Document Open Access Logo

Small Independent Sets Versus Small Separator in Geometric Intersection Graphs

Authors Malory Marin , Rémi Watrigant

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2026.137.pdf
  • Filesize: 1.58 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Subexponential Algorithms
  • Unit Disk Graphs
  • 2-Subcoloring
  • Two-Sets Cut-Uncut

Metrics

Abstract

While most classical NP-hard graph problems cannot be solved in time 2^o(n) on general graphs under the Exponential Time Hypothesis (ETH), many exhibit the square-root phenomenon and admit optimal algorithms running in time 2^O(√n) on certain geometric intersection graphs, such as planar graphs or unit disk graphs. In 2018, de Berg et al. developed a general algorithmic framework for such problems on intersection graphs of similarly sized fat objects in ℝ^d, achieving running times of the form 2^O(n^{1-1/d}), along with matching lower bounds under ETH.
In this paper, we identify problems that do not exhibit the square-root phenomenon, yet still admit subexponential algorithms on intersection graphs of similarly sized fat objects in ℝ^d, for every fixed dimension d ⩾ 2. We introduce the notion of a weak square-root phenomenon: problems that can be solved in time 2^Õ(n^{1-1/(d+1)}), and for which matching lower bounds hold under ETH. We develop both an algorithmic framework and a corresponding lower bound framework. As concrete examples, we show that the problems 2-Subcoloring and Two Sets Cut-Uncut exhibit this behavior.
Our algorithms rely on a new win-win structural theorem, which can be informally stated as follows: every such graph admits a sublinear separator whose removal leaves connected components with sublinear independence number. To facilitate the design of these algorithms, we introduce a new graph parameter, the α-modulator number, which generalizes both the independence number and the vertex cover number.

Cite As Get BibTex

Malory Marin and Rémi Watrigant. Small Independent Sets Versus Small Separator in Geometric Intersection Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 137:1-137:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.137

Author Details

Malory Marin
  • ENS de Lyon, CNRS, Université Claude Bernard Lyon 1, LIP, UMR 5668, 69342, Lyon cedex 07, France
Rémi Watrigant
  • Université Claude Bernard Lyon 1, ENS de Lyon, CNRS, LIP, UMR 5668, 69342, Lyon cedex 07, France

References

  1. Jochen Alber and Jiří Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. Journal of Algorithms, 52(2):134-151, 2004. URL: https://doi.org/10.1016/J.JALGOR.2003.10.001.
  2. Michael O Albertson, Robert E Jamison, Stephen T Hedetniemi, and Stephen C Locke. The subchromatic number of a graph. In Annals of Discrete Mathematics, volume 39, pages 33-49. Elsevier, 1989. URL: https://doi.org/10.1016/0012-365X(89)90196-9.
  3. Matthias Bentert, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach, and Tuukka Korhonen. Two-Sets Cut-Uncut on Planar Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024), volume 297 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.22.
  4. Matthias Bentert, Fedor V Fomin, Fanny Hauser, and Saket Saurabh. The parameterized complexity landscape of two-sets cut-uncut. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2408.13543.
  5. Marthe Bonamy, Édouard Bonnet, Nicolas Bousquet, Pierre Charbit, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, Florian Sikora, and Stéphan Thomassé. Eptas and subexponential algorithm for maximum clique on disk and unit ball graphs. Journal of the ACM (JACM), 68(2):1-38, 2021. URL: https://doi.org/10.1145/3433160.
  6. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more. SIAM J. Comput., 49(4):772-810, 2020. URL: https://doi.org/10.1137/18M1166869.
  7. Christopher A Cullenbine, R Kevin Wood, and Alexandra M Newman. Theoretical and computational advances for network diversion. Networks, 62(3):225-242, 2013. URL: https://doi.org/10.1002/NET.21514.
  8. Norman D Curet. The network diversion problem. Military Operations Research, pages 35-44, 2001. Google Scholar
  9. Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and Jakub Onufry Wojtaszczyk. Solving the 2-disjoint connected subgraphs problem faster than 2 n. Algorithmica, 70(2):195-207, 2014. URL: https://doi.org/10.1007/S00453-013-9796-X.
  10. Clément Dallard, Fedor V Fomin, Petr A Golovach, Tuukka Korhonen, and Martin Milanič. Computing tree decompositions with small independence number. ACM Transactions on Algorithms, 22(1):1-25, 2025. URL: https://doi.org/10.1145/3767730.
  11. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. ii. tree-independence number. Journal of Combinatorial Theory, Series B, 164:404-442, 2024. URL: https://doi.org/10.1016/J.JCTB.2023.10.006.
  12. Andreas Darmann and Janosch Döcker. On a simple hard variant of not-all-equal 3-sat. Theoretical Computer Science, 815:147-152, 2020. URL: https://doi.org/10.1016/J.TCS.2020.02.010.
  13. Mark de Berg, Hans L Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C van der Zanden. A framework for eth-tight algorithms and lower bounds in geometric intersection graphs. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 574-586, 2018. URL: https://doi.org/10.1145/3188745.3188854.
  14. Ozgur Erken. A branch-and-bound algorithm for the network diversion problem. PhD thesis, Monterey, California. Naval Postgraduate School, 2002. Google Scholar
  15. Jirí Fiala, Klaus Jansen, Van Bang Le, and Eike Seidel. Graph subcolorings: complexity and algorithms. SIAM Journal on Discrete Mathematics, 16(4):635-650, 2003. URL: https://doi.org/10.1137/S0895480101395245.
  16. Fedor V Fomin, Petr A Golovach, Nikola Jedličková, Jan Kratochvíl, Danil Sagunov, and Kirill Simonov. Path cover, hamiltonicity, and independence number: An fpt perspective. arXiv preprint, 2024. URL: https://arxiv.org/abs/2403.05943.
  17. Fedor V Fomin, Petr A Golovach, Laure Morelle, and Dimitrios M Thilikos. H-planarity and parametric extensions: when modulators act globally. In Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4584-4601. SIAM, 2026. URL: https://doi.org/10.1137/1.9781611978971.166.
  18. Fedor V Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, hitting and packing cycles in subexponential time on unit disk graphs. Discrete & Computational Geometry, 62(4):879-911, 2019. URL: https://doi.org/10.1007/S00454-018-00054-X.
  19. Fedor V Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1563-1575. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.124.
  20. John Gimbel and Chris Hartman. Subcolorings and the subchromatic number of a graph. Discrete Mathematics, 272(2-3):139-154, 2003. URL: https://doi.org/10.1016/S0012-365X(03)00177-8.
  21. Chris Gray, Frank Kammer, Maarten Löffler, and Rodrigo I Silveira. Removing local extrema from imprecise terrains. Computational Geometry, 45(7):334-349, 2012. URL: https://doi.org/10.1016/J.COMGEO.2012.02.002.
  22. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: https://doi.org/10.1006/JCSS.2001.1774.
  23. Benjamin S Kallemyn. Modeling Network Interdiction Tasks. Doctoral thesis, Air Force Institute of Technology, Air University, Wright-Patterson Air Force Base, Ohio, 2015. Google Scholar
  24. Iyad Kanj, Christian Komusiewicz, Manuel Sorge, and Erik Jan Van Leeuwen. Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs. Journal of Computer and System Sciences, 92:22-47, 2018. URL: https://doi.org/10.1016/J.JCSS.2017.08.002.
  25. Walter Kern, Barnaby Martin, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Disjoint paths and connected subgraphs for h-free graphs. Theoretical Computer Science, 898:59-68, 2022. URL: https://doi.org/10.1016/J.TCS.2021.10.019.
  26. Tomohiro Koana, Nidhi Purohit, and Kirill Simonov. Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024), volume 321 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.IPEC.2024.10.
  27. Chungmok Lee, Donghyun Cho, and Sungsoo Park. A combinatorial benders decomposition algorithm for the directed multiflow network diversion problem. Military Operations Research, 24(1):23-40, 2019. Google Scholar
  28. Malory Marin and Rémi Watrigant. Subcoloring of (Unit) Disk Graphs. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025), volume 345 of Leibniz International Proceedings in Informatics (LIPIcs), pages 74:1-74:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.MFCS.2025.74.
  29. Dániel Marx and Anastasios Sidiropoulos. The limited blessing of low dimensionality: when 1-1/d is the best possible exponent for d-dimensional geometric problems. In Proceedings of the thirtieth annual symposium on Computational geometry, pages 67-76, 2014. Google Scholar
  30. Pascal Ochem. 2-subcoloring is np-complete for planar comparability graphs. Information Processing Letters, 128:46-48, 2017. URL: https://doi.org/10.1016/J.IPL.2017.08.004.
  31. Daniël Paulusma and Johan MM van Rooij. On partitioning a graph into two connected subgraphs. Theoretical computer science, 412(48):6761-6769, 2011. URL: https://doi.org/10.1016/J.TCS.2011.09.001.
  32. Jan Arne Telle and Yngve Villanger. Connecting terminals and 2-disjoint connected subgraphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 418-428. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-45043-3_36.
  33. Pim van’t Hof, Daniël Paulusma, and Gerhard J Woeginger. Partitioning graphs into connected parts. Theoretical computer science, 410(47-49):4834-4843, 2009. URL: https://doi.org/10.1016/J.TCS.2009.06.028.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact