Karger (STOC 1995) gave the first FPTAS for the network (un)reliability problem, setting in motion research over the next three decades that obtained increasingly faster running times, eventually leading to a $\tilde{O}(n^2)$-time algorithm (Karger, STOC 2020). This represented a natural culmination of this line of work because the algorithmic techniques used can enumerate $\Theta(n^2)$ (near)-minimum cuts. In this paper, we go beyond this quadratic barrier and obtain a faster algorithm for the network unreliability problem. Our algorithm runs in $m^{1+o(1)} + \tilde{O}(n^{1.5})$ time. Our main contribution is a new estimator for network unreliability in very reliable graphs. These graphs are usually the bottleneck for network unreliability since the disconnection event is elusive. Our estimator is obtained by defining an appropriate importance sampling subroutine on a dual spanning tree packing of the graph. To complement this estimator for very reliable graphs, we use recursive contraction for moderately reliable graphs. We show that an interleaving of sparsification and contraction can be used to obtain a better parametrization of the recursive contraction algorithm that yields a faster running time matching the one obtained for the very reliable case.

翻译：Karger（STOC 1995）给出了首个用于网络（不可靠性）问题的FPTAS，这启动了随后三十年的研究，逐步获得了更快的运行时间，最终得出一个$\tilde{O}(n^2)$时间算法（Karger，STOC 2020）。这代表了该研究方向的自然顶点，因为所使用的算法技术能够枚举$\Theta(n^2)$个（近）最小割。在本文中，我们超越了这一二次障碍，为网络不可靠性问题获得了更快的算法。我们的算法运行时间为$m^{1+o(1)} + \tilde{O}(n^{1.5})$。我们的主要贡献是为高可靠图提出了一种新的网络不可靠性估计器。这些图通常是网络不可靠性问题的瓶颈，因为断连事件难以捉摸。该估计器通过在图的其对偶生成树打包上定义适当的重点采样子例程来获得。为了补充适用于高可靠图的这一估计器，我们对中等可靠图使用了递归收缩。我们证明，稀疏化与收缩的交错使用可以为递归收缩算法提供更优的参数化，从而获得与高可靠情况匹配的更快运行时间。