Abstract
The general Art Gallery Problem (AGP) consists in finding the minimum number of guards sufficient to ensure the visibility coverage of an art gallery represented by a polygon. The AGP is a well known \(\mathbb{NP}\)-hard problem and, for this reason, all algorithms proposed so far to solve it are unable to guarantee optimality except in special cases. In this paper, we present a new method for solving the Art Gallery Problem by iteratively generating upper and lower bounds while seeking to reach an exact solution. Notwithstanding that convergence remains an important open question, our algorithm has been successfully tested on a very large collection of instances from publicly available benchmarks. Tests were carried out for several classes of instances totalizing more than a thousand hole-free polygons with sizes ranging from 20 to 1000 vertices. The proposed algorithm showed a remarkable performance, obtaining provably optimal solutions for every instance in a matter of minutes on a standard desktop computer. To our knowledge, despite the AGP having been studied for four decades within the field of computational geometry, this is the first time that an exact algorithm is proposed and extensively tested for this problem. Future research directions to expand the present work are also discussed.
This research was supported by FAPESP – Fundação de Amparo à Pesquisa do Estado de São Paulo, CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico and Faepex/Unicamp.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Amit, Y., Mitchell, J.S.B., Packer, E.: Locating guards for visibility coverage of polygons. In: ALENEX, New Orleans, Lousiana (January 2007)
Avis, D., Toussaint, G.T.: An efficient algorithm for decomposing a polygon into star-shaped polygons. Pattern Recognition 13(6), 395–398 (1981)
Bottino, A., Laurentini, A.: A nearly optimal sensor placement algorithm for boundary coverage. Pattern Recognition 41(11), 3343–3355 (2008)
Bottino, A., Laurentini, A.: A nearly optimal algorithm for covering the interior of an art gallery. Pattern Recognition 44(5), 1048–1056 (2011)
CGAL. Computational Geometry Algorithms Library, www.cgal.org (last access January 2012)
Chvátal, V.: A combinatorial theorem in plane geometry. Journ. of Combin. Theory Series B 18, 39–41 (1975)
Chwa, K.-Y., Jo, B.-C., Knauer, C., Moet, E., van Oostrum, R., Shin, C.-S.: Guarding art galleries by guarding witnesses. Intern. Journal of Computational Geometry and Applications 16(02n03), 205–226 (2006)
Couto, M.C., de Rezende, P.J., de Souza, C.C.: An exact algorithm for minimizing vertex guards on art galleries. International Transactions in Operational Research 18(4), 425–448 (2011)
Couto, M.C., de Souza, C.C., de Rezende, P.J.: An exact and efficient algorithm for the orthogonal art gallery problem. In: Proc. of the XX Brazilian Symp. on Comp. Graphics and Image Processing, pp. 87–94. IEEE Computer Society (2007)
Eidenbenz, S.: Approximation algorithms for terrain guarding. Inf. Process. Lett. 82(2), 99–105 (2002)
Ghosh, S.K.: Approximation algorithms for art gallery problems. In: Proc. Canadian Inform. Process. Soc. Congress (1987)
Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, New York (2007)
Ghosh, S.K.: Approximation algorithms for art gallery problems in polygons. Discrete Applied Mathematics 158(6), 718–722 (2010)
Honsberger, R.: Mathematical Gems II. The Dolciani Mathematical Expositions, vol. 2. MAA (1976)
Kröller, A., Baumgartner, T., Fekete, S.P., Moeini, M., Schmidt, C.: Practical solutions and bounds for art gallery problems (August 2012), http://ismp2012.mathopt.org/show-abs?abs=1046
Kröller, A., Baumgartner, T., Fekete, S.P., Schmidt, C.: Exact solutions and bounds for general art gallery problems. J. Exp. Algorithmics 17(1), 2.3:2.1–2.3:2.23 (2012)
Lee, D.T., Lin, A.: Computational complexity of art gallery problems. IEEE Transactions on Information Theory 32(2), 276–282 (1986)
O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, New York (1987)
Shermer, T.: Recent results in art galleries. Proceedings of the IEEE 80(9), 1384–1399 (1992)
Urrutia, J.: Art gallery and illumination problems. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 973–1027. North-Holland (2000)
XPRESS. Xpress Optimization Suite (2009), http://www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-Suite.aspx (access January 2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tozoni, D.C., de Rezende, P.J., de Souza, C.C. (2013). The Quest for Optimal Solutions for the Art Gallery Problem: A Practical Iterative Algorithm. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds) Experimental Algorithms. SEA 2013. Lecture Notes in Computer Science, vol 7933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38527-8_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-38527-8_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38526-1
Online ISBN: 978-3-642-38527-8
eBook Packages: Computer ScienceComputer Science (R0)