Given a graph and two fixed vertices $s$ and $t$, the Replacement Path Problem (RP) is to compute for every edge $e$, the distance between $s$ and $t$ when $e$ is removed. There are two natural extensions to RP: (1) Single Source Replacement Paths (SSRP): Given a graph $G$ and a source node $s$, compute for every vertex $v$ and every edge $e$ the $s$-$v$ distance in $G \setminus \{e\}$. That is, we do not fix the target anymore. (2) $2$-Fault Replacement Paths (2-FRP): Given a graph $G$ and two nodes $s$ and $t$, compute for every pair of edges $e, e'$ the $s$-$t$ distance in $G \setminus \{e, e'\}$. That is, we consider two failures instead of one. Previously, there was no known reduction between SSRP and 2-FRP. It seemed plausible that 2-FRP would be computationally harder because there are no settings where 2-FRP admits a faster algorithm than SSRP. In directed unweighted graphs there is a provable gap in complexity, and in undirected graphs many of the known 2-FRP algorithms in a variety of settings are much slower than those for SSRP in the same setting. The main contribution of this paper is a tight reduction from undirected $2$-FRP to undirected SSRP, showing that contrary to prior intuition, 2-FRP is not harder than SSRP. As our reduction is weight-preserving, we obtain the first algorithms for $2$-FRP that match the best-known runtimes for SSRP: (1) $\tilde{O}(M n^ω)$ for weights in $[1, M]$ [GVW19], improving upon $O(Mn^{2.87})$ [CZ24]; (2) $n^3/2^{Ω(\sqrt{\log n})}$ for weights in $[1, \text{poly}(n)]$ [GVW19], improving over the previous $n^3\text{polylog}(n)$ running time [VWWX22]; (3) $\tilde{O}(mn^{1/2}+n^{2})$ combinatorial time for unweighted graphs [CC19], and more generally for rational weights in $[1, 2]$ [CM20], improving upon $\tilde{O}(n^{3-1/18})$ [CZ24]. We complement these upper bounds with tight lower bounds under fine-grained hypotheses.

翻译：给定一个图及两个固定顶点 $s$ 和 $t$，替换路径问题（RP）要求计算每条边 $e$ 被移除后 $s$ 与 $t$ 之间的距离。RP 有两个自然扩展：（1）单源替换路径（SSRP）：给定图 $G$ 和源节点 $s$，计算每个顶点 $v$ 及每条边 $e$ 在 $G \setminus \{e\}$ 中的 $s$-$v$ 距离，即不再固定目标节点；（2）$2$-故障替换路径（2-FRP）：给定图 $G$ 及两个节点 $s$ 和 $t$，计算每对边 $e, e'$ 在 $G \setminus \{e, e'\}$ 中的 $s$-$t$ 距离，即考虑两次而非一次故障。此前，SSRP 与 2-FRP 之间不存在已知的约简关系。直观上，2-FRP 的计算难度可能更大，因为现有场景中 2-FRP 的算法速度均未超过 SSRP。在有向无权重图中，两者存在可证明的复杂度差距；而在无向图中，多种设置下的已知 2-FRP 算法均远慢于同设置下的 SSRP 算法。本文的主要贡献在于建立了从无向 $2$-FRP 到无向 SSRP 的紧约简，表明与先前的直觉相反，2-FRP 并不比 SSRP 更难。由于该约简保持权重不变，我们首次得到了与 SSRP 最优运行时间相匹配的 2-FRP 算法：（1）权重范围为 $[1, M]$ 时，运行时间为 $\tilde{O}(M n^ω)$ [GVW19]，优于此前 $O(Mn^{2.87})$ [CZ24]；（2）权重范围为 $[1, \text{poly}(n)]$ 时，运行时间为 $n^3/2^{Ω(\sqrt{\log n})}$ [GVW19]，优于此前 $n^3\text{polylog}(n)$ [VWWX22]；（3）无权重图及更一般地权重范围为 $[1, 2]$ 的有理权重图 [CM20] 中，组合算法运行时间为 $\tilde{O}(mn^{1/2}+n^{2})$ [CC19]，优于此前 $\tilde{O}(n^{3-1/18})$ [CZ24]。我们通过细粒度假设下的紧下界补充了这些上界结果。