Algorithms and Lower Bounds for Replacement Paths under Multiple Edge Failure

This paper considers a natural fault-tolerant shortest paths problem: for some constant integer f, given a directed weighted graph with no negative cycles and two fixed vertices s and t, compute (either explicitly or implicitly) for every tuple of f edges, the distance from s to t if these edges fail. We call this problem f-Fault Replacement Paths (f FRP).We first present an $\tilde{O}(n^{3}$) time algorithm for 2FRP in n-vertex directed graphs with arbitrary edge weights and no negative cycles. As 2FRP is a generalization of the well-studied Replacement Paths problem (RP) that asks for the distances between s and t for any single edge failure, 2FRP is at least as hard as RP. Since RP in graphs with arbitrary weights is equivalent in a fine-grained sense to All-Pairs Shortest Paths (APSP) [Vassilevska Williams and Williams FOCS’10, J. ACM’18], 2FRP is at least as hard as APSP, and thus a substantially subcubic time algorithm in the number of vertices for 2FRP would be a breakthrough. Therefore, our algorithm in $\tilde{O}(n^{3})$ time is conditionally nearly optimal. Our algorithm immediately implies an $\tilde{O}(n^{f+1})$ time algorithm for the more general f FRP problem, giving the first improvement over the straightforward $O(n^{f+2})$ time algorithm.Then we focus on the restriction of 2FRP to graphs with small integer weights bounded by M in absolute values. We show that similar to $\mathrm{R}\mathrm{P}, 2\mathrm{F}\mathrm{R}\mathrm{P}$ has a substantially subcubic time algorithm for small enough M. Using the current best algorithms for rectangular matrix multiplication, we obtain a randomized algorithm that runs in $\tilde{O}(M^{2/3}n^{2.9153})$ time. This immediately implies an improvement over our $\tilde{O}(n^{f+1})$ time arbitrary weight algorithm for all $f\gt1$. We also present a data structure variant of the algorithm that can trade off pre-processing and query time. In addition to the algebraic algorithms, we also give an $n^{8/3-o(1)}$ conditional lower bound for combinatorial 2FRP algorithms in directed unweighted graphs, and more generally, combinatorial lower bounds for the data structure version of $fF\mathrm{R}\mathrm{P}$.