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.

Hyperbolicity, Degeneracy, and Expansion of Random Intersection Graphs

“…We used the NetworkX Python package [19] to generate our random intersection graphs (using the uniform random intersection graph method), and the SageMath software system [14] to compute the hyperbolicity [6,9,16], degeneracy [1,38] and diameter [10,11,27,45] of the generated graphs. The measurements of the p-centered coloring number (presented below) were executed using the implementation available in [37].…”

Recognition of $C_4$-Free and 1/2-Hyperbolic Graphs

“…Indeed, it is straightforward by using Definition 1 to compute the graph hyperbolicity δ(G) in Θ(n 4 )-time (see [11] and [21] for practical and theoretical improvements of the complexity). Also, note that δ(G) is always a half-integer (w.r.t.…”

“…So far, the best known algorithm for determining the hyperbolicity of a graph has an O(n 3.69 )-time complexity [21]. This is however prohibitive for graphs with tens of thousands of nodes such as the graph of the Autonomous Systems of the Internet, road maps, etc.…”

“…However, to the best of our knowledge, there was no known algorithmic application to it prior to this work. We are more interested in a reduction such as the recent one in [21], where the authors proved an equivalence between the problems of finding a 2-approximation for the hyperbolicity and the (max, min)-matrix multiplication. A recent work [19] further exploits the relation between both problems, yielding constant-factor approximations for the hyperbolicity in subcubic-time.…”

“…In Section 4, we finally reduce both problems to the problem of the rectangular matrix product that was defined in [31]. This allows us to solve both of them in O(n 3.26 )-time, which beats the previous records established in [21,37].…”