We present near-optimal algorithms for detecting small vertex cuts in the CONGEST model of distributed computing. Despite extensive research in this area, our understanding of the vertex connectivity of a graph is still incomplete, especially in the distributed setting. To this date, all distributed algorithms for detecting cut vertices suffer from an inherent dependency in the maximum degree of the graph, $\Delta$. Hence, in particular, there is no truly sub-linear time algorithm for this problem, not even for detecting a single cut vertex. We take a new algorithmic approach for vertex connectivity which allows us to bypass the existing $\Delta$ barrier. As a warm-up to our approach, we show a simple $\widetilde{O}(D)$-round randomized algorithm for computing all cut vertices in a $D$-diameter $n$-vertex graph. This improves upon the $O(D+\Delta/\log n)$-round algorithm of [Pritchard and Thurimella, ICALP 2008]. Our key technical contribution is an $\widetilde{O}(D)$-round randomized algorithm for computing all cut pairs in the graph, improving upon the state-of-the-art $O(\Delta \cdot D)^4$-round algorithm by [Parter, DISC '19]. Note that even for the considerably simpler setting of edge cuts, currently $\widetilde{O}(D)$-round algorithms are known only for detecting pairs of cut edges. Our approach is based on employing the well-known linear graph sketching technique [Ahn, Guha and McGregor, SODA 2012] along with the heavy-light tree decomposition of [Sleator and Tarjan, STOC 1981]. Combining this with a careful characterization of the survivable subgraphs, allows us to determine the connectivity of $G \setminus \{x,y\}$ for every pair $x,y \in V$, using $\widetilde{O}(D)$-rounds. We believe that the tools provided in this paper are useful for omitting the $\Delta$-dependency even for larger cut values.

翻译：我们提出了在分布式计算CONGEST模型中检测小顶点割的近最优算法。尽管该领域已有大量研究，但我们对图顶点连通性的理解仍不完整，尤其在分布式场景下。迄今为止，所有用于检测割顶点的分布式算法均受限于图的最大度$\Delta$的固有关联性。因此，特别地，尚不存在真正的亚线性时间算法来解决该问题，甚至对于检测单个割顶点也是如此。我们采用了一种新的顶点连通性算法方法，能够突破现有的$\Delta$障碍。作为该方法的预热，我们展示了一个简单的$\widetilde{O}(D)$轮随机算法，用于计算所有直径为$D$的$n$顶点图中的割顶点。这改进了[Pritchard and Thurimella, ICALP 2008]中$O(D+\Delta/\log n)$轮算法。我们的关键技术贡献是一个$\widetilde{O}(D)$轮随机算法，用于计算图中所有割点对，改进了[Parter, DISC '19]提出的当前最优$O(\Delta \cdot D)^4$轮算法。值得注意的是，即使对于更简单的边割设定，目前已知的$\widetilde{O}(D)$轮算法也仅能检测边割对。我们的方法基于采用著名的线性图草图技术[Ahn, Guha and McGregor, SODA 2012]以及[Sleator and Tarjan, STOC 1981]的重-轻树分解。结合对可存活子图的精细刻画，我们能够为每一对$x,y \in V$确定$G \setminus \{x,y\}$的连通性，仅需$\widetilde{O}(D)$轮。我们相信本文提供的工具对于消除更大割值场景下的$\Delta$依赖性同样有效。