Abstract
Given a set of rectangles we are asked to pack as many of them as possible into a bigger rectangle. The rectangles packed may not overlap and may not be rotated. This problem is NP-hard in the strong sense even for packing squares into a square. We establish the relationship between the asymptotic worst-case ratio and the (absolute) worst-case ratio for the problem. It is proved that there exists an asymptotic FPTAS, and thus a PTAS, for packing squares into a rectangle. We give an approximation algorithm with asymptotic ratio of at most two for packing rectangles, and further show a simple (2+ε)-approximation algorithm.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baker, B.S., Calderbank, A.R., Coffman, E.G., Lagarias, J.C.: Approximation algorithms for maximizing the number of squares packed into a rectangle. SIAM J. on Algebraic and Discrete Methods 4, 383–397 (1983)
Bansal, N., Sviridenko, M.: New approximability and inapproximability results for 2-dimensional bin packing. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 189–196 (2004)
Caprara, A., Monaci, M.: On the two-dimensional knapsack problem. Operations Research Letters 32, 5–14 (2004)
Correa, J.R., Kenyon, C.: Approximation schemes for multidimensional packing. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 179–188 (2004)
Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. Journal of ACM 32, 130–136 (1985)
Jansen, K., Zhang, G.: On rectangle packing: maximizing benefits. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 197–206 (2004)
Kenyon, C., Remila, E.: A near-optimal solution to a two-dimensional cutting stock problem. Mathematics of Operations Research 25, 645–656 (2000)
Leung, J.Y.-T., Tam, T.W., Wong, C.S., Young, G.H., Chin, F.Y.L.: Packing squares into a square. J. Parallel Distrib. Comput. 10, 271–275 (1990)
Schiermeyer, I.: Reverse-fit: a 2-optimal algorithm for packing rectangles. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 290–299. Springer, Heidelberg (1994)
Steinberg, A.: A strip-packing algorithm with absolute performance bound 2. SIAM J. Computing 26, 401–409 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jansen, K., Zhang, G. (2004). Maximizing the Number of Packed Rectangles. In: Hagerup, T., Katajainen, J. (eds) Algorithm Theory - SWAT 2004. SWAT 2004. Lecture Notes in Computer Science, vol 3111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27810-8_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-27810-8_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22339-9
Online ISBN: 978-3-540-27810-8
eBook Packages: Springer Book Archive