# Guarding Orthogonal Art Galleries Using Sliding Cameras: Algorithmic and Hardness Results

• Stephane Durocher
• Saeed Mehrabi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)

## Abstract

Let P be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment s ⊆ P as its trajectory. The camera can see a point p ∈ P if there exists a point q ∈ s such that pq is a line segment normal to s that is completely contained in P. In the minimum-cardinality sliding cameras problem, the objective is to find a set S of sliding cameras of minimum cardinality to guard P (i.e., every point in P can be seen by some sliding camera in S) while in the minimum-length sliding cameras problem the goal is to find such a set S so as to minimize the total length of trajectories along which the cameras in S travel.

In this paper, we first settle the complexity of the minimum-length sliding cameras problem by showing that it is polynomial tractable even for orthogonal polygons with holes, answering a question posed by Katz and Morgenstern [9]. Next we show that the minimum-cardinality sliding cameras problem is NP-hard when P is allowed to have holes, which partially answers another question posed by Katz and Morgenstern [9].

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Amit, Y., Mitchell, J.S.B., Packer, E.: Locating guards for visibility coverage of polygons. Int. J. Comput. Geometry Appl. 20(5), 601–630 (2010)
2. 2.
Biedl, T.C., Irfan, M.T., Iwerks, J., Kim, J., Mitchell, J.S.B.: The art gallery theorem for polyominoes. Disc. & Comp. Geom. 48(3), 711–720 (2012)
3. 3.
Chvátal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory, Ser. B 18, 39–41 (1975)
4. 4.
Fekete, S.P., Mitchell, J.S.B.: Terrain decomposition and layered manufacturing. Int. J. of Comp. Geom. & App. 11(6), 647–668 (2001)
5. 5.
Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Disc. App. Math. 30(1), 29–42 (1991)
6. 6.
Hoffmann, F.: On the rectilinear art gallery problem. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 717–728. Springer, Heidelberg (1990)
7. 7.
Kahn, J., Klawe, M.M., Kleitman, D.J.: Traditional galleries require fewer watchmen. SIAM J. on Algebraic Disc. Methods 4(2), 194–206 (1983)
8. 8.
Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
9. 9.
Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Int. J. of Comp. Geom. & App. 21(2), 241–250 (2011)
10. 10.
Katz, M.J., Roisman, G.S.: On guarding the vertices of rectilinear domains. Comput. Geom. 39(3), 219–228 (2008)
11. 11.
König, D.: Gráfok és mátrixok. Matematikai és Fizikai Lapok 38, 116–119 (1931)
12. 12.
Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. on Inf. Theory 32(2), 276–282 (1986)
13. 13.
Lingas, A., Wasylewicz, A., Żyliński, P.: Linear-time 3-approximation algorithm for the r-star covering problem. In: Nakano, S.-i., Rahman, M. S. (eds.) WALCOM 2008. LNCS, vol. 4921, pp. 157–168. Springer, Heidelberg (2008)
14. 14.
Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. In: Proc. ACM SoCG, pp. 211–223 (1988)Google Scholar
15. 15.
O’Rourke, J.: Art gallery theorems and algorithms. Oxford University Press (1987)Google Scholar
16. 16.
Schuchardt, D., Hecker, H.-D.: Two NP-hard art-gallery problems for ortho-polygons. Math. Logic Quarterly 41(2), 261–267 (1995)
17. 17.
Urrutia, J.: Art gallery and illumination problems. In: Handbook of Comp. Geom., pp. 973–1027. North-Holland (2000)Google Scholar
18. 18.
Worman, C., Keil, J.M.: Polygon decomposition and the orthogonal art gallery problem. Int. J. of Comp. Geom. & App. 17(2), 105–138 (2007)

© Springer-Verlag Berlin Heidelberg 2013

## Authors and Affiliations

• Stephane Durocher
• 1
• Saeed Mehrabi
• 1
1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada