Abstract
We address the problem of packing of a set of n weighted rectangles into a single rectangle so that the total weight of the packed rectangles is maximized. We consider the case of large resources, that is, the single rectangle is \({\it \Omega}(1/\varepsilon^{3})\) times larger than any rectangle to be packed, for small ε> 0. We present an algorithm which finds a packing of a subset of rectangles with the total weight at least (1 − ε) times the optimum. The running time of the algorithm is polynomial in n and 1/ε. As an application we present a (2 + ε)-approximation algorithm for a special case of the advertisement placement problem.
Supported by EU-Project CRESCCO IST-2001-33135, and by EU-Project AEOLUS 015964.
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
Adler, M., Gibbons, P., Matias, Y.: Scheduling space-sharing for internet advertising. Journal of Scheduling (1998)
Baker, B.S., Brown, D.J., Katseff, H.P.: A 5/4 algorithm for two dimensional packing. Journal of Algorithms 2, 348–368 (1981)
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 Journal on Algebraic and Discrete Methods 4, 383–397 (1983)
Baker, B.S., Coffman, E.G., Rivest, R.L.: Orthogonal packings in two dimensions. SIAM Journal on Computing 9, 846–855 (1980)
Bansal, N., Sviridenko, M.: New approximability and inapproximability results for 2-dimensional bin packing. In: Proceedings 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 189–196 (2004)
Caprara, A.: Packing 2-dimensional bins in harmony. In: Proceedings 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 490–499 (2002)
Chung, F.R.K., Garey, M.R., Johnson, D.S.: On packing two-dimensional bins. SIAM Journal on Algebraic and Discrete Methods 3, 66–76 (1982)
Coffman Jr., E.G., Garey, M.R., Johnson, D.S., Tarjan, R.E.: Performance bounds for level-oriented two-dimensional packing algorithms. SIAM Journal on Computing 9, 808–826 (1980)
Correa, J.R., Kenyon, C.: Approximation schemes for multidimensional packing. In: Proceedings 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 179–188 (2004)
Ferreira, C.E., Miyazawa, F.K., Wakabayashi, Y.: Packing squares into squares. Perquisa Operacional 19, 349–355 (1999)
Freund, A., Naor, J.: Approximating the advertisement placement problem. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 415–424. Springer, Heidelberg (2002)
Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. Freeman, San Francisco (1979)
Gilmore, P.C., Gomory, R.E.: Multistage cutting stock problems of two and more dimensions. Operations Research 13, 94–120 (1965)
Golan, I.: Performance bounds for orthogonal, oriented two-dimensional packing algorithms. SIAM Journal on Computing 10, 571–582 (1981)
Grigoriadis, M.D., Khachiyan, L.G., Porkolab, L., Villavicencio, J.: Approximate max-min resource sharing for structured concave optimization. SIAM Journal on Optimization 41, 1081–1091 (2001)
Jansen, K.: Approximation algorithms for the general max-min resource sharing problem: faster and simpler. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 311–322. Springer, Heidelberg (2004)
Jansen, K., Zhang, G.: Maximizing the number of packed rectangles. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 362–371. Springer, Heidelberg (2004)
Jansen, K., Zhang, G.: On rectangle packing: maximizing benefits. In: Proceedings 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 197–206 (2004)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack problems. Springer, Heidelberg (2004)
Kenyon, C., Rémila, E.: Approximate strip-packing. In: Proceedings 37th Annual Symposium on Foundations of Computer Science (FOCS), pp. 31–36 (1996)
Lawler, E.: Fast approximation algorithms for knapsack problems. Mathematics of Operations Research 4, 339–356 (1979)
Leung, J.Y.-T., Tam, T.W., Wong, C.S., Young, G.H., Chin, F.Y.L.: Packing squares into a square. Journal of Parallel and Distributed Computing 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)
Seiden, S., van Stee, R.: New bounds for multi-dimentional packing. Algorithmica 36(3), 261–293 (2003)
Sleator, D.D.: A 2.5 times optimal algorithm for bin packing in two dimensions. Informatin Processing Letters (10), 37–40 (1980)
Steinberg, A.: A strip-packing algorithm with absolute performance bound 2. SIAM Journal on Computing 26(2), 401–409 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fishkin, A.V., Gerber, O., Jansen, K. (2005). On Efficient Weighted Rectangle Packing with Large Resources. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_103
Download citation
DOI: https://doi.org/10.1007/11602613_103
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30935-2
Online ISBN: 978-3-540-32426-3
eBook Packages: Computer ScienceComputer Science (R0)