We show the first near-linear time randomized algorithms for listing all minimum vertex cuts of polylogarithmic size that separate the graph into at least three connected components (also known as shredders) and for finding the most shattering one, i.e., the one maximizing the number of connected components. Our algorithms break the quadratic time bound by Cheriyan and Thurimella (STOC'96) for both problems that has stood for more than two decades. Our work also removes a bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95). Note that it is necessary to list only minimum vertex cuts that separate the graph into at least three components because there can be an exponential number of minimum vertex cuts in general. To obtain near-linear time algorithms, we have extended techniques in local flow algorithms developed by Forster et al. (SODA'20) to list shredders on a local scale. We also exploit fast queries to a pairwise vertex connectivity oracle subject to vertex failures (Long and Saranurak FOCS'22, Kosinas ESA'23). This is the first application of connectivity oracles subject to vertex failures to speed up a static graph algorithm.

翻译：我们提出了首个近线性时间随机算法，用于列举所有将图分离为至少三个连通分量（亦称粉碎器）的多对数规模最小顶点割，并用于寻找最具分裂性的割——即使得连通分量数量最大化的割。我们的算法突破了Cheriyan和Thurimella（STOC'96）针对这两个问题提出的、持续二十余年的二次时间界限。本研究还消除了顶点连通性增强问题（Jordan '95）实现近线性时间算法的瓶颈。需要注意的是，仅需列举将图分离为至少三个分量的最小顶点割，因为一般情况下最小顶点割的数量可能呈指数级增长。为实现近线性时间算法，我们扩展了Forster等人（SODA'20）开发的局部流算法技术，以在局部范围内列举粉碎器。同时，我们利用针对顶点失效的成对顶点连通性预言机快速查询技术（Long和Saranurak FOCS'22, Kosinas ESA'23）。这是首次应用面向顶点失效的连通性预言机来加速静态图算法。