Illuminating Both Sides of Line Segments

  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2098)

Abstract

What is the minimum number of light sources that can collectively illuminate both sides of n disjoint line segments in the plane? We prove that this optimization problem is NP-hard. The worst case analysis shows, however, that ⌊4(n + 1)/5⌋ light sources are always enough and sometimes necessary for all n ≥2.

This problem was motivated by an open problem posed by Czyzowicz et al.: what is the minimal number of light sources that can collectively illuminate any set of n disjoint segments from one side at least.

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References

  1. 1.
    Baker, B. S.: Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Math. 41 (1994) 153–180MATHGoogle Scholar
  2. 2.
    Berge, C.: Sur le couplage maximum d’un graphe. C. R. Acad. Sciences, Paris 247 (1958) 258–259MATHMathSciNetGoogle Scholar
  3. 3.
    Berman, F., Johnson, D., Leighton, T., Shor, P. W., Snyder, L.: Generalized planar matching. J. Algorithms 11 (1990) 153–184.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Czyzowicz, J., Rival, I., Urrutia, J.: Galleries, light matchings, and visibility graphs. In: Galleries, light matchings and visibility graphs. Algorithms and data structures (Ottawa, ON, 1989), Lecture Notes in Comput. Sci. 382, Springer-Verlag, Berlin, (1989) 316–324Google Scholar
  5. 5.
    Czyzowicz, J., Rivera-Campo, E., Urrutia, J.: Illuminating rectangles and triangles in the plane. J. Combin. Theory Ser. B (1993) 1–17Google Scholar
  6. 6.
    Czyzowicz, J., Rivera-Campo, E., Santoro, N., Urrutia, J., Zaks, J.: Tight bounds for the rectangular art gallery problem, in: Graph-theoretic concepts in computer science (Fischbachau, 1991), Lecture Notes in Comput. Sci. 570, Springer-Verlag, Berlin, (1992) 105–112Google Scholar
  7. 7.
    Czyzowicz, J., Rivera-Campo, E., Urrutia, J., Zaks, J.: On illuminating line segments in the plane. Discrete Math. 137 (1995) 147–153CrossRefMathSciNetGoogle Scholar
  8. 8.
    Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximately of some art gallery problems, in: Proceedings 10th Canadian Conference on Computational Geometry (Montréal, 1998)Google Scholar
  9. 9.
    Fejex, Tóth, L.: Illumination of convex disks. Acta Math. Acad. Sci. Hungar. 29 (1977) 355–360CrossRefMathSciNetGoogle Scholar
  10. 10.
    Hartvigsen, D., Hell, P.: The k-piece packing problem. Manuscript (2000)Google Scholar
  11. 11.
    Hoffmann, F.: On the rectilinear Art Gallery Problem. in: Proc ICALP, Lecture Notes in Comput. Sci. 90, Springer-Verlag (1990) 717–728Google Scholar
  12. 12.
    Kaneko, A.: A necessary and sufficient condition for the existence of a path factor every component of which is a path of length at least two. SubmittedGoogle Scholar
  13. 13.
    Kano, M., Katona, G. Y., Király, Z.: On path-factors and subfactors of graphs without paths of length one. SubmittedGoogle Scholar
  14. 14.
    Karp, R. M.: Reducibility among combinatorial problems, in: Complexity of Computer Computations (R. Miller and J. Thatcher, eds.). Plenum Press, New York (1972) 85–103Google Scholar
  15. 15.
    Kirkpatrick, D. G., Hell, P.: On the completeness of a generalized matching problem, in: Proc. 10th Ann. ACM Symp. on Theory of Computing (San Diego, Calif., 1978), ACM, New York (1978) 240–245Google Scholar
  16. 16.
    Lee, D. T., Lin, A. K.: Computational complexity of art gallery problems. IEEE Trans. Inform. Theory 32 (1986) 276–282MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Nishizeki, T., Baybars, I.: Lower bounds on the cardinality of the maximum matchings of planar graphs. Discrete Math. 28 (1979) 255–267MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    O’Rourke, J.: Open problems in the combinatorics of visibility and illumination, in: Advances in Discrete and Computational Geometry (B. Chazelle, J. E. Goodman, and R. Pollack, eds.) (Contemporary Mathematics), AMS, Providence (1998) 237–243Google Scholar
  19. 19.
    O’Rourke, J.: Art gallery theorems and algorithms. The International Series of Monographs on Computer Science, Oxford University Press, New York (1987)Google Scholar
  20. 20.
    Tóth, Cs. D.: Illumination in the presence of opaque line segments in the plane. SubmittedGoogle Scholar
  21. 21.
    Tóth, Cs. D.: Illuminating disjoint line segments in the plane. SubmittedGoogle Scholar
  22. 22.
    Urrutia, J.: Art Gallery and Illumination Problems. in: Handbook on Computational Geometry (J. R. Sack, J. Urrutia eds.), Elsevier Science Publishers, Amsterdam (2000) 973–1027CrossRefGoogle Scholar
  23. 23.
    Zaks, J.: A note on illuminating line segments in the plane. Manuscript (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Csaba D. Tóth
    • 1
  1. 1.ETH ZürichInstitut fü Theoretische InformatikZürichSwitzerland

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