突破网络不可靠性的二次时间壁垒
Beyond the Quadratic Time Barrier for Network Unreliability
AI总结：
本文突破网络不可靠性问题的二次时间壁垒，提出结合对偶生成树重要性采样与递归收缩的新FPTAS，将时间复杂度降至$m^{1+o(1)} + \tilde{O}(n^{1.5})$。
摘要：
Karger（STOC 1995）提出了首个针对网络（不）可靠度问题的FPTAS（完全多项式时间近似方案），推动了此后三十年的相关研究，算法运行速度不断提升，最终催生了时间复杂度为$\tilde{O}(n^2)$的算法（Karger, STOC 2020）。这一成果曾被视为该研究方向的自然顶点，因为所用的算法技术能够枚举$\theta(n^2)$个（近）最小割。在本文中，我们突破了这一二次时间壁垒，得到了一种更快的网络不可靠性问题FPTAS。我们的算法运行时间为$m^{1+o(1)} + \tilde{O}(n^{1.5})$。\n我们的核心贡献是提出了一种适用于高可靠图的网络不可靠性新估计量。由于断连事件难以观测，这类图通常是网络不可靠性计算的瓶颈。我们通过在图的对偶生成树填充上定义合适的重要性采样子程序来得到该估计量。为补充高可靠图的估计量，我们对中等可靠图采用递归收缩方法。我们证明，将稀疏化与收缩交错使用，可以对递归收缩算法实现更好的参数化，从而获得与高可靠情形相匹配的更快运行时间。
英文摘要：
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 $Θ(n^2)$ (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.
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