Abstract
The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NP-hard, even for polygons without holes [3]. We propose a polynomial-time approximation algorithm for this problem for polygons with or without holes that achieves an approximation ratio of O(log n), where n is the number of vertices in the input polygon. To obtain this result, we first show that an optimum solution of a restricted version of this problem, where the vertices of the convex polygons may only lie on a certain grid, contains at most three times as many convex polygons as the optimum solution of the unrestricted problem. As a second step, we use dynamic programming to obtain a convex polygon which is maximum with respect to the number of “basic triangles” that are not yet covered by another convex polygon. We obtain a solution that is at most a logarithmic factor off the optimum by iteratively applying our dynamic programming algorithm. Furthermore, we show that Minimum Convex Cover is APX-hard, i.e., there exists a constant δ > 0 such that no polynomial-time algorithm can achieve an approximation ratio of 1 + δ. We obtain this result by analyzing and slightly modifying an already existing reduction [3].
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
S. Arora and C. Lund; Hardness of Approximations; in: Approximation Algorithms for NP-Hard Problems (ed. Dorit Hochbaum), PWS Publishing Company, pp. 399–446, 1996.
B. Chazelle, D.P. Dobkin; Optimal Convex Decompositions; Computational Geometry, pp. 63–133, Elsevier Science B. V., 1985.
J. C. Culberson and R. A. Reckhow; Covering Polygons is Hard; Journal of Algorithms, Vol. 17, No. 1, pp. 2–44, July 1994.
D. P. Dobkin, H. Edelsbrunner, and M. H. Overmars; Searching for Empty Convex Polygons; Algorithmica, 5, pp. 561–571, 1990.
S. Eidenbenz, C. Stamm, and P. Widmayer; Inapproximability of some Art Gallery Problems; Proc. 10th Canadian Conf. Computational Geometry, pp. 64–65, 1998.
S. Eidenbenz; Inapproximability Results for Guarding Polygons without Holes; Lecture Notes in Computer Science, Vol. 1533 (ISAAC’98), p. 427–436, 1998.
D.S. Franzblau; Performance Guarantees on a Sweep Line Heuristic for Covering Rectilinear Polygons with Rectangles; SIAM J. Discrete Math., Vol 2,3, pp. 307–321, 1989.
S. Ghosh; Approximation Algorithms for Art Gallery Problems; Proc. of the Canadian Information Processing Society Congress, 1987.
J. Gudmundsson, C. Levcopoulos; Close Approximations of Minimum Rectangular Coverings; J. Comb. Optimization, Vol. 4, No. 4, pp. 437–452, 1999.
D. Hochbaum; Approximating Covering and Packing Problems: Set Cover, Vertex Cover, Independent Set, and Related Problems; in: Approximation Algorithms for NP-Hard Problems (ed. Dorit Hochbaum), PWS Publishing Company, pp. 94–143, 1996.
V.S.A. Kumar, H. Ramesh; Covering Rectilinear Polygons with Axis-parallel Rectangles; Proc. STOC’99, pp. 445–454, 1999.
A. Lingas; The Power of Non-Rectilinear Holes; Proc. 9th Colloquium on Automata, Languages, and Programming, pp. 369–383, 1982.
A. Lingas, V. Soltan; Minimum Convex Partition of a Polygon with Holes by Cuts in Given Directions; Theory of Comuting Systems, Vol. 31, pp. 507–538, 1998.
J. O’Rourke and K. J. Supowit; Some NP-hard Polygon Decomposition Problems; IEEE Transactions on Information Theory, Vol IT-29, No. 2, 1983.
T. Shermer; Recent results in Art Galleries; Proc. of the IEEE, 1992.
J. Urrutia; Art Gallery and Illumination Problems; in Handbook on Computational Geometry, ed. J.-R. Sack and J. Urrutia, Elsevier, Chapter 22, pp. 973–1027, 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Eidenbenz, S., Widmayer, P. (2001). An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee. In: auf der Heide, F.M. (eds) Algorithms — ESA 2001. ESA 2001. Lecture Notes in Computer Science, vol 2161. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44676-1_28
Download citation
DOI: https://doi.org/10.1007/3-540-44676-1_28
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42493-2
Online ISBN: 978-3-540-44676-7
eBook Packages: Springer Book Archive