Given a simple weighted directed graph $G = (V, E, \omega)$ on $n$ vertices as well as two designated terminals $s, t\in V$, our goal is to compute the shortest path from $s$ to $t$ avoiding any pair of presumably failed edges $f_1, f_2\in E$, which is a natural generalization of the classical replacement path problem which considers single edge failures only. This dual failure replacement paths problem was recently studied by Vassilevska Williams, Woldeghebriel and Xu [FOCS 2022] who designed a cubic time algorithm for general weighted digraphs which is conditionally optimal; in the same paper, for unweighted graphs where $\omega \equiv 1$, the authors presented an algebraic algorithm with runtime $\tilde{O}(n^{2.9146})$, as well as a conditional lower bound of $n^{8/3-o(1)}$ against combinatorial algorithms. However, it was unknown in their work whether fast matrix multiplication is necessary for a subcubic runtime in unweighted digraphs. As our primary result, we present the first truly subcubic combinatorial algorithm for dual failure replacement paths in unweighted digraphs. Our runtime is $\tilde{O}(n^{3-1/18})$. Besides, we also study algebraic algorithms for digraphs with small integer edge weights from $\{-M, -M+1, \cdots, M-1, M\}$. As our secondary result, we obtained a runtime of $\tilde{O}(Mn^{2.8716})$, which is faster than the previous bound of $\tilde{O}(M^{2/3}n^{2.9144} + Mn^{2.8716})$ from [Vassilevska Williams, Woldeghebriela and Xu, 2022].

翻译：给定一个包含 $n$ 个顶点的简单加权有向图 $G = (V, E, \omega)$ 以及两个指定端点 $s, t\in V$，我们的目标是计算从 $s$ 到 $t$ 避开任意一对假设故障边 $f_1, f_2\in E$ 的最短路径，这是经典替换路径问题（仅考虑单边故障）的自然推广。该双故障替换路径问题近期被 Vassilevska Williams、Woldeghebriel 和 Xu [FOCS 2022] 研究，他们为一般加权有向图设计了一个条件最优的三次时间算法；在同一论文中，对于 $\omega \equiv 1$ 的无权图，作者提出了一个运行时间为 $\tilde{O}(n^{2.9146})$ 的代数算法，并给出了组合算法的一个条件下界 $n^{8/3-o(1)}$。然而，他们的工作未能明确在无权有向图中实现亚三次运行时是否需要快速矩阵乘法。作为我们的主要结果，我们提出了无权有向图中双故障替换路径问题的首个真正亚三次组合算法。我们的运行时间为 $\tilde{O}(n^{3-1/18})$。此外，我们还研究了边权为 $\{-M, -M+1, \cdots, M-1, M\}$ 中小整数的有向图的代数算法。作为我们的次要结果，我们获得了运行时间 $\tilde{O}(Mn^{2.8716})$，这比 [Vassilevska Williams、Woldeghebriela 和 Xu, 2022] 中之前的界 $\tilde{O}(M^{2/3}n^{2.9144} + Mn^{2.8716})$ 更快。