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Abstract

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme (\(\mathcal{PTAS}\)) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided \(\mathcal{P} \not= \mathcal{NP}\).

Keywords

two-dimensional bin packing strip packing rectangle packing approximation algorithm absolute worst-case ratio 

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References

  1. 1.
    Bansal, N.: Personal communication (2008)Google Scholar
  2. 2.
    Bansal, N., Caprara, A., Sviridenko, M.: Improved approximation algorithms for multidimensional bin packing problems. In: FOCS: Proc. 47th IEEE Symposium on Foundations of Computer Science, pp. 697–708 (2006)Google Scholar
  3. 3.
    Bansal, N., Caprara, A., Sviridenko, M.: A structural lemma in 2-dimensional packing, and its implications on approximability, IBM Research Division, RC24468, W0801-070 (2008), http://domino.research.ibm.com/library/cyberdig.nsf/index.html
  4. 4.
    Bansal, N., Correa, J.R., Kenyon, C., Sviridenko, M.: Bin packing in multiple dimensions - inapproximability results and approximation schemes. Mathematics of Operations Research 31(1), 31–49 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Caprara, A.: Packing d-dimensional bins in d stages. Mathematics of Operations Research 33(1), 203–215 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Caprara, A., Lodi, A., Monaci, M.: Fast approximation schemes for two-stage, two-dimensional bin packing. Mathematics of Operations Research 30(1), 150–172 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chlebík, M., Chlebíková, J.: Inapproximability results for orthogonal rectangle packing problems with rotations. In: CIAC: Proc. 6th Conference on Algorithms and Complexity, pp. 199–210 (2006)Google Scholar
  8. 8.
    Harren, R., van Stee, R.: Absolute approximation ratios for packing rectangles into bins. Journal of Scheduling (to appear, 2009)Google Scholar
  9. 9.
    Jansen, K., Solis-Oba, R.: New approximability results for 2-dimensional packing problems. In: MFCS: Proc. 32nd International Symposium on Mathematical Foundations of Computer Science, pp. 103–114 (2007)Google Scholar
  10. 10.
    Jansen, K., van Stee, R.: On strip packing with rotations. In: STOC: Proc. 37th ACM Symposium on Theory of Computing, pp. 755–761 (2005)Google Scholar
  11. 11.
    Jansen, K., Zhang, G.: Maximizing the total profit of rectangles packed into a rectangle. Algorithmica 47(3), 323–342 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kenyon, C., Rémila, E.: A near optimal solution to a two-dimensional cutting stock problem. Mathematics of Operations Research 25(4), 645–656 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Leung, J.Y.-T., Tam, T.W., Wong, C.S., Young, G.H., Chin, F.Y.: Packing squares into a square. Journal of Parallel and Distributed Computing 10(3), 271–275 (1990)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    Steinberg, A.: A strip-packing algorithm with absolute performance bound 2. SIAM Journal on Computing 26(2), 401–409 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    van Stee, R.: An approximation algorithm for square packing. Operations Research Letters 32(6), 535–539 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zhang, G.: A 3-approximation algorithm for two-dimensional bin packing. Operations Research Letters 33(2), 121–126 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Rolf Harren
    • 1
  • Rob van Stee
    • 1
  1. 1.Max-Planck-Institut für Informatik (MPII)SaarbrückenGermany

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