Abstract
The three art gallery problems Vertex Guard, Edge Guard and Point Guard are known to be NP-hard 8. Approximation algorithms for Vertex Guard and Edge Guard with a logarithmic ratio were proposed in 7. We prove that for each of these problems, there exists a constant ∈ > 0, such that no polynomial time algorithm can guarantee an approximation ratio of 1 + ∈ unless P = NP. We obtain our results by proposing gap-preserving reductions, based on reductions from 8. Our results are the first inapproximability results for these problems.
This work is partially supported by the Swiss National Science Foundation
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References
P. Bose, T. Shermer, G. Toussaint and B. Zhu; Guarding Polyhedral Terrains; Computational Geometry 7, pp. 173–185, Elsevier Science B. V., 1997.
S. Arora, C. Lund; Hardness of Approximations; in: Approximation Algorithms for NP-Hard Problems (ed. Dorit Hochbaum), PWS Publishing Company, pp. 399–446, 1996
D. P. Bovet and P. Crescenzi; Introduction to the Theory of Complexity; Prentice Hall, 1993
J. C. Culberson and R. A. Reckhow; Covering Polygons is Hard; Proc. 29th Symposium on Foundations of Computer Science, 1988
St. Eidenbenz, Ch. Stamm, P. Widmayer; Positioning Guards at Fixed Height above a Terrain-an Optimum Inapproximability Result; to appear in Proceedings ESA, 1998
St. Eidenbenz, Ch. Stamm, P. Widmayer; Inapproximability of some Art Gallery Problems; to appear in Proceedings CCCG, 1998
S. Ghosh; Approximation Algorithms for Art Gallery Problems; Proc. of the Canadian Information Processing Society Congress, 1987
D. T. Lee and A. K. Lin; Computational Complexity of Art Gallery Problems; in IEEE Trans. Info. Th, pp. 276–282, IT-32 1986
B. Nilsson; Guarding Art Galleries-Methods for Mobile Guards; doctoral thesis, Department of Computer Science, Lund University, 1994
J. O’Rourke and K. J. Supowit; Some NP-hard Polygon Decomposition Problems; IEEE Transactions on Information Theory, Vol IT-29, No. 2, 1983
J. O’Rourke; Art Gallery Theorems and Algorithms; Oxford University Press, New York 1987
C.H. Papadimitriou and M. Yannakakis; Optimization, Approximation, and Complexity Classes; Proc. 20th ACM Symposium on the Theory of Computing, 1988.
T. Shermer; Recent Results in Art Galleries; Proc. of the IEEE, 1992
J. Urrutia; Art Gallery and Illumination Problems; to appear in Handbook on Computational Geometry, edited by J.-R. Sack and J. Urrutia, 1998
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© 1998 Springer-Verlag Berlin Heidelberg
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Eidenbenz, S. (1998). Inapproximability Results for Guarding Polygons without Holes. In: Chwa, KY., Ibarra, O.H. (eds) Algorithms and Computation. ISAAC 1998. Lecture Notes in Computer Science, vol 1533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49381-6_45
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DOI: https://doi.org/10.1007/3-540-49381-6_45
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