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Clique-Based Separators for Geometric Intersection Graphs

Authors Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, Leonidas Theocharous

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Author Details

Mark de Berg
  • Department of Computer Science, TU Eindhoven, The Netherlands
Sándor Kisfaludi-Bak
  • Institute for Theoretical Studies, ETH Zürich, Switzerland
Morteza Monemizadeh
  • Department of Computer Science, TU Eindhoven, The Netherlands
Leonidas Theocharous
  • Department of Computer Science, TU Eindhoven, The Netherlands

Funding

The work in this paper is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.

Cite AsGet BibTex

Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, and Leonidas Theocharous. Clique-Based Separators for Geometric Intersection Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.22

Abstract

Let F be a set of n objects in the plane and let 𝒢^{×}(F) be its intersection graph. A balanced clique-based separator of 𝒢^{×}(F) is a set 𝒮 consisting of cliques whose removal partitions 𝒢^{×}(F) into components of size at most δ n, for some fixed constant δ < 1. The weight of a clique-based separator is defined as ∑_{C ∈ 𝒮}log (|C|+1). Recently De Berg et al. (SICOMP 2020) proved that if S consists of convex fat objects, then 𝒢^{×}(F) admits a balanced clique-based separator of weight O(√n). We extend this result in several directions, obtaining the following results. - Map graphs admit a balanced clique-based separator of weight O(√n), which is tight in the worst case. - Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3} log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(√n log n). - Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3} log n). - Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(√n + r log(n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for MAXIMUM INDEPENDENT SET (and, hence, VERTEX COVER), for FEEDBACK VERTEX SET, and for q-Coloring for constant q in these graph classes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational geometry
  • intersection graphs
  • separator theorems

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References

  1. B. Aronov, M.T. Berg, de, E. Ezra, and M. Sharir. Improved bounds for the union of locally fat objects in the plane. SIAM Journal on Computing, 43(2):543-572, 2014. URL: https://doi.org/10.1137/120891241.
  2. Boaz Ben-Moshe, Olaf A. Hall-Holt, Matthew J. Katz, and Joseph S. B. Mitchell. Computing the visibility graph of points within a polygon. In Proc. 20th ACM Symposium on Computational Geometry, pages 27-35. ACM, 2004. URL: https://doi.org/10.1145/997817.997825.
  3. Édouard Bonnet and Pawel Rzazewski. Optimality program in segment and string graphs. Algorithmica, 81(7):3047-3073, 2019. URL: https://doi.org/10.1007/s00453-019-00568-7.
  4. Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178-189, 2003. URL: https://doi.org/10.1016/S0196-6774(02)00294-8.
  5. Zhi-Zhong Chen. Approximation algorithms for independent sets in map graphs. J. Algorithms, 41(1):20-40, 2001. URL: https://doi.org/10.1006/jagm.2001.1178.
  6. Zhi-Zhong Chen, Michelangelo Grigni, and Christos H. Papadimitriou. Map graphs. J. ACM, 49(2):127-138, 2002. URL: https://doi.org/10.1145/506147.506148.
  7. Kenneth L. Clarkson and Peter W. Shor. Application of random sampling in computational geometry, II. Discret. Comput. Geom., 4:387-421, 1989. URL: https://doi.org/10.1007/BF02187740.
  8. 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 J. Comput., 49:1291-1331, 2020. URL: https://doi.org/10.1137/20M1320870.
  9. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications (3rd Edition). Springer, 2008. URL: https://www.worldcat.org/oclc/227584184.
  10. Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, and Leonidas Theocharous. Clique-based separators for geometric intersection graphs, 2021. URL: http://arxiv.org/abs/2109.09874.
  11. Hristo Djidjev and Shankar M. Venkatesan. Reduced constants for simple cycle graph separation. Acta Informatica, 34(3):231-243, 1997. URL: https://doi.org/10.1007/s002360050082.
  12. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Decomposition of map graphs with applications. In Proc. 46th International Colloquium on Automata, Languages, and Programming, (ICALP), pages 60:1-60:15, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.60.
  13. Jacob Fox, János Pach, and Csaba D. Tóth. A bipartite strengthening of the crossing lemma. J. Comb. Theory, Ser. B, 100(1):23-35, 2010. URL: https://doi.org/10.1016/j.jctb.2009.03.005.
  14. Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. SIAM J. Comput., 46(6):1712-1744, 2017. URL: https://doi.org/10.1137/16M1079336.
  15. Klara Kedem, Ron Livne, János Pach, and Micha Sharir. On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discret. Comput. Geom., 1:59-70, 1986. URL: https://doi.org/10.1007/BF02187683.
  16. Sándor Kisfaludi-Bak. Hyperbolic intersection graphs and (quasi)-polynomial time. In Proc. 31st ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1621-1638, 2020. URL: https://doi.org/10.1137/1.9781611975994.100.
  17. Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. How does object fatness impact the complexity of packing in d dimensions? In 30th International Symposium on Algorithms and Computation, ISAAC 2019, volume 149 of LIPIcs, pages 36:1-36:18, 2019. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2019.36.
  18. James R. Lee. Separators in region intersection graphs. In 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, volume 67 of LIPIcs, pages 1:1-1:8, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.1.
  19. Richard J. Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math, 36(2):177-189, 1977. URL: https://doi.org/doi/10.1137/0136016.
  20. Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using voronoi diagrams. In Proc. 23rd Annual European Symposium on Algorithms (ESA), volume 9294 of Lecture Notes in Computer Science, pages 865-877. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_72.
  21. Jirí Matoušek. Near-optimal separators in string graphs. Comb. Probab. Comput., 23(1):135-139, 2014. URL: https://doi.org/10.1017/S0963548313000400.
  22. Gary L. Miller, Shang-Hua Teng, William P. Thurston, and Stephen A. Vavasis. Separators for sphere-packings and nearest neighbor graphs. J. ACM, 44(1):1-29, 1997. URL: https://doi.org/10.1145/256292.256294.
  23. János Pach and Micha Sharir. Geometric incidences. In János Pach, editor, Towards a Theory of Geometric Graphs (Contemporary Mathematics, Vol. 342), pages 185-223. Amer. Math. Soc., 2004. Google Scholar
  24. Rom Pinchasi. A finite family of pseudodiscs must include a “small” pseudodisc. SIAM Journal on Discrete Mathematics, 28:1930-1934, October 2014. URL: https://doi.org/10.1137/130949750.
  25. Ricky Pollack, Micha Sharir, and Günter Rote. Computing the geodesic center of a simple polygon. Discret. Comput. Geom., 4(6):611-626, 1989. URL: https://doi.org/10.1007/BF02187751.
  26. Rajiv Raman and Saurabh Ray. Constructing planar support for non-piercing regions. Discrete & Computational Geometry, pages 1-25, 2020. Google Scholar
  27. Warren D. Smith and Nicholas C. Wormald. Geometric separator theorems & applications. In 39th Annual Symposium on Foundations of Computer Science (FOCS), pages 232-243. IEEE Computer Society, 1998. URL: https://doi.org/10.1109/SFCS.1998.743449.
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