Computing the Gromov hyperbolicity of a discrete metric space

Abstract

We give exact and approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We observe that computing the Gromov hyperbolicity from a fixed base-point reduces to a (max,min) matrix product. Hence, using the (max,min) matrix product algorithm by Duan and Pettie, the fixed base-point hyperbolicity can be determined in O(n^2.69) time. It follows that the Gromov hyperbolicity can be computed in O(n^3.69) time, and a 2-approximation can be found in O(n^2.69) time. We also give a (2 log_2 n)-approximation algorithm that runs in O(n^2) time, based on a tree-metric embedding by Gromov. We also show that hyperbolicity at a fixed base-point cannot be computed in O(n^2.05) time, unless there exists a faster algorithm for (max,min) matrix multiplication than currently known.