Faster Algorithms for Dual-Failure Replacement Paths

Given a simple weighted directed graph $G = (V, E, ω)$ 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 $ω\equiv 1$, 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].