A fine-grained dichotomy for the center problem on Gromov hyperbolic graphs
For graph theorists and algorithm designers, this work precisely characterizes the boundary of tractability for computing graph centers in hyperbolic graphs.
The paper resolves the fine-grained complexity of the center problem on Gromov hyperbolic graphs by providing a linear-time algorithm for 1/2-hyperbolic graphs and proving that no such algorithm exists for 1-hyperbolic graphs under the Hitting Set Conjecture, closing all open cases.
A vertex in a graph is called central if it minimizes its maximum distance to the other vertices. The radius of a graph $G$ is the largest distance between a central vertex and the other vertices, and it is denoted by $rad(G)$. In the center problem, we are asked to find a central vertex. We study the fine-grained complexity of the center problem on graphs with small Gromov hyperbolicity. Roughly, the Gromov hyperbolicity of a graph represents how close, locally, it is to a tree, from a metric point of view. It has applications in the design of approximation algorithms. In particular, there is a linear-time algorithm that for every $δ$-hyperbolic graph $G$ outputs some vertex at distance at most $rad(G) + 5δ$ to the other vertices [Chepoi et al, SoCG'08]. However, a linear-time algorithm for computing a central vertex is known only for $0$-hyperbolic graphs, whereas its existence was ruled out for $2$-hyperbolic graphs under the Hitting Set Conjecture of [Abboud et al, SODA'16]. Our main contribution in the paper is a linear-time algorithm for computing a central vertex in the class of $\frac 1 2$-hyperbolic graphs. Furthermore, we rule out the existence of such an algorithm for $1$-hyperbolic graphs, under the Hitting Set Conjecture, thus completely settling all the cases left open.