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Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs

Authors Lech Duraj , Filip Konieczny , Krzysztof Potępa

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Computational geometry
Keywords
  • Graph Diameter
  • Geometric Intersection Graphs
  • Vapnik-Chervonenkis Dimension

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Abstract

We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs.
We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in 𝒪{(k ⋅ n^{1-1/d} ⋅ m ⋅ polylog(n))} time complexity for the graph on n vertices and m edges, where k is the diameter and d is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an 𝒪{(n^{7/4} ⋅ polylog(n))} parameterized diameter algorithm for unit square intersection graph of size n, as well as a more general algorithm for convex polygon intersection graphs.

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Lech Duraj, Filip Konieczny, and Krzysztof Potępa. Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.51

Author Details

Lech Duraj
  • Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Poland
Filip Konieczny
  • Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Poland
Krzysztof Potępa
  • Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Poland

Funding

All authors were supported by National Science Center of Poland grant 2019/34/E/ST6/ 00443.
  • Konieczny, Filip: During the preparation of this article, Filip Konieczny was a participant of the tutoring programme under the Excellence Initiative at the Jagiellonian University.

Acknowledgements

We would like to thank the reviewers for helping to improve our paper with their suggestions.

References

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