Document

# On r-Guarding Thin Orthogonal Polygons

## File

LIPIcs.ISAAC.2016.17.pdf
• Filesize: 0.52 MB
• 13 pages

## Cite As

Therese Biedl and Saeed Mehrabi. On r-Guarding Thin Orthogonal Polygons. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ISAAC.2016.17

## Abstract

Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point p to guard a point q if and only if the minimum axis-aligned rectangle spanned by p and q is inside the polygon. A simple proof shows that this problem is NP-hard on orthogonal polygons with holes, even if the polygon is thin. If there are no holes, then a thin polygon becomes a tree polygon in the sense that the so-called dual graph of the polygon is a tree. It was known that finding the minimum set of r-guards is polynomial for tree polygons (and in fact for all orthogonal polygons), but the run-time was ~O(n^17). We show here that with a different approach one can find the minimum set of r-guards can be found in tree polygons in linear time, answering a question posed by Biedl et al. (SoCG 2011). Furthermore, the approach is much more general, allowing to specify subsets of points to guard and guards to use, and it generalizes to polygons with h holes or thickness K, becoming fixed-parameter tractable in h + K.
##### Keywords
• Art Gallery Problem
• Orthogonal Polygons
• r-Guarding
• Treewidth
• Fixed-parameter Tractable

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. T. Biedl, M. T. Irfan, J. Iwerks, J. Kim, and J. S. B. Mitchell. Guarding polyominoes. In Proc. of the ACM Symp. on Computational Geometry (SoCG'11), pages 387-396, 2011.
2. T. C. Biedl and S. Mehrabi. On r-guarding thin orthogonal polygons. CoRR, abs/1604.07100, 2016. URL: http://arxiv.org/abs/1604.07100.
3. B. Chazelle. Triangulating a simple polygon in linear time. Disc. Comp. Geom., 6(5):485-524, 1991.
4. V. Chvátal. A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B, 18:39-41, 1975. URL: http://dx.doi.org/10.1137/S0097539796302531.
5. B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990.
6. J. Culberson and R. A. Reckhow. Orthogonally convex coverings of orthogonal polygons without holes. Journal of Computer and System Sciences, 39(2):166-204, 1989.
7. M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Mark, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, Heidelberg, Germany, 2015.
8. S. Durocher and S. Mehrabi. Guarding orthogonal art galleries using sliding cameras: algorithmic and hardness results. In Proceedings of Mathematical Foundations of Computer Science (MFCS 2013), volume 8087 of LNCS, pages 314-324, 2013.
9. S. Eidenbenz, C. Stamm, and P. Widmayer. Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1):79-113, 2001. URL: http://dx.doi.org/10.1007/s00453-001-0040-8.
10. D. S. Franzblau and D. J. Kleitman. 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), pages 167-174, 1984.
11. M. R. Garey and D. S. Johnson. The Rectilinear Steiner Tree Problem is NP-complete. SIAM Journal of Applied Mathematics, 32:826-834, 1977.
12. S. K. Ghosh. Approximation algorithms for art gallery problems in polygons. Disc. App. Math., 158(6):718-722, 2010.
13. F. Kammer and T. Tholey. Approximate tree decompositions of planar graphs in linear time. In Proc. of the ACM-SIAM Symp. on Discrete Algorithms (SODA 2012), pages 683-698, 2012.
14. G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16:4-32, 1996.
15. M. J. Katz and G. Morgenstern. Guarding orthogonal art galleries with sliding cameras. Int. J. Comput. Geometry Appl., 21(2):241-250, 2011.
16. J. M. Keil. Minimally covering a horizontally convex orthogonal polygon. In Proceedings of the ACM Symposium on Computational Geometry (SoCG 1986), pages 43-51, 1986.
17. E. Krohn and B. J. Nilsson. Approximate guarding of monotone and rectilinear polygons. Algorithmica, 66(3):564-594, 2013.
18. D. T. Lee and Arthur K. Lin. Computational complexity of art gallery problems. IEEE Trans. on Information Theory, 32(2):276-282, 1986. URL: http://dx.doi.org/10.1109/TIT.1986.1057165.
19. A. Lingas, A. Wasylewicz, and P. Zylinski. Linear-time 3-approximation algorithm for the r-star covering problem. Int. J. Comput. Geometry Appl., 22(2):103-142, 2012.
20. R. Motwani, A. Raghunathan, and H. Saran. Perfect graphs and orthogonally convex covers. SIAM J. Discrete Math., 2(3):371-392, 1989.
21. J. O'Rourke. Art Gallery Theorems and Algorithms. The International Series of Monographs on Computer Science. Oxford University Press, New York, NY, 1987.
22. L. Palios and P. Tzimas. Minimum r-star cover of class-3 orthogonal polygons. In Proceedings of the International Workshop on Combinatorial Algorithms (IWOCA 2014), volume 8986 of LNCS, pages 286-297. Springer, 2015.
23. S. Poljak. A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae, 15(2):307-309, 1974.
24. D. Schuchardt and H.-D. Hecker. Two NP-hard art-gallery problems for ortho-polygons. Mathematical Logic Quarterly, 41(2):261-267, 1995.
25. P. J. Slater. R-domination in graphs. J. ACM, 23(3):446-450, 1976.
26. A. P. Tomás. Guarding thin orthogonal polygons is hard. In Proceedings of Fundamentals of Computation Theory (FCT 2013), volume 8070 of LNCS, pages 305-316, 2013.
27. C. Worman and J. M. Keil. Polygon decomposition and the orthogonal art gallery problem. Int. J. Comput. Geometry Appl., 17(2):105-138, 2007. URL: http://dx.doi.org/10.1142/S0218195907002264.