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Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

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Abstract

In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. The main contribution of this paper is a new characterization of the hyperbolicity of graphs, via a new parameter which we call rooted insize. This characterization has algorithmic implications in the field of large-scale network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in embedding an undirected graph into hyperbolic space with minimum distortion (Verbeek and Suri, in Symposium on Computational Geometry, ACM, New York, 2014). The hyperbolicity of a graph can be computed in polynomial-time, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm (with an additive constant 1) for computing the hyperbolicity of an n-vertex graph \(G=(V,E)\) in optimal time \(O(n^2)\) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space. We also show that a similar characterization of hyperbolicity holds for all geodesic metric spaces endowed with a geodesic spanning tree. Along the way, we prove that any complete geodesic metric space (Xd) has such a geodesic spanning tree.

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Notes

  1. 1.

    The \({\widehat{O}}(\cdot )\) notation hides polyloglog factors.

  2. 2.

    Informally, \((y|z)_w\) can be viewed as half the detour you make, when going over w to get from y to z.

  3. 3.

    In all algorithmic results, we assume the word-RAM model.

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Acknowledgements

We are grateful to the referees of the journal and short versions of the paper for a careful reading and many useful comments and suggestions. The research of J.C., V.C., and Y.V. was supported by ANR project DISTANCIA (ANR-17-CE40-0015).

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Correspondence to Victor Chepoi.

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An extended abstract [9] of this paper has appeared in the proceedings of SoCG 2018.

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Chalopin, J., Chepoi, V., Dragan, F.F. et al. Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs. Discrete Comput Geom (2019). https://doi.org/10.1007/s00454-019-00107-9

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Keywords

  • Gromov hyperbolicity
  • Negative curvature
  • Geodesic triangle
  • Rooted insize
  • Geodesic spanning tree
  • Fast approximation algorithm

Mathematics Subject Classification

  • 05C85
  • 05C12