The hypergraph unreliability problem asks for the probability that a hypergraph gets disconnected when every hyperedge fails independently with a given probability. For graphs, the unreliability problem has been studied over many decades, and multiple fully polynomial-time approximation schemes are known starting with the work of Karger (STOC 1995). In contrast, prior to this work, no non-trivial result was known for hypergraphs (of arbitrary rank). In this paper, we give quasi-polynomial time approximation schemes for the hypergraph unreliability problem. For any fixed $\varepsilon \in (0, 1)$, we first give a $(1+\varepsilon)$-approximation algorithm that runs in $m^{O(\log n)}$ time on an $m$-hyperedge, $n$-vertex hypergraph. Then, we improve the running time to $m\cdot n^{O(\log^2 n)}$ with an additional exponentially small additive term in the approximation.

翻译：超图不可靠性问题要求计算当每条超边以给定概率独立失效时，超图失去连通的概率。对于图而言，不可靠性问题已历经数十年的研究，自Karger（STOC 1995）的开创性工作以来，已提出多种完全多项式时间近似方案。然而，在此工作之前，针对（任意秩的）超图尚无任何非平凡结果。本文给出了超图不可靠性问题的拟多项式时间近似方案。对于任意固定的$\varepsilon \in (0, 1)$，我们首先提出一个$(1+\varepsilon)$-近似算法，该算法在$m$条超边、$n$个顶点的超图上运行时间为$m^{O(\log n)}$。随后，我们将运行时间改进至$m\cdot n^{O(\log^2 n)}$，并在近似中增加一个指数级小的加性项。