Nearly Optimal Approximate Dual-Failure Replacement Paths

Given a directed graph G = (V, E, ω) on n vertices with positive edge weights 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 f1, f2 ∈ 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, Woldeghebriela and Xu, 2022] who designed a O(n³) time algorithm for general weighted digraphs which is conditionally optimal. In the same paper, they also showed that the cubic time barrier can be bypassed by a O(M²/³n^2.9146) time algorithm for input graphs with small integer edge weights from {−M, −M + 1, · · ·, M − 1, M}. In this paper, we study the natural question whether sub-cubic time algorithms exist for general weighted digraphs when approximation is allowed. As our main result, we show that (1 + є)-approximations of all dual-failure replacement paths can be computed in O∊(n²) time which is nearly optimal.