Algorithms and Lower Bounds for Replacement Paths under Multiple Edge Failure - Williams, Virginia Vassilevska; Woldeghebriel, Eyob; Xu, Yinzhan (Proceedings / annual Symposium on Foundations of Computer Science, 2022)
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 RP, 2FRP 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 fFRP.