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A Fine-Grained Dichotomy for the Center Problem on Gromov Hyperbolic Graphs

Author Guillaume Ducoffe

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LIPIcs.ICALP.2026.82.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Center problem
  • Gromov hyperbolicity
  • Fine-grained complexity in P
  • Graph algorithms

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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 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.

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Guillaume Ducoffe. A Fine-Grained Dichotomy for the Center Problem on Gromov Hyperbolic Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 82:1-82:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.82

Author Details

Guillaume Ducoffe
  • National Institute for Research and Development in Informatics, Bucharest, Romania
  • Faculty of Mathematics and Computer Science, University of Bucharest, Romania

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