Abstract
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.
This work was initiated at the yearly research retreat of the Algorithmics and Computational Complexity (AKT) group of TU Berlin, held in in Krölpa, Thuringia, Germany, from April 3rd till April 9th, 2016.
T. Fluschnik — Supported by the DFG, project DAMM (NI 369/13-2).
C. Komusiewicz — Supported by the DFG, project MAGZ (KO 3669/4-1).
A. Nichterlein — Supported by a postdoctoral fellowship of the DAAD while at Durham University.
N. Talmon — Supported by a postdoctoral fellowship from I-CORE ALGO.
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Fluschnik, T., Komusiewicz, C., Mertzios, G.B., Nichterlein, A., Niedermeier, R., Talmon, N. (2017). When Can Graph Hyperbolicity Be Computed in Linear Time?. In: Ellen, F., Kolokolova, A., Sack, JR. (eds) Algorithms and Data Structures. WADS 2017. Lecture Notes in Computer Science(), vol 10389. Springer, Cham. https://doi.org/10.1007/978-3-319-62127-2_34
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