, Volume 66, Issue 3, pp 564–594 | Cite as

Approximate Guarding of Monotone and Rectilinear Polygons

  • Erik A. Krohn
  • Bengt J. NilssonEmail author


We show that vertex guarding a monotone polygon is NP-hard and construct a constant factor approximation algorithm for interior guarding monotone polygons. Using this algorithm we obtain an approximation algorithm for interior guarding rectilinear polygons that has an approximation factor independent of the number of vertices of the polygon. If the size of the smallest interior guard cover is OPT for a rectilinear polygon, our algorithm produces a guard set of size O(OPT 2).


Computational geometry Art gallery problems Monotone polygons Rectilinear polygons Approximation algorithms 


  1. 1.
    Aggarwal, A.: The art gallery theorem: its variations, applications and algorithmic aspects. Ph.D. thesis, Johns Hopkins University (1984) Google Scholar
  2. 2.
    Brodén, B., Hammar, M., Nilsson, B.J.: Guarding lines and 2-link polygons is APX-hard. In: Proc. 13th Canadian Conference on Computational Geometry, CCCG’01, pp. 45–48 (2001) Google Scholar
  3. 3.
    Cardano, G.: Artis Magnæ, Sive de Regulis Algebraicis Liber Unus (1545). (English translation reprinted by Dover Publications in 1993 as Ars Magna or The Rules of Algebra) Google Scholar
  4. 4.
    Chazelle, B.: Triangulating a simple polygon in linear time. In: Proc. 31st Symposium on Foundations of Computer Science, pp. 220–230 (1990) CrossRefGoogle Scholar
  5. 5.
    Chen, D.Z., Estivill-Castro, V., Urrutia, J.: Optimal guarding of polygons and monotone chains. In: Proc. 7th Canadian Conference on Computational Geometry, CCCG’95, pp. 133–138 (1995) Google Scholar
  6. 6.
    Chin, W., Ntafos, S.: Optimum Watchman routes. Inf. Process. Lett. 28, 39–44 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chvátal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory, B 13(6), 395–398 (1975) Google Scholar
  8. 8.
    Clarkson, K.L., Varadarajan, K.: Improved approximation algorithms for geometric set cover. In: Proc. 21st ACM Symposium on Computational Geometry (2005) Google Scholar
  9. 9.
    Culberson, J.C., Reckhow, R.A.: Covering polygons is hard. In: Proc. 29th Symposium on Foundations of Computer Science, pp. 601–611 (1988) Google Scholar
  10. 10.
    de Berg, M.: On rectilinear link distance. Comput. Geom., Theory Appl. 1(1), 13–34 (1991) zbMATHCrossRefGoogle Scholar
  11. 11.
    Deshpande, A., Kim, T., Demaine, E.D., Sarna, S.E.: A pseudopolynomial time O(logn)-approximation algorithm for art gallery problems. In: Proc. 10th Workshop on Algorithms and Data Structures, WADS’07. Lecture Notes in Computer Science, vol. 4619, pp. 163–174. Springer, Berlin (2007) CrossRefGoogle Scholar
  12. 12.
    Efrat, A., Har-Peled, S.: Guarding galleries and terrains. Inf. Process. Lett. 100, 238–245 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Eidenbenz, S.: Inapproximability results for guarding polygons without holes. In: Proc. 9th Annual International Symposium on Algorithms and Computation, pp. 427–436 (1998) CrossRefGoogle Scholar
  14. 14.
    Eidenbenz, S.: Inapproximability of visibility problems on polygons and terrains. Ph.D. thesis, ETH, Zurich (2000) Google Scholar
  15. 15.
    ElGindy, H., Avis, D.: A linear algorithm for computing the visibility polygon from a point. J. Algorithms 2, 186–197 (1981) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fejes Tóth, L.: Illumination of convex discs. Acta Math. Acad. Sci. Hung. 29, 355–360 (1977) zbMATHCrossRefGoogle Scholar
  17. 17.
    Fisk, S.: A short proof of Chvátal’s Watchman theorem. J. Comb. Theory, B 24, 374 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979) zbMATHGoogle Scholar
  19. 19.
    Ghosh, S.K.: Approximation algorithms for art gallery problems. In: Proceedings of the Canadian Information Processing Society Congress (1987) Google Scholar
  20. 20.
    Goodman, J.E., O’Rourke, J. (eds.): Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (1997) zbMATHGoogle Scholar
  21. 21.
    Hoffmann, F., Kaufmann, M., Kriegel, K.: The art gallery theorem for polygons with holes. In: Proc. 32nd IEEE Symposium on the Foundations of Computer Science, pp. 39–48 (1991) Google Scholar
  22. 22.
    Joe, B., Simpson, R.B.: Correction to Lee’s visibility polygon algorithm. BIT 27, 458–473 (1987) zbMATHCrossRefGoogle Scholar
  23. 23.
    Katz, M.J., Roisman, G.S.: On guarding rectilinear domains. In: Proc. 10th Scandinavian Workshop on Algorithm Theory. Lecture Notes in Computer Science, vol. 4059, pp. 220–231. Springer, Berlin (2006) Google Scholar
  24. 24.
    King, J., Krohn, E.: Terrain guarding is NP-hard. In: Proc. 21st ACM-SIAM Symposium on Discrete Algorithms, SODA’10, pp. 1580–1593 (2010) Google Scholar
  25. 25.
    Lee, D.T.: Visibility of a simple polygon. Comput. Vis. Graph. Image Process. 22, 207–221 (1983) zbMATHCrossRefGoogle Scholar
  26. 26.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theory IT-32, 276–282 (1986) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lee, D.T., Preparata, F.P.: An optimal algorithm for finding the kernel of a polygon. J. ACM 26, 415–421 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Levcopoulos, C.: Heuristics for minimum decompositions of polygons. Ph.D. thesis, University of Linköping, Linköping, Sweden (1987) Google Scholar
  29. 29.
    Levcopoulos, C., Lingas, A.: Covering polygons with minimum number of rectangles. In: Proc. 1st Symposium on the Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 166, pp. 63–72. Springer, Berlin (1984) Google Scholar
  30. 30.
    Nilsson, B.J.: Guarding art galleries—methods for mobile guards. Ph.D. thesis, Lund University (1995) Google Scholar
  31. 31.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, London (1987) zbMATHGoogle Scholar
  32. 32.
    Sack, J.R., Urrutia, J. (eds.): Handbook on Computational Geometry. Elsevier, Amsterdam (1999) Google Scholar
  33. 33.
    Shermer, T.C.: Recent results in art galleries. Proc. IEEE September, 1384–1399 (1992) CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Wisconsin—OshkoshOshkoshUSA
  2. 2.Department of Computer ScienceMalmö UniversityMalmöSweden

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