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Charting the Landscape of Diameter Computation on Geometric Intersection Graphs in the Plane

Authors Timothy M. Chan , Hsien-Chih Chang , Jie Gao , Sándor Kisfaludi-Bak , Hung Le , Da Wei Zheng

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LIPIcs.ICALP.2026.54.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • String graphs
  • Fine-grained complexity

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Abstract

Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in 2D, the problem is solvable in truly subquadratic time [Chan et al., 2025], while for other objects, including unit segments and equilateral triangles in 2D or unit balls and axis-parallel unit cubes in 3D, there is no truly subquadratic time algorithm under the Orthogonal Vector (OV) hypothesis [Bringmann et al., 2022]. 
We undertake a comprehensive study of computing the diameter of geometric intersection graphs for various types of objects. We discover many new irregularities, showing that the landscape is extremely nuanced: the source of hardness is a combination of the object type, the true diameter value, and how the objects intersect with each other. Our highlighted results for the 2D case include:  
1) The diameter of non-degenerate, axis-aligned line segments can be computed in truly subquadratic time. Previous hardness result [Bringmann et al., 2022] for line segments applies only to degenerate instances. On the other hand, for the degenerate case, we show that a truly subquadratic time algorithm exists when the true diameter is constant. 
2) An almost-linear-time algorithm for unit-square graphs of constant diameter. Previous algorithms [Duraj et al., 2024; Chan et al., 2025] rely on succinct representation assuming bounded VC-dimension; for such a strategy Ω(n^{7/4}) time is an inherent barrier. 
3) An Õ(n^{4/3})-time algorithm to decide if the diameter of a unit-disk graph is at most 2. This improves upon the recent algorithm with running time Õ(n^{2-1/9}) [Chan et al., 2025]. 
4) Deciding if the diameter of intersection graphs of fat triangles or line segments is at most 2 is truly subquadratic-hard under fine-grained complexity assumptions. Previous lower bounds [Bringmann et al., 2022] only hold when deciding if diameter is at most 3.  Our findings are presented in a pair of papers. This paper focuses solely on the 2D case, while the companion paper is devoted to higher-dimensional cases.

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Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, Hung Le, and Da Wei Zheng. Charting the Landscape of Diameter Computation on Geometric Intersection Graphs in the Plane. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.54

Author Details

Timothy M. Chan
  • Siebel School of Computing and Data Science, University of Illinois Urbana-Champaign, IL, USA
Hsien-Chih Chang
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Jie Gao
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Sándor Kisfaludi-Bak
  • Department of Computer Science, Aalto University, Espoo, Finland
Hung Le
  • Manning CICS, University of Massachusetts Amherst, MA, USA
Da Wei Zheng
  • Institute of Science and Technology Austria, Klosterneuburg, Austria

Funding

  • Chan, Timothy M.: Supported by NSF grant CCF-2224271.
  • Chang, Hsien-Chih: Supported by NSF CAREER award CCF-2443017.
  • Gao, Jie: Supported by NSF DMS-2220271, DMS-2311064, IIS-2229876, CCF-2118953, CNS-2515159.
  • Kisfaludi-Bak, Sándor: Supported by the Research Council of Finland, Grant 363444.
  • Le, Hung: Supported by an NSF grant CCF-2517033 and an NSF CAREER Award CCF-2237288.
  • Zheng, Da Wei: This project has received funding from the Austrian Science Fund (FWF) grant https://www.doi.org/10.55776/I5982. For open access purposes, the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.

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