal independent sets (MIS) and ruling sets, in power graphs. We work in the standard CONGEST model of distributed message passing, where the communication network is abstracted as a graph $G$. Typically, the problem instance in CONGEST is identical to the communication network $G$, that is, we perform the symmetry breaking in $G$. In this work, we consider a setting where the problem instance corresponds to a power graph $G^k$, where each node of the communication network $G$ is connected to all of its $k$-hop neighbors. Our main contribution is a deterministic polylogarithmic time algorithm for computing $k$-ruling sets of $G^k$, which (for $k>1$) improves exponentially on the current state-of-the-art runtimes. The main technical ingredient for this result is a deterministic sparsification procedure which may be of independent interest. On top of being a natural family of problems, ruling sets (in power graphs) are well-motivated through their applications in the powerful shattering framework [BEPS JACM'16, Ghaffari SODA'19] (and others). We present randomized algorithms for computing maximal independent sets and ruling sets of $G^k$ in essentially the same time as they can be computed in $G$. We also revisit the shattering algorithm for MIS [BEPS JACM'16] and present different approaches for the post-shattering phase. Our solutions are algorithmically and analytically simpler (also in the LOCAL model) than existing solutions and obtain the same runtime as [Ghaffari SODA'16].

翻译：在本文中，我们提出了针对幂图中经典对称破缺问题——最大独立集（MIS）和统治集——的高效分布式算法。我们采用标准的分布式消息传递CONGEST模型，其中通信网络被抽象为图$G$。通常，CONGEST模型中的问题实例与通信网络$G$相同，即我们在$G$中执行对称破缺。在本研究中，我们考虑问题实例对应于幂图$G^k$的情形，其中通信网络$G$的每个节点连接到其所有$k$跳邻居。我们的主要贡献是计算$G^k$的$k$-统治集的一种确定性多对数时间算法，该算法（当$k>1$时）在运行时间上比当前最优结果实现了指数级改进。该结果的核心技术要素是一种确定性的稀疏化过程，这一过程可能具有独立的研究价值。除了这是一类自然的问题族外，（幂图中的）统治集通过其在强大的分片框架[BEPS JACM'16, Ghaffari SODA'19]等中的应用得到了充分动机。我们提出了随机算法，用于计算$G^k$的最大独立集和统治集，其运行时间与在$G$中计算这些集合的时间本质相同。我们还重新审视了用于MIS的分片算法[BEPS JACM'16]，并为分片后阶段提出了不同方法。我们的解决方案在算法和分析上（在LOCAL模型中也是如此）比现有解决方案更简单，并获得了与[Ghaffari SODA'16]相同的运行时间。