Simplified and Faster Algorithms for the Directed Single/Dual-Failure Replacement Paths via Modulo-Layer Hitting Sets

We study the replacement paths problem in unweighted directed graphs with $n$ vertices and $m$ edges under one or two edge failures. We give simpler and faster algorithms for both settings. Both algorithms are based on a new technique, which we call modulo-layer hitting sets. For the single-failure setting, we obtain a simple deterministic combinatorial algorithm running in $O(m\sqrt{n})$ time, which is nearly optimal under the combinatorial Boolean Matrix Multiplication (BMM) conjecture. In addition to removing the logarithmic overhead of the deterministic $\tilde{O}(m\sqrt{n})$-time algorithm of Alon, Chechik, and Cohen~[ICALP'19], our algorithm admits a much simpler implementation and analysis. For the dual-failure setting, we obtain a randomized combinatorial algorithm running in $\tilde{O}(n^{3-1/4})$ time. This improves upon the previous $\tilde{O}(n^{3-1/18})$-time combinatorial algorithm of Chechik and Zhang~[ICALP'24].