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 FPTAS 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：卡格尔（STOC 1995）给出了网络（不可靠性）问题的首个FPTAS，推动了接下来三十年的研究，逐步获得更快的运行时间，最终实现了$\tilde{O}(n^2)$时间的算法（卡格尔，STOC 2020）。这代表了该研究方向的自然顶点，因为所采用的算法技术可以枚举$\Theta(n^2)$个（近）最小割。在本文中，我们突破了这一二次障碍，为网络不可靠性问题获得了更快的FPTAS。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性图中的网络不可靠性提出了一种新的估计器。这些图通常是网络不可靠性的瓶颈，因为断连事件难以捕捉。该估计器通过在图的偶对生成树包络上定义适当的重要性采样子程序获得。为了补充这一针对高可靠性图的估计器，我们对中等可靠性图采用递归收缩。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好的参数化，从而获得与高可靠性情况相匹配的更快运行时间。