Near-optimal distributed computation of small vertex cuts

We present near-optimal algorithms for detecting small vertex cuts in the \({\textsf{CONGEST}}\) 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.