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On Efficient Weighted Rectangle Packing with Large Resources

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Algorithms and Computation (ISAAC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3827))

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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.

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References

  1. Adler, M., Gibbons, P., Matias, Y.: Scheduling space-sharing for internet advertising. Journal of Scheduling (1998)

    Google Scholar 

  2. Baker, B.S., Brown, D.J., Katseff, H.P.: A 5/4 algorithm for two dimensional packing. Journal of Algorithms 2, 348–368 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  3. 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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baker, B.S., Coffman, E.G., Rivest, R.L.: Orthogonal packings in two dimensions. SIAM Journal on Computing 9, 846–855 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  5. 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)

    Google Scholar 

  6. Caprara, A.: Packing 2-dimensional bins in harmony. In: Proceedings 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 490–499 (2002)

    Google Scholar 

  7. 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)

    Article  MATH  MathSciNet  Google Scholar 

  8. 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)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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)

    Google Scholar 

  10. Ferreira, C.E., Miyazawa, F.K., Wakabayashi, Y.: Packing squares into squares. Perquisa Operacional 19, 349–355 (1999)

    Google Scholar 

  11. 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)

    Chapter  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  13. Gilmore, P.C., Gomory, R.E.: Multistage cutting stock problems of two and more dimensions. Operations Research 13, 94–120 (1965)

    Article  MATH  Google Scholar 

  14. Golan, I.: Performance bounds for orthogonal, oriented two-dimensional packing algorithms. SIAM Journal on Computing 10, 571–582 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  15. 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)

    Article  MathSciNet  Google Scholar 

  16. 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)

    Chapter  Google Scholar 

  17. 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)

    Chapter  Google Scholar 

  18. Jansen, K., Zhang, G.: On rectangle packing: maximizing benefits. In: Proceedings 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 197–206 (2004)

    Google Scholar 

  19. Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack problems. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  20. Kenyon, C., Rémila, E.: Approximate strip-packing. In: Proceedings 37th Annual Symposium on Foundations of Computer Science (FOCS), pp. 31–36 (1996)

    Google Scholar 

  21. Lawler, E.: Fast approximation algorithms for knapsack problems. Mathematics of Operations Research 4, 339–356 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  22. 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)

    Article  MathSciNet  Google Scholar 

  23. 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)

    Chapter  Google Scholar 

  24. Seiden, S., van Stee, R.: New bounds for multi-dimentional packing. Algorithmica 36(3), 261–293 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sleator, D.D.: A 2.5 times optimal algorithm for bin packing in two dimensions. Informatin Processing Letters (10), 37–40 (1980)

    Google Scholar 

  26. Steinberg, A.: A strip-packing algorithm with absolute performance bound 2. SIAM Journal on Computing 26(2), 401–409 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Liverpool, Chadwick Building, Peach Street, Liverpool, L69 7ZF, UK

    Aleksei V. Fishkin

  2. Institute of Computer Science, University of Kiel, 24118, Kiel, Germany

    Olga Gerber & Klaus Jansen

Authors
  1. Aleksei V. Fishkin
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  2. Olga Gerber
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  3. Klaus Jansen
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Editor information

Editors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon , Hong Kong

    Xiaotie Deng

  2. Department of Computer Science, University of Texas at Dallas, EE/CS Building, 75083, Richardson, TX, USA

    Ding-Zhu Du

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© 2005 Springer-Verlag Berlin Heidelberg

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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

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  • 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)

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