Optimum Inapproximability Results for Finding Minimum Hidden Guard Sets in Polygons and Terrains
- First Online:
We study the problem Minimum Hidden Guard Set, which consists of positioning a minimum number of guards in a given polygon or terrain such that no two guards see each other and such that every point in the polygon or on the terrain is visible from at least one guard. By constructing a gap-preserving reduction from Maximum 5-Ocurrence-3-Satisfiability, we show that this problem cannot be approximated by a polynomial-time algorithm with an approximation ratio of n1−ε for any ε > 0, unless NP = P, where n is the number of polygon or terrain vertices. The result even holds for input polygons without holes. This separates the problem from other visibility problems such as guarding and hiding, where strong inapproximability results only hold for polygons with holes. Furthermore, we show that an approximation algorithm achieves a matching approximation ratio of n.
Unable to display preview. Download preview PDF.
- 1.S. Arora, C. Lund; Hardness of Approximations; in: Approximation Algorithms for NP-Hard Problems (ed. Dorit Hochbaum), PWS Publishing Company, pp. 399–446, 1996.Google Scholar
- 2.P. Crescenzi, V. Kann; A Compendium of NP Optimization Problems; in the book by G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, M. Protasi, Complexity and Approximation. Combinatorial Optimization Problems and their Approximability Properties, Springer-Verlag, Berlin, 1999; also available in an online-version at: http://www.nada.kth.se/theory/compendium/compendium.html.Google Scholar
- 3.S. Eidenbenz, C. Stamm, and P. Widmayer; Inapproximability of some Art Gallery Problems; Proc. 10th Canadian Conf. on Computational Geometry, pp. 64–65, 1998.Google Scholar
- 7.S. Eidenbenz; How Many People Can Hide in a Terrain?; Lecture Notes in Computer Science, Vol. 1741 (ISAAC’99), pp. 184–194, 1999.Google Scholar
- 8.S. Ghosh; Approximation Algorithms for Art Gallery Problems; Proc. of the Canadian Information Processing Society Congress, 1987.Google Scholar
- 9.D. T. Lee and A. K. Lin; Computational Complexity of Art Gallery Problems; IEEE Trans. Info. Th, pp. 276–282, IT-32, 1986.Google Scholar
- 10.M. van Kreveld; Digital Elevation Models and TIN Algorithms; in: Algorithmic Foundations of Geographic Information Systems (ed. van Kreveld et al.), LNCS tutorial vol. 1340, pp. 37–78, Springer, 1997.Google Scholar
- 12.T. Shermer; Recent results in Art Galleries; Proc. of the IEEE, 1992.Google Scholar
- 13.J. Urrutia; Art Gallery and Illumination Problems; in: Handbook on Computational Geometry, J.R. Sack, J. Urrutia (eds.), North Holland, pp. 973–1127, 2000.Google Scholar