When Can Graph Hyperbolicity Be Computed in Linear Time?

  • Till FluschnikEmail author
  • Christian Komusiewicz
  • George B. Mertzios
  • André Nichterlein
  • Rolf Niedermeier
  • Nimrod Talmon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


Hyperbolicity measures, in terms of (distance) metrics, 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 practical algorithms for computing the hyperbolicity number of a 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 instance, we show that hyperbolicity can be computed in time \(2^{O(k)} + O(n +m)\) (m being the number of graph edges, k being the size of a vertex cover) while at the same time, unless the SETH fails, there is no \(2^{o(k)}n^2\)-time algorithm.


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  1. 1.
    Abboud, A., Vassilevska Williams, V., Wang, J.R.: Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In: Proc. 27th SODA, pp. 377–391. SIAM (2016)Google Scholar
  2. 2.
    Abu-Ata, M., Dragan, F.F.: Metric tree-like structures in real-world networks: an empirical study. Networks 67(1), 49–68 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borassi, M., Coudert, D., Crescenzi, P., Marino, A.: On computing the hyperbolicity of real-world graphs. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 215–226. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48350-3_19 CrossRefGoogle Scholar
  4. 4.
    Borassi, M., Crescenzi, P., Habib, M.: Into the square: On the complexity of some quadratic-time solvable problems. Electronic Notes in Theoretical Computer Science 322, 51–67 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications, vol. 3. SIAM (1999)Google Scholar
  6. 6.
    Brinkmann, G., Koolen, J.H., Moulton, V.: On the hyperbolicity of chordal graphs. Annals of Combinatorics 5(1), 61–69 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cohen, N., Coudert, D., Lancin, A.: On computing the Gromov hyperbolicity. ACM Journal of Experimental Algorithmics 20, 1.6:1–1.6:18 (2015)Google Scholar
  8. 8.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14(4), 926–934 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Damaschke P.: Induced subraph isomorphism for cographs is NP-complete. In: Möhring R.H. (eds) WG 1990. LNCS, vol 484, pp. 72–78. Springer, Heidelberg (1991)Google Scholar
  10. 10.
    Doucha, M., Kratochvíl, J.: Cluster vertex deletion: A parameterization between vertex cover and clique-width. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 348–359. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-32589-2_32 CrossRefGoogle Scholar
  11. 11.
    Eisenbrand, F., Grandoni, F.: On the complexity of fixed parameter clique and dominating set. Theoretical Computer Science 326(1–3), 57–67 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fellows, M., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science 410(1), 53–61 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fournier, H., Ismail, A., Vigneron, A.: Computing the Gromov hyperbolicity of a discrete metric space. Information Processing Letters 115(6–8), 576–579 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman (1979)Google Scholar
  15. 15.
    Giannopoulou, A.C., Mertzios, G.B., Niedermeier, R.: Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs. In: Proc. 10th IPEC, vol. 43 of LIPIcs, pp. 102–113. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  16. 16.
    Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, vol. 8, pp. 75–263. MSRI Publ., Springer New York (1987)Google Scholar
  17. 17.
    Mitsche, D., Pralat, P.: On the hyperbolicity of random graphs. The Electronic Journal of Combinatorics 21(2), P2.39 (2014)Google Scholar
  18. 18.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall (1982)Google Scholar
  19. 19.
    Williams, R., Yu, H.: Finding orthogonal vectors in discrete structures. In: Proc. 25th SODA, pp. 1867–1877. SIAM (2014)Google Scholar
  20. 20.
    Williams, V.V., Wang, J.R., Williams, R., Yu, H.: Finding four-node subgraphs in triangle time. In: Proc. 26th SODA, pp. 1671–1680. SIAM (2015)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Till Fluschnik
    • 1
    Email author
  • Christian Komusiewicz
    • 2
  • George B. Mertzios
    • 3
  • André Nichterlein
    • 1
    • 3
  • Rolf Niedermeier
    • 1
  • Nimrod Talmon
    • 4
  1. 1.Institut Für Softwaretechnik Und Theoretische InformatikTU BerlinBerlinGermany
  2. 2.Institut Für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  3. 3.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  4. 4.Weizmann Institute of ScienceRehovotIsrael

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