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∈ V, our goal is to compute the shortest path from s to t avoiding any pair of presumably failed edges f_1, f_2∈ 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≡ 1, the authors presented an algebraic algorithm with runtime Õ(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 Õ(n^3-1/18). Besides, we also study algebraic algorithms for digraphs with small integer edge weights from {-M, -M+1, ⋯, M-1, M}. As our secondary result, we obtained a runtime of Õ(Mn^2.8716), which is faster than the previous bound of Õ(M^2/3n^2.9144 + Mn^2.8716) from [Vassilevska Williams, Woldeghebriela and Xu, 2022].