Approximation Algorithms for the Gromov Hyperbolicity of Discrete Metric Spaces

This paper discusses new approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We give a (1 + epsilon)-approximation algorithm with running time (O) over tilde(epsilon (1)n(1 vertical bar omega)), where O(n(omega)) = O(n(2.373)) is the time complexity of matrix multiplications. Here an alpha-approximation delta' means delta' <= delta' <= alpha delta' for the Gromov hyperbolicity delta*. We also give a (2 + epsilon)-approximation algorithm with running time (O) over tilde(epsilon(-1)n(omega)). These are faster than the previous O(n((5+omega)/2))-time algorithm for the exact solution and the O(n((3+omega)/2))-time algorithm for a 2-approximation [Fournier, Ismail and Vigneron 2012], which directly perform (max, min)-product of matrices.