Skip to main content
SpringerLink
Log in

On Orthogonally Guarding Orthogonal Polygons with Bounded Treewidth

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

There exist many variants of guarding an orthogonal polygon in an orthogonal fashion: sometimes a guard can see within a rectangle, along a staircase, or along an orthogonal path with at most k bends. In this paper, we study all these guarding models for the special case of orthogonal polygons that have bounded treewidth in some sense. As our main result, we show that the problem of finding the minimum number of guards in all these models becomes linear-time solvable on polygons with bounded treewidth. We complement our main result by giving some hardness results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. This run-time can surely be improved, but since our algorithm is dominated by the run-time of solving the problem we reduce ours to (see Sect. 4.5 for more details), we did not want to explore this further.

  2. We use “\(L_1\)” to emphasize that this path must be orthogonal; the concept would make sense for non-orthogonal paths but we do not have any results for them.

References

  1. Biedl, T., Mehrabi, S.: On r-guarding thin orthogonal polygons. In: 27th International Symposium on Algorithms and Computation (ISAAC 2016), Vol. 64 of LIPIcs, pp. 17:1–17:13 (D2016)

  2. Biedl, T., Mehrabi, S.: On guarding orthogonal polygons with bounded treewidth. In: Canadian Conference on Computational Geometry (CCCG 2017), Ottawa, Ontario (2017)

  3. O’Rourke, J.: Art Gallery Theorems and Algorithms. The International Series of Monographs on Computer Science. Oxford University Press, New York (1987)

    MATH  Google Scholar 

  4. Chvátal, V.: A combinatorial theorem in plane geometry. J. Combin. Theory, Ser. B 18, 39–41 (1975)

    Article  MathSciNet  Google Scholar 

  5. Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theory 32(2), 276–282 (1986)

    Article  MathSciNet  Google Scholar 

  6. Schuchardt, D., Hecker, H.-D.: Two NP-hard art-gallery problems for ortho-polygons. Math. Logic Q. 41(2), 261–267 (1995)

    Article  MathSciNet  Google Scholar 

  7. Krohn, E., Nilsson, B.J.: Approximate guarding of monotone and rectilinear polygons. Algorithmica 66(3), 564–594 (2013)

    Article  MathSciNet  Google Scholar 

  8. Tomás, A.P.: Guarding thin orthogonal polygons is hard. In: Proceedings of Fundamentals of Computation Theory (FCT 2013), Vol. 8070 of LNCS, pp. 305–316 (2013)

  9. Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximability results for guarding polygons and terrains. Algorithmica 31(1), 79–113 (2001)

    Article  MathSciNet  Google Scholar 

  10. Ghosh, S.K.: Approximation algorithms for art gallery problems in polygons. Discret. Appl. Math. 158(6), 718–722 (2010)

    Article  MathSciNet  Google Scholar 

  11. Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Int. J. Comput. Geom. Appl. 21(2), 241–250 (2011)

    Article  MathSciNet  Google Scholar 

  12. Durocher, S., Mehrabi, S.: Guarding orthogonal art galleries using sliding cameras: algorithmic and hardness results. In: Proceedings of Mathematical Foundations of Computer Science (MFCS 2013), Vol. 8087 of LNCS, pp. 314–324 (2013)

  13. Durocher, S., Filtser, O., Fraser, R., Mehrabi, A.D., Mehrabi, S.: Guarding orthogonal art galleries with sliding cameras. Comput. Geom. 65, 12–26 (2017)

    Article  MathSciNet  Google Scholar 

  14. Mehrabi, S.: Geometric optimization problems on orthogonal polygons: hardness results and approximation algorithms, Ph.D. thesis, University of Manitoba, Winnipeg, Canada (August 2015)

  15. Franzblau, D.S., Kleitman, D.J.: An algorithm for constructing regions with rectangles: independence and minimum generating sets for collections of intervals. In: Proceedings of the ACM Symposium on Theory of Computing (STOC 1984), pp. 167–174 (1984)

  16. Motwani, R., Raghunathan, A., Saran, H.: Perfect graphs and orthogonally convex covers. SIAM J. Discret. Math. 2(3), 371–392 (1989)

    Article  MathSciNet  Google Scholar 

  17. King, J., Krohn, E.: Terrain guarding is np-hard. SIAM J. Comput. 40(5), 1316–1339 (2011)

    Article  MathSciNet  Google Scholar 

  18. Daescu, O., Friedrichs, S., Malik, H., Polishchuk, V., Schmidt, C.: Altitude terrain guarding and guarding uni-monotone polygons. Comput. Geom. 84, 22–35 (2019)

    Article  MathSciNet  Google Scholar 

  19. Durocher, S., Li, P.C., Mehrabi, S.: Guarding orthogonal terrains. In: Proceedings of the 27th Canadian Conference on Computational Geometry, CCCG 2015, Kingston, Ontario, Canada, August 10–12, 2015, Queen’s University, Ontario, Canada (2015)

  20. Worman, C., Keil, J.M.: Polygon decomposition and the orthogonal art gallery problem. Int. J. Comput. Geom. Appl. 17(2), 105–138 (2007)

    Article  MathSciNet  Google Scholar 

  21. Biedl, T.C., Irfan, M.T., Iwerks, J., Kim, J., Mitchell, J.S.B.: The art gallery theorem for polyominoes. Discret. Comput. Geom. 48(3), 711–720 (2012)

    Article  MathSciNet  Google Scholar 

  22. Arkin, E.M., Fekete, S.P., Islam, K., Meijer, H., Mitchell, J.S.B., Rodríguez, Y.N., Polishchuk, V., Rappaport, D., Xiao, H.: Not being (super)thin or solid is hard: a study of grid hamiltonicity. Comput. Geom. 42(6–7), 582–605 (2009)

    Article  MathSciNet  Google Scholar 

  23. Keil, J.M.: Minimally covering a horizontally convex orthogonal polygon. In: Proceedings of the ACM Symposium on Computational Geometry (SoCG 1986), pp. 43–51 (1986)

  24. Lingas, A., Wasylewicz, A., Zylinski, P.: Linear-time 3-approximation algorithm for the r-star covering problem. Int. J. Comput. Geom. Appl. 22(2), 103–142 (2012)

    Article  MathSciNet  Google Scholar 

  25. Culberson, J., Reckhow, R.A.: Orthogonally convex coverings of orthogonal polygons without holes. J. Comput. Syst. Sci. 39(2), 166–204 (1989)

    Article  MathSciNet  Google Scholar 

  26. Palios, L., Tzimas, P.: Minimum r-star cover of class-3 orthogonal polygons. In: Proceedings of the International Workshop on Combinatorial Algorithms (IWOCA 2014), Vol. 8986 of LNCS, Springer, pp. 286–297 (2015)

  27. Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. In: ACM Symposium on Computational Geometry (SoCG 1988), pp. 211–223 (1988)

  28. Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. J. Comput. Syst. Sci. 40(1), 19–48 (1990)

    Article  MathSciNet  Google Scholar 

  29. Biedl, T., Mehrabi, S.: Grid obstacle representations with connections to staircase guarding. In: Graph Drawing and Network Visualization (GD’17), LNCS, Springer-Verlag, (2017), To appear

  30. Gewali, L., Ntafos, S.C.: Covering grids and orthogonal polygons with periscope guards. Comput. Geom. 2, 309–334 (1992)

    Article  MathSciNet  Google Scholar 

  31. Shermer, T.: Covering and guarding polygons using \(L_k\)-sets. Geom. Dedicata. 37, 183–203 (1991)

    Article  MathSciNet  Google Scholar 

  32. Biedl, T., Chan, T.M., Lee, S., Mehrabi, S., Montecchiani, F., Vosoughpour, H.: On guarding orthogonal polygons with sliding cameras. In: International Conference and Workshops on Algorithms and Computation (WALCOM 2017), Vol. 10167 of LNCS, Springer, pp. 54–65 (2017)

  33. Bandyapadhyay, S., Roy, A.B.: Effectiveness of local search for art gallery problems. In: proceedings of the 15th International Symposium on Algorithms and Data Structures (WADS 2017), St. John’s, NL, Canada, pp. 49–60 (2017)

  34. Slater, P.J.: R-domination in graphs. J. ACM 23(3), 446–450 (1976)

    Article  Google Scholar 

  35. Courcelle, B.: The monadic second-order logic of graphs I: recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

    Article  MathSciNet  Google Scholar 

  36. Courcelle, B.: On the expression of graph properties in some fragments of monadic second-order logic. In: DIAMCS Workshop on Descriptive Complexity and Finite Models, American Mathematical Society, pp. 33–57 (1997)

  37. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

    Article  MathSciNet  Google Scholar 

  38. Chazelle, B.: Triangulating a simple polygon in linear time. Discret. Comput. Geom. 6(5), 485–524 (1991)

    Article  MathSciNet  Google Scholar 

  39. Kammer, F., Tholey, T.: Approximate tree decompositions of planar graphs in linear time. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA 2012), pp. 683–698 (2012)

  40. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991)

    Article  MathSciNet  Google Scholar 

  41. Garey, M.R., Johnson, D.S.: The rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 32, 826–834 (1977)

    Article  MathSciNet  Google Scholar 

  42. Poljak, S.: A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae 15(2), 307–309 (1974)

    MathSciNet  MATH  Google Scholar 

  43. Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996)

    Article  MathSciNet  Google Scholar 

  44. Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)

    Article  MathSciNet  Google Scholar 

  45. Kovaleva, S., Spieksma, F.C.R.: Primal-dual approximation algorithms for a packing-covering pair of problems. RAIRO - Oper. Res. 36(1), 53–71 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Funding

Funding was provided by Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations

  1. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada

    Therese Biedl

  2. School of Computer Science, Carleton University, Ottawa, Canada

    Saeed Mehrabi

Authors
  1. Therese Biedl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saeed Mehrabi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saeed Mehrabi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preliminary versions of the results of this paper appeared in at the 27th International Symposium on Algorithms and Computation (ISAAC 2016) [1] and at the 28th Canadian Conference on Computational Geometry (CCCG 2017) [2]. This work is supported in part by NSERC, and was done while the second author was at the University of Waterloo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biedl, T., Mehrabi, S. On Orthogonally Guarding Orthogonal Polygons with Bounded Treewidth. Algorithmica 83, 641–666 (2021). https://doi.org/10.1007/s00453-020-00769-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-020-00769-5

Keywords

Search

Navigation