Document Open Access Logo

ETH-Tight Algorithm for Cycle Packing on Unit Disk Graphs

Authors Shinwoo An, Eunjin Oh

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.7.pdf
  • Filesize: 0.93 MB
  • 15 pages

Document Identifiers

Author Details

Shinwoo An
  • POSTECH, Pohang, South Korea
Eunjin Oh
  • POSTECH, Pohang, South Korea

Funding

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (No.RS-2023-00209069).

Cite AsGet BibTex

Shinwoo An and Eunjin Oh. ETH-Tight Algorithm for Cycle Packing on Unit Disk Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.7

Abstract

In this paper, we consider the Cycle Packing problem on a unit disk graph defined as follows. Given a unit disk graph G with n vertices and an integer k, the goal is to find a set of k vertex-disjoint cycles of G if it exists. Our algorithm runs in time 2^O(√k) n^O(1). This improves the 2^O(√klog k) n^O(1)-time algorithm by Fomin et al. [SODA 2012, ICALP 2017]. Moreover, our algorithm is optimal assuming the exponential-time hypothesis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Unit disk graphs
  • cycle packing
  • tree decomposition
  • parameterized algorithm

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Shinwoo An and Eunjin Oh. Feedback Vertex Set on Geometric Intersection Graphs. In Proceedings of the 32nd International Symposium on Algorithms and Computation (ISAAC 2021), pages 47:1-47:12, 2021. Google Scholar
  2. Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, and Jie Xue. True contraction decomposition and almost eth-tight bipartization for unit-disk graphs. In 38th International Symposium on Computational Geometry (SoCG 2022), pages 11:1-11:16, 2022. Google Scholar
  3. Sujoy Bhore, Paz Carmi, Sudeshna Kolay, and Meirav Zehavi. Parameterized study of steiner tree on unit disk graphs. Algorithmica, 85:133-152, 2023. Google Scholar
  4. Hans L Bodlaender, Eelko Penninkx, and Richard B Tan. A linear kernel for the k-disjoint cycle problem on planar graphs. In Proceedings of the 19th International Symposium on Algorithms and Computation (ISAAC 2008), pages 306-317, 2008. Google Scholar
  5. Sergio Cabello and Miha Jejčič. Shortest paths in intersection graphs of unit disks. Computational Geometry, 48(4):360-367, 2015. Google Scholar
  6. Timothy M Chan. Finding triangles and other small subgraphs in geometric intersection graphs. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 1777-1805, 2023. Google Scholar
  7. Timothy M Chan and Dimitrios Skrepetos. Approximate shortest paths and distance oracles in weighted unit-disk graphs. Journal of Computational Geometry, 10(2):3-20, 2019. Google Scholar
  8. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Joham MM van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS 2011), pages 150-159, 2011. Google Scholar
  9. Mark de Berg, Hans L Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C van der Zanden. A framework for Exponential-Time-Hypothesis-tight algorithms and lower bounds in geometric intersection graphs. SIAM Journal on Computing, 49(6):1291-1331, 2020. Google Scholar
  10. Hristo N Djidjev. Weighted graph separators and their applications. In Proceedings of the 5th Annual European Symposium on Algorithms (ESA 1997), pages 130-143, 1997. Google Scholar
  11. Frederic Dorn, Fedor V Fomin, and Dimitrios M Thilikos. Catalan structures and dynamic programming in H-minor-free graphs. Journal of Computer and System Sciences, 78(5):1606-1622, 2012. Google Scholar
  12. Frederic Dorn, Eelko Penninkx, Hans L Bodlaender, and Fedor V Fomin. Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica, 58(3):790-810, 2010. Google Scholar
  13. Rodney G Downey and Michael R Fellows. Fundamentals of parameterized complexity, volume 4. Springer, 2013. Google Scholar
  14. Louis Esperet, Sébastien Julliot, and Arnaud de Mesmay. Distributed coloring and the local structure of unit-disk graphs. Theoretical Computer Science, 944:113674, 2023. Google Scholar
  15. Fedor V Fomin, Petr A Golovach, Tanmay Inamdar, and Saket Saurabh. ETH tight algorithms for geometric intersection graphs: Now in polynomial space. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), pages 21:1-21:16, 2021. Google Scholar
  16. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, hitting and packing cycles in subexponential time on unit disk graphs. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), pages 65:1-65:15, 2017. Google Scholar
  17. Fedor V Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Decomposition of map graphs with applications. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), pages 60:1-60:15, 2019. Google Scholar
  18. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. ETH-tight algorithms for long path and cycle on unit disk graphs. In Proceedings of the 36th International Symposium on Computational Geometry (SoCG 2020), pages 44:1-44:18, 2020. Google Scholar
  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 (SODA 2012), pages 1563-1575, 2012. Google Scholar
  20. Zachary Friggstad and Mohammad R Salavatipour. Approximability of packing disjoint cycles. Algorithmica, 60(2):395-400, 2011. Google Scholar
  21. Sariel Har-Peled and Amir Nayyeri. A simple algorithm for computing a cycle separator. CoRR, abs/1709.08122, 2017. Google Scholar
  22. Refael Hassin and Shlomi Rubinstein. An approximation algorithm for maximum triangle packing. Discrete applied mathematics, 154(6):971-979, 2006. Google Scholar
  23. Weizhi Hong, Yingli Ran, and Zhao Zhang. Parallel algorithms for minimum general partial dominating set and maximum budgeted dominating set in unit disk graph. Theoretical Computer Science, 932:13-20, 2022. Google Scholar
  24. Sangram K Jena and Gautam K Das. Vertex-edge domination in unit disk graphs. Discrete Applied Mathematics, 319:351-361, 2022. Google Scholar
  25. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Routing in unit disk graphs. Algorithmica, 80(3):830-848, 2018. Google Scholar
  26. Ken-ichi Kawarabayashi and Bruce Reed. Odd cycle packing. In Proceedings of the forty-second ACM symposium on Theory of computing (STOC 2010), pages 695-704, 2010. Google Scholar
  27. Philip N Klein, Shay Mozes, and Christian Sommer. Structured recursive separator decompositions for planar graphs in linear time. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing (STOC 2013), pages 505-514, 2013. Google Scholar
  28. Michael Krivelevich, Zeev Nutov, Mohammad R Salavatipour, Jacques Verstraete, and Raphael Yuster. Approximation algorithms and hardness results for cycle packing problems. ACM Transactions on Algorithms, 3(4):48:1-48:21, 2007. Google Scholar
  29. Daniel Lokshtanov, Amer E Mouawad, Saket Saurabh, and Meirav Zehavi. Packing cycles faster than Erdös-Pósa. SIAM Journal on Discrete Mathematics, 33(3):1194-1215, 2019. Google Scholar
  30. Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. Subexponential parameterized algorithms on disk graphs. In 33rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2022), pages 2005-2031, 2022. Google Scholar
  31. Dániel Marx. Parameterized complexity of independence and domination on geometric graphs. In Proceedings of the 2nd International Conference on Parameterized and Exact Computation (IWPEC 2006), pages 154-165, 2006. Google Scholar
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact