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 been unimproved for more than two decades. Our work also removes an impor- tant bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95) and finding an even-length cycle in a directed graph, a problem shown to be equivalent to many other fundamental problems (Vazirani and Yannakakis '90, Robertson et al. '99). 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：我们首次给出随机化近似线性时间算法，用于枚举将图分割成至少三个连通分量（又称碎裂割）的多对数大小最小顶点割，并找出最碎裂的割（即最大化连通分量数量的割）。我们的算法突破了Cheriyan和Thurimella（STOC'96）为这两个问题设定的二次时间界限——该界限已二十余年未被改进。本工作还消除了顶点连通性增广问题（Jordan '95）与有向图中寻找偶长环问题（该问题已被证明等价于许多其他基础问题，Vazirani和Yannakakis '90, Robertson等人 '99）向近似线性时间算法发展的关键瓶颈。注意，仅需枚举将图分割成至少三个分量的最小顶点割，因为一般情形下最小顶点割的数量可能呈指数级。为获得近似线性时间算法，我们扩展了Forster等人（SODA'20）开发的局部流算法技术，在局部尺度上枚举碎裂割。我们还利用顶点失效场景下成对顶点连通性预言的快速查询技术（Long和Saranurak FOCS'22, Kosinas ESA'23）。这是连通性预言在顶点失效场景下加速静态图算法的首次应用。