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Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets

Authors Tim Planken , Torsten Ueckerdt

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Tim Planken
  • University of Birmingham, UK
Torsten Ueckerdt
  • Karlsruhe Institute of Technology, Germany

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Tim Planken and Torsten Ueckerdt. Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.74

Abstract

A range family ℛ is a family of subsets of ℝ^d, like all halfplanes, or all unit disks. Given a range family ℛ, we consider the m-uniform range capturing hypergraphs ℋ(V,ℛ,m) whose vertex-sets V are finite sets of points in ℝ^d with any m vertices forming a hyperedge e whenever e = V ∩ R for some R ∈ ℛ. Given additionally an integer k ≥ 2, we seek to find the minimum m = m_ℛ(k) such that every ℋ(V,ℛ,m) admits a polychromatic k-coloring of its vertices, that is, where every hyperedge contains at least one point of each color. Clearly, m_ℛ(k) ≥ k and the gold standard is an upper bound m_ℛ(k) = O(k) that is linear in k. A t-shallow hitting set in ℋ(V,ℛ,m) is a subset S ⊆ V such that 1 ≤ |e ∩ S| ≤ t for each hyperedge e; i.e., every hyperedge is hit at least once but at most t times by S. We show for several range families ℛ the existence of t-shallow hitting sets in every ℋ(V,ℛ,m) with t being a constant only depending on ℛ. This in particular proves that m_ℛ(k) ≤ tk = O(k) in such cases, improving previous polynomial bounds in k. Particularly, we prove this for the range families of all axis-aligned strips in ℝ^d, all bottomless and topless rectangles in ℝ², and for all unit-height axis-aligned rectangles in ℝ².

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypergraphs
Keywords
  • geometric hypergraphs
  • range spaces
  • polychromatic coloring
  • shallow hitting sets

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References

  1. The geometric hypergraph zoo. URL: https://coge.elte.hu/cogezoo.html.
  2. Eyal Ackerman, Balázs Keszegh, and Mate Vizer. Coloring points with respect to squares. Discrete & Computational Geometry, 58(4):757-784, 2017. URL: https://doi.org/10.1007/s00454-017-9902-y.
  3. Greg Aloupis, Jean Cardinal, Sébastien Collette, Shinji Imahori, Matias Korman, Stefan Langerman, Oded Schwartz, Shakhar Smorodinsky, and Perouz Taslakian. Colorful strips. Graphs and Combinatorics, 27(3):327-339, 2011. URL: https://doi.org/10.1007/s00373-011-1014-5.
  4. Andrei Asinowski, Jean Cardinal, Nathann Cohen, Sébastien Collette, Thomas Hackl, Michael Hoffmann, Kolja Knauer, Stefan Langerman, Michał Lasoń, Piotr Micek, Günter Rote, and Torsten Ueckerdt. Coloring hypergraphs induced by dynamic point sets and bottomless rectangles. In Frank Dehne, Roberto Solis-Oba, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures, pages 73-84, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-642-40104-6_7.
  5. Béla Bollobás, David Pritchard, Thomas Rothvoss, and Alex Scott. Cover-decomposition and polychromatic numbers. SIAM Journal on Discrete Mathematics, 27(1):240-256, 2013. URL: https://doi.org/10.1137/110856332.
  6. Balázs Bursics, Bence Csonka, and Luca Szepessy. Hitting sets and colorings of hypergraphs, 2023. URL: https://arxiv.org/abs/2307.12154.
  7. Jean Cardinal, Kolja Knauer, Piotr Micek, Dömötör Pálvölgyi, Torsten Ueckerdt, and Narmada Varadarajan. Colouring bottomless rectangles and arborescences. Computational Geometry, 115:102020, 2023. URL: https://doi.org/10.1016/j.comgeo.2023.102020.
  8. Jean Cardinal, Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. Making octants colorful and related covering decomposition problems. SIAM Journal on Discrete Mathematics, 28(4):1948-1959, 2014. URL: https://doi.org/10.1137/140955975.
  9. Jean Cardinal, Kolja B. Knauer, Piotr Micek, and Torsten Ueckerdt. Making triangles colorful. Journal of Computational Geometry, 4(1):240-246, 2013. URL: https://doi.org/10.20382/jocg.v4i1a10.
  10. Vera Chekan and Torsten Ueckerdt. Polychromatic colorings of unions of geometric hypergraphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 144-157. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-15914-5_11.
  11. Xiaomin Chen, János Pach, Mario Szegedy, and Gábor Tardos. Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles. Random Structures & Algorithms, 34(1):11-23, 2009. URL: https://doi.org/10.1002/rsa.20246.
  12. Matt Gibson and Kasturi Varadarajan. Decomposing coverings and the planar sensor cover problem. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 159-168, 2009. URL: https://doi.org/10.1109/FOCS.2009.54.
  13. Balázs Keszegh, Nathan Lemons, and Dömötör Pálvölgyi. Online and quasi-online colorings of wedges and intervals. Order, 33(3):389-409, 2016. URL: https://doi.org/10.1007/s11083-015-9374-8.
  14. Balázs Keszegh and Dömötör Pálvölgyi. More on decomposing coverings by octants. Journal of Computational Geometry, 6(1):300-315, 2015. URL: https://doi.org/10.20382/jocg.v6i1a13.
  15. Balázs Keszegh and Dömötör Pálvölgyi. An abstract approach to polychromatic coloring: shallow hitting sets in ABA-free hypergraphs and pseudohalfplanes. Journal of Computational Geometry, 10(1):1-26, 2019. URL: https://doi.org/10.20382/jocg.v10i1a1.
  16. Balázs Keszegh. Coloring half-planes and bottomless rectangles. Computational Geometry, 45(9):495-507, 2012. The 19th Canadian Conference on Computational Geometry (CCCG2007) held in Ottawa, Canada on August 20-22, 2007. URL: https://doi.org/10.1016/j.comgeo.2011.09.004.
  17. István Kovács. Indecomposable coverings with homothetic polygons. Discrete & Computational Geometry, 53(4):817-824, 2015. URL: https://doi.org/10.1007/s00454-015-9687-9.
  18. János Pach. Decomposition of multiple packing and covering. 2. Kolloquium über Diskrete Geometrie, pages 169-178, 1980. URL: https://infoscience.epfl.ch/record/129388.
  19. János Pach. Covering the plane with convex polygons. Discrete & Computational Geometry, 1(1):73-81, 1986. URL: https://doi.org/10.1007/BF02187684.
  20. János Pach and Dömötör Pálvölgyi. Unsplittable coverings in the plane. Advances in Mathematics, 302:433-457, 2016. URL: https://doi.org/10.1016/j.aim.2016.07.011.
  21. János Pach, Dömötör Pálvölgyi, and Géza Tóth. Survey on decomposition of multiple coverings. In Imre Bárány, Károly J. Böröczky, Gábor Fejes Tóth, and János Pach, editors, Geometry - Intuitive, Discrete, and Convex: A Tribute to László Fejes Tóth, pages 219-257. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-642-41498-5_9.
  22. János Pach, Gábor Tardos, and Géza Tóth. Indecomposable coverings. In Jin Akiyama, William Y. C. Chen, Mikio Kano, Xueliang Li, and Qinglin Yu, editors, Discrete Geometry, Combinatorics and Graph Theory, pages 135-148, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-70666-3_15.
  23. Dömötör Pálvölgyi. Indecomposable coverings with unit discs. arXiv preprint v1, 2013. URL: https://arxiv.org/abs/1310.6900.
  24. Dömötör Pálvölgyi and Géza Tóth. Convex polygons are cover-decomposable. Discrete & Computational Geometry, 43(3):483-496, 2010. URL: https://doi.org/10.1007/s00454-009-9133-y.
  25. Tim Planken and Torsten Ueckerdt. Polychromatic colorings of geometric hypergraphs via shallow hitting sets. arXiv preprint, 2023. URL: https://arxiv.org/abs/2310.19982.
  26. Tim Planken and Torsten Ueckerdt. Shallow hitting edge sets in uniform hypergraphs. arXiv preprint, 2023. URL: https://arxiv.org/abs/2307.05757.
  27. Alexandre Rok. Combinatorial Properties of Graphs and Hypergraphs Arising in Geometry. PhD thesis, Ben-Gurion University of the Negev, 2019. URL: https://cris.bgu.ac.il/en/studentTheses/combinatorial-properties-of-graphs-and-hypergraphs-arising-in-geo.
  28. Alexandre Rok, Oded Schwartz, and Shakhar Smorodinsky. Polychromatic coloring of axis-parallel strips. unpublished manuscript, 2019. Google Scholar
  29. Shakhar Smorodinsky and Yelena Yuditsky. Polychromatic coloring for half-planes. Journal of Combinatorial Theory, Series A, 119(1):146-154, 2012. URL: https://doi.org/10.1016/j.jcta.2011.07.001.
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