A fine-grained dichotomy for the center problem on Gromov hyperbolic graphs

2026-05-02

AI Summary

This study investigates the fine-grained complexity of the graph center problem in Gromov hyperbolic graphs—specifically, finding a vertex that minimizes the maximum distance to all other vertices. By integrating tools from metric geometry, Gromov hyperbolicity theory, and conditional lower bound techniques, the authors present the first linear-time algorithm for 1/2-hyperbolic graphs. Moreover, under the Hitting Set conjecture, they establish that no linear-time algorithm exists for 1-hyperbolic graphs. This work provides a complete complexity classification of the problem across all remaining values of the hyperbolicity constant δ, achieving tight alignment between upper and lower bounds.

Abstract

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.

Problem

• center problem
• Gromov hyperbolicity
• fine-grained complexity
• central vertex
• graph radius

Innovation

• fine-grained complexity
• Gromov hyperbolicity
• graph center
• linear-time algorithm
• Hitting Set Conjecture