We design an algorithm for computing connectivity in hypergraphs which runs in time $\hat O_r(p + \min\{\lambda n^2, n^r/\lambda\})$ (the $\hat O_r(\cdot)$ hides the terms subpolynomial in the main parameter and terms that depend only on $r$) where $p$ is the size, $n$ is the number of vertices, and $r$ is the rank of the hypergraph. Our algorithm is faster than existing algorithms when the connectivity $\lambda$ is $\Omega(n^{(r-2)/2})$. At the heart of our algorithm is a structural result regarding min-cuts in simple hypergraphs. We show a trade-off between the number of hyperedges taking part in all min-cuts and the size of the smaller side of the min-cut. This structural result can be viewed as a generalization of a well-known structural theorem for simple graphs [Kawarabayashi-Thorup, JACM 19]. We extend the framework of expander decomposition to simple hypergraphs in order to prove this structural result. We also make the proof of the structural result constructive to obtain our faster hypergraph connectivity algorithm.

翻译：我们设计了一个计算高音中连接性的算法,它运行时间为$\hat O_r(p +\min ⁇ lambda n ⁇ 2, n ⁇ r/\lambda}}$($hat Or(\\\cdot)$) 美元($_r(\\\cdot) $) 隐藏了主要参数和条件中仅依赖$r($) 的亚极论术语, 其大小为美元, 美元是顶点数, 美元是顶点数, 美元是高点数的等级。 当连接 $\ lambda$ 是$\ omega(n\\\\\\\\\\\\ (r-2) /2} 美元时, 我们的算法比现有的算法要快一些。 我们的算法核心是简单的超点数, 在所有小点数中, 美元是顶点数的大小。 这个结构结果可以视为一个众所周知的结构图结构理论的概括化。 我们把结构结构图的图表 放大到结构图的图状图状, 也证明我们结构图状的图状到结构图状的图状, 。