Abstract
In the Art Gallery problem, given is a polygonal gallery and the goal is to guard the gallery’s interior or walls with a number of guards that must be placed strategically in the interior, on walls or on corners of the gallery. Here we consider a more realistic version: exhibits now have size and may have different costs. Moreover the meaning of guarding is relaxed: we use a new concept, that of watching an expensive art item, i.e. overseeing a part of the item. The main result of the paper is that the problem of maximizing the total value of a guarded weighted boundary is APX-complete. This is shown by an appropriate gap-preserving reduction from the Max-5-occurrence-3-Sat problem. We also show that this technique can be applied to a number of maximization variations of the art gallery problem. In particular we consider the following problems: given a polygon with or without holes and k available guards, maximize a) the length of walls guarded and b) the total cost of paintings watched or overseen. We prove that all the above problems are APX-complete.
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.
References
Lee, D., Lin, A., Computational complexity of art gallery problems, IEEE Trans. Inform. Theory 32, 276–282, 1986.
O’Rourke, J., Art Gallery Theorems and Algorithms, Oxford Univ. Press, New York, 1987.
Shermer, T., Recent results in Art Galleries, Proc. of the IEEE, 1992.
Urrutia, J., Art gallery and Illumination Problems, Handbook on Comput. Geometry, 1998.
Eidenbenz, S., (In-)Approximability of Visibility Problems on Polygons and Terrains, PhD Thesis, ETH Zurich, 2000.
Eidenbenz, S., Inapproximability Results for Guarding Polygons without Holes, Lecture notes in Computer Science, Vol. 1533 (ISAAC’98), p. 427–436, 1998.
Eidenbenz, S., Stamm, C., Widmayer, P., Inapproximability of some Art Gallery Problems, Proc. 10th Canadian Conf. Computational Geometry (CCCG’98), pp. 64–65, 1998.
Ghosh, S., Approximation algorithms for Art Gallery Problems, Proc. of the Canadian Information Processing Society Congress, pp. 429–434, 1987.
Arora, S., Probabilistic Checking of Proofs and the Hardness of Approximation Problems, PhD thesis, Berkeley, 1994.
Arora, S., Lund, C., Hardness of Approximations; in: Approximation Algorithms for NP-Hard Problems (Dorit Hochbaum ed.), pp. 399–446, PWS Publishing Company, 1996.
Hochbaum, D., Approximation Algorithms for NP-Hard Problems, PWS, 1996.
Markou, E., Fragoudakis, C., Zachos, S., Approximating Visibility Problems within a constant, 3rd Workshop on Approximation and Randomization Algorithms in Communication Networks, Rome pp. 91–103, 2002.
Laurentini A., Guarding the walls of an art gallery, The Visual Computer Journal, (1999) 15:265–278.
Deneen L. L., Joshi S., Treasures in an art gallery, Proc. 4th Canadian Conf. Computational Geometry, pp. 17–22, 1992.
Carlsson S., Jonsson H., Guarding a Treasury, Proc. 5th Canadian Conf. Computational Geometry, pp. 85–90, 1993.
Markou, E., Zachos, S., Fragoudakis, C., Optimizing guarding costs for Art Galleries by choosing placement of art pieces and guards (submitted)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Markou, E., Zachos, S., Fragoudakis, C. (2003). Maximizing the Guarded Boundary of an Art Gallery Is APX-Complete. In: Petreschi, R., Persiano, G., Silvestri, R. (eds) Algorithms and Complexity. CIAC 2003. Lecture Notes in Computer Science, vol 2653. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44849-7_10
Download citation
DOI: https://doi.org/10.1007/3-540-44849-7_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40176-6
Online ISBN: 978-3-540-44849-5
eBook Packages: Springer Book Archive