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When Can Graph Hyperbolicity be Computed in Linear Time?

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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 \(\varepsilon >0\).

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Notes

  1. Cographs are the graphs without induced \(P_4\)s [10]. The parameter distance to cographs is upper-bounded by the parameter distance to cluster graph [15] and thus also by the parameter vertex cover number.

  2. The Strong Exponential Time Hypothesis [23, 24] states that there is no \(O((2-\epsilon )^n)\)-time algorithm deciding k-SAT, where \(\epsilon >0\) is independent of k and n denotes the number of variables.

  3. This case is often left undefined in the literature. Our definition, however, allows to consider also disconnected graphs.

  4. Note that there is also a slightly different definition where graphs that we call \(\delta \)-hyperbolic are called \(2\delta \)-hyperbolic [8, 28]; we follow the definition of Brinkmann et al. [7].

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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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Authors and Affiliations

  1. Algorithmics and Computational Complexity, Fakultät IV, TU Berlin, Berlin, Germany

    Till Fluschnik, André Nichterlein & Rolf Niedermeier

  2. Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Christian Komusiewicz

  3. Department of Computer Science, Durham University, Durham, UK

    George B. Mertzios

  4. Ben-Gurion University, Beersheba, Israel

    Nimrod Talmon

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  1. Till Fluschnik
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  2. Christian Komusiewicz
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Correspondence to Till Fluschnik.

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This work was initiated at the yearly research retreat of the Algorithmics and Computational Complexity (AKT) group of TU Berlin, held in Krölpa, Thuringia, Germany, from April 3rd till April 9th, 2016. An extended abstract appeared in the Proceedings of the 15th International Symposium on Algorithms and Data Structures (WADS ’17), volume 10389, pages 397–408. Springer, 2017. This version contains additional details and full proofs.

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Fluschnik, T., Komusiewicz, C., Mertzios, G.B. et al. When Can Graph Hyperbolicity be Computed in Linear Time?. Algorithmica 81, 2016–2045 (2019). https://doi.org/10.1007/s00453-018-0522-6

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