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PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Undirected Replacement Paths: Dual Fault Reduces to Singl...
Jakob Nogler, Virginia Vassilevska Williams · 2026-05-04 · via cs.DS updates on arXiv.org

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.