Faster Algorithms for Dual-Failure Replacement Paths

Shiri Chechik, Tianyi Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Given a simple weighted directed graph G = (V, E, ω) on n vertices as well as two designated terminals s, t ∈ V , our goal is to compute the shortest path from s to t avoiding any pair of presumably failed edges f1, f2 ∈ E, which is a natural generalization of the classical replacement path problem which considers single edge failures only. This dual failure replacement paths problem was recently studied by Vassilevska Williams, Woldeghebriel and Xu [FOCS 2022] who designed a cubic time algorithm for general weighted digraphs which is conditionally optimal; in the same paper, for unweighted graphs where ω ≡ 1, the authors presented an algebraic algorithm with runtime Õ(n2.9146), as well as a conditional lower bound of n8/3−o(1) against combinatorial algorithms. However, it was unknown in their work whether fast matrix multiplication is necessary for a subcubic runtime in unweighted digraphs. As our primary result, we present the first truly subcubic combinatorial algorithm for dual failure replacement paths in unweighted digraphs. Our runtime is Õ(n3−1/18). Besides, we also study algebraic algorithms for digraphs with small integer edge weights from {−M, −M +1, · · ·, M −1, M}. As our secondary result, we obtained a runtime of Õ(Mn2.8716), which is faster than the previous bound of Õ(M2/3n2.9144 + Mn2.8716) from [Vassilevska Williams, Woldeghebriela and Xu, 2022].

Original languageEnglish
Title of host publication51st International Colloquium on Automata, Languages, and Programming, ICALP 2024
EditorsKarl Bringmann, Martin Grohe, Gabriele Puppis, Ola Svensson
ISBN (Electronic)9783959773225
DOIs
StatePublished - Jul 2024
Event51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 - Tallinn, Estonia
Duration: 8 Jul 202412 Jul 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume297

Conference

Conference51st International Colloquium on Automata, Languages, and Programming, ICALP 2024
Country/TerritoryEstonia
CityTallinn
Period8/07/2412/07/24

Keywords

  • graph algorithms
  • replacement paths
  • shortest paths

All Science Journal Classification (ASJC) codes

  • Software

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