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Gromov hyperbolicity and related graph-parameters represent how close (locally) the shortest-path metric of a graph is to a tree metric. The Gromov hyperbolicity of a given graph can be computed in polynomial-time, however it is unlikely that it can be done faster than in supercubic time. This makes this parameter difficult to compute, or even to approximate, on large graphs.In this paper, we provide a simple factor 8 approximation algorithm for computing Gromov hyperbolicity of an n-vertex unweighted graph G=(V,E) in optimal time O(n^2) (given the distance matrix of G as the input).  This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity: thinness, slimness, and insize.  We also present efficient algorithms for exact computation of these parameters. To our best knowledge, the latter parameters were not known to be polynomial-time computable until this note.

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs