Colorful Strips

  • Greg Aloupis
  • Jean Cardinal
  • Sébastien Collette
  • Shinji Imahori
  • Matias Korman
  • Stefan Langerman
  • Oded Schwartz
  • Shakhar Smorodinsky
  • Perouz Taslakian
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)

Abstract

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k−1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4ln k + ln d) points.

We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k−1) + 1. Lower bounds are also given for all of the above problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations.

From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N.: A simple algorithm for edge-coloring bipartite multigraphs. Inf. Proc. Lett. 85(6), 301–302 (2003)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. John Wiley, Chichester (2000)MATHGoogle Scholar
  3. 3.
    Aloupis, G., Cardinal, J., Collette, S., Langerman, S., Orden, D., Ramos, P.: Decomposition of multiple coverings into more parts. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, SODA 2009 (2009)Google Scholar
  4. 4.
    Aloupis, G., Cardinal, J., Collette, S., Langerman, S., Smorodinsky, S.: Coloring geometric range spaces. Discrete & Computational Geometry 41(2), 348–362 (2009)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Alspach, B.: The wonderful walecki construction. Bull. Inst. Combin. Appl. 52, 7–20 (2008)MATHMathSciNetGoogle Scholar
  6. 6.
    Buchsbaum, A., Efrat, A., Jain, S., Venkatasubramanian, S., Yi, K.: Restricted strip covering and the sensor cover problem. In: ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 (2007)Google Scholar
  7. 7.
    Cole, R., Ost, K., Schirra, S.: Edge-coloring bipartite multigraphs in O(E log D) time. Combinatorica 21(1), 5–12 (2001)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Even, G., Lotker, Z., Ron, D., Smorodinsky, S.: Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM Journal on Computing 33(1), 94–136 (2004)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  10. 10.
    Haxell, P., Szabó, T., Tardos, G.: Bounded size components: partitions and transversals. J. Comb. Theory Ser. B 88(2), 281–297 (2003)MATHCrossRefGoogle Scholar
  11. 11.
    Mani, P., Pach, J.: Decomposition problems for multiple coverings with unit balls (manuscript) (1986)Google Scholar
  12. 12.
    Moser, R.A.: A constructive proof of the lovász local lemma. In: Proc. of the ACM symposium on Theory of computing, New York, NY, USA, pp. 343–350 (2009)Google Scholar
  13. 13.
    Pach, J.: Decomposition of multiple packing and covering. In: 2. Kolloq. über Diskrete Geom., pp. 169–178. Inst. Math. Univ. Salzburg, Salzburg (1980)Google Scholar
  14. 14.
    Pach, J.: Decomposition of multiple packing and covering. In: 2. Kolloquium Uber Diskrete Geometrie, pp. 169–178. Inst. Math. Univ. Salzburg, Salzburg (1980)Google Scholar
  15. 15.
    Pach, J.: Covering the plane with convex polygons. Discrete & Computational Geometry 1, 73–81 (1986)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Pach, J., Tardos, G., Tóth, G.: Indecomposable coverings. In: Akiyama, J., Chen, W.Y.C., Kano, M., Li, X., Yu, Q. (eds.) CJCDGCGT 2005. LNCS, vol. 4381, pp. 135–148. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Pach, J., Tóth, G.: Decomposition of multiple coverings into many parts. In: Proc. of the ACM Symposium on Computational Geometry, pp. 133–137 (2007)Google Scholar
  18. 18.
    Smorodinsky, S.: On the chromatic number of some geometric hypergraphs. SIAM Journal on Discrete Mathematics 21(3), 676–687 (2007)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Stanton, R.G., Cowan, D.D., James, L.O.: Some results on path numbers. In: Louisiana Conference on Combin., Graph Theory and Computing (1970)Google Scholar
  20. 20.
    Tardos, G., Tóth, G.: Multiple coverings of the plane with triangles. Discrete & Computational Geometry 38(2), 443–450 (2007)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Greg Aloupis
    • 1
  • Jean Cardinal
    • 1
  • Sébastien Collette
    • 1
  • Shinji Imahori
    • 2
  • Matias Korman
    • 3
  • Stefan Langerman
    • 1
  • Oded Schwartz
    • 4
  • Shakhar Smorodinsky
    • 5
  • Perouz Taslakian
    • 1
  1. 1.Université Libre de BruxellesBrusselsBelgium
  2. 2.Graduate School of EngineeringNagoya UniversityNagoyaJapan
  3. 3.Graduate School of Information Sciences (GSIS)Tohoku UniversityJapan
  4. 4.Departments of MathematicsTechnische Universität BerlinBerlinGermany
  5. 5.Ben-Gurion UniversityBe’er ShevaIsrael

Personalised recommendations