We design an algorithm for computing connectivity in hypergraphs which runs in time $\hat O_r(p + \min\{\lambda^{\frac{r-3}{r-1}} n^2, n^r/\lambda^{r/(r-1)}\})$ (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 the rank is constant and the connectivity $\lambda$ is $\omega(1)$. 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压中连接性的算法,它运行时间为$\hat O_r(p +\min ⁇ lambda ⁇ frac{r-3 ⁇ r-1}n ⁇ 2, nr/\lambda ⁇ r/(r-1) ⁇ _(r-1)$($hat Or(cdot)$),它隐藏了主要参数和条件中仅取决于$$(美元)的亚极论术语,其中美元大小为美元,美元是顶点的数量,美元是高点的等级。当等级不变时,我们的算法比现有的算法要快得多。当等级不变时,而连接$\lambda$是$\lambda$\r\(r-1)\(r-1)\\\\\\\\\\美元)。 在我们的算法核心部分是简单的高点图中小切点的结构性结果。我们展示了所有小点的超点数与小点的大小之间的交易。这种结构结果可以被视为一个众所周知的结构结构结构结构结构的概括化结构图,用于简单图表的图状[KAAAA] 也证明我们的结构性结构结构图的图的图状结构结构结构图状的扩展的扩展结果。