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Algorithmica

pp 1–30 | Cite as

When Can Graph Hyperbolicity be Computed in Linear Time?

  • Till Fluschnik
  • Christian Komusiewicz
  • George B. Mertzios
  • André Nichterlein
  • Rolf Niedermeier
  • Nimrod Talmon
Article
  • 8 Downloads

Abstract

Hyperbolicity is a distance-based measure of how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms used in practice for computing the hyperbolicity number of an n-vertex graph have running time \(O(n^4)\). Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For example, we show that hyperbolicity can be computed in \(2^{O(k)} + O(n +m)\) time (where m and k denote the number of edges and the size of a vertex cover in the input graph, respectively) while at the same time, unless the Strong Exponential Time Hypothesis (SETH) fails, there is no \(2^{o(k)}\cdot n^{2-\varepsilon }\)-time algorithm for every \(\varepsilon >0\).

Keywords

Parameterized complexity Polynomial-time algorithm FPT in P Strong Exponential Time Hypothesis Vertex cover number Cographs 

Notes

Acknowledgements

We are grateful to the anonymous reviewers of WADS’17 and Algorithmica for their comments. TF acknowledges support by the DFG, Projects DAMM (NI 369/13-2) and TORE (NI 369/18). CK acknowledges support by the DFG, Project MAGZ (KO 3669/4-1). GM acknowledges support by the EPSRC Grant EP/P020372/1. AN acknowledges support by a postdoctoral fellowship of the DAAD while at Durham University. NT acknowledges support by a postdoctoral fellowship from I-CORE ALGO.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Till Fluschnik
    • 1
  • Christian Komusiewicz
    • 2
  • George B. Mertzios
    • 3
  • André Nichterlein
    • 1
  • Rolf Niedermeier
    • 1
  • Nimrod Talmon
    • 4
  1. 1.Algorithmics and Computational Complexity, Fakultät IVTU BerlinBerlinGermany
  2. 2.Fachbereich Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany
  3. 3.Department of Computer ScienceDurham UniversityDurhamUK
  4. 4.Ben-Gurion UniversityBeershebaIsrael

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