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Undirected Replacement Paths: Dual Fault Reduces to Single Source

Authors Jakob Nogler , Virginia Vassilevska Williams

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LIPIcs.ICALP.2026.144.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Divide and conquer
Keywords
  • Single Source Replacement Paths
  • Dualt Fault Replacement Paths
  • Fine-Grained Complexity

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Abstract

Given a graph and two 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:  
- 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⧵e. That is, we do not fix the target anymore. 
- 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⧵e,e'. That is, there are two failures instead of one. 
Previously, there was no known formal 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 get new algorithms for 2-FRP that match the best-known runtimes for SSRP:  
(a) 𝒪̃(M n^ω) for weights in [1..M] [Grandoni and Vassilevska Williams, FOCS 2012 & TALG 2019], improving upon 𝒪(Mn^{2.87}) [Chechik, Zhang, ICALP 2024]; 
(b) n³/2^Ω(√{log n}) for weights in [1..poly(n)] [Grandoni and Vassilevska Williams, FOCS 2012 & TALG 2019], improving over the previous n³polylog(n) running time [Vassilevska W., Woldeghebriel and Xu, FOCS 2022]; 
(c) 𝒪̃(mn^{1/2} + n²) combinatorial time for unweighted graphs [Chechik and Cohen, SODA 2019], and more generally for rational weights in [1,2] [Chechik and Magen, ICALP 2020], improving upon 𝒪̃(n^{3-1/18}) [Chechik, Zhang ICALP 2024]. 
We complement these upper bounds with tight lower bounds under fine-grained hypotheses.

Cite As Get BibTex

Jakob Nogler and Virginia Vassilevska Williams. Undirected Replacement Paths: Dual Fault Reduces to Single Source. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 144:1-144:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.144

Author Details

Jakob Nogler
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Virginia Vassilevska Williams
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Nogler, Jakob: Supported by the Akamai Presidential Fellowship.
  • Vassilevska Williams, Virginia: Supported by NSF Grant CCF-2330048, BSF Grant 2024233 and a Simons Investigator Award.

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