Skip to main content
Springer Nature Link
Log in

Two for One: Tight Approximation of 2D Bin Packing

  • Conference paper
Algorithms and Data Structures (WADS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5664))

Included in the following conference series:

  • 1507 Accesses

Abstract

In this paper, we study the two-dimensional geometrical bin packing problem (2DBP): given a list of rectangles, provide a packing of all these into the smallest possible number of 1×1 bins without rotating the rectangles.

We present a 2-approximate algorithm, which improves over the previous best known ratio of 3, matches the best results for the rotational case and also matches the known lower bound of approximability. Our approach makes strong use of a recently-discovered PTAS for a related knapsack problem and a new algorithm that can pack instances into OPT + 2 bins for any constant OPT.

Work supported by EU project “AEOLUS: Algorithmic Principles for Building Efficient Overlay Computers”, EU contract number 015964, and DFG project JA612/12-1, “Design and analysis of approximation algorithms for two- and threedimensional packing problems”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Baker, B.S., Coffman Jr., E.G., Rivest, R.L.: Orthogonal packings in two dimensions. SIAM J. Comput. 9(4), 846–855 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Steinberg, A.: A strip-packing algorithm with absolute performance bound 2. SIAM J. Comput. 26(2), 401–409 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kenyon, C., Rémila, E.: A near-optimal solution to a two-dimensional cutting stock problem. Math. Oper. Res. 25(4), 645–656 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jansen, K., Solis-Oba, R.: New approximability results for 2-dimensional packing problems. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 103–114. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  6. Jansen, K., Zhang, G.: On rectangle packing: maximizing benefits. In: Munro, J.I. (ed.) SODA 2004, pp. 204–213. SIAM, Philadelphia (2004)

    Google Scholar 

  7. 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(3), 271–275 (1990)

    Article  MathSciNet  Google Scholar 

  8. Jansen, K., Prädel, L.: How to maximize the total area of rectangles packed into a rectangle? Technical Report 0908, Christian-Albrechts-Universität zu Kiel (2009)

    Google Scholar 

  9. Caprara, A.: Packing 2-dimensional bins in harmony. In: IEEE (ed.) FOCS 2002, pp. 490–499. IEEE Computer Society Press, Los Alamitos (2002)

    Google Scholar 

  10. Bansal, N., Sviridenko, M.: New approximability and inapproximability results for 2-dimensional bin packing. In: Munro, J.I. (ed.) SODA 2004, pp. 196–203. SIAM, Philadelphia (2004)

    Google Scholar 

  11. Zhang, G.: A 3-approximation algorithm for two-dimensional bin packing. Oper. Res. Lett. 33(2), 121–126 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bansal, N., Caprara, A., Sviridenko, M.: Improved approximation algorithms for multidimensional bin packing problems. In: IEEE (ed.) FOCS 2006, pp. 697–708. IEEE Computer Society, Los Alamitos (2006)

    Google Scholar 

  13. Harren, R., van Stee, R.: Packing rectangles into 2opt bins using rotations. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 306–318. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  14. Freund, A., Naor, J.: Approximating the advertisement placement problem. J. Scheduling 7(5), 365–374 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in robotics and VLSI. In: Fontet, M., Mehlhorn, K. (eds.) STACS 1984. LNCS, vol. 166, pp. 55–62. Springer, Heidelberg (1984)

    Chapter  Google Scholar 

  16. Jansen, K., Zhang, G.: Maximizing the total profit of rectangles packed into a rectangle. Algorithmica 47(3), 323–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jansen, K., Solis-Oba, R.: A polynomial time approximation scheme for the square packing problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 184–198. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  18. Bansal, N., Caprara, A., Sviridenko, M.: A structural lemma in 2-dimensional packing, and its implications on approximability. Technical Report RC24468, IBM (2008)

    Google Scholar 

  19. Bansal, N., Caprara, A., Jansen, K., Prädel, L., Sviridenko, M.: A structural lemma in 2-dimensional packing, and its implications on approximability (unpublished manuscript) (2009)

    Google Scholar 

  20. Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM Journal of Applied Mathematics 17(2), 416–429 (1969)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Christian-Albrechts-Universität zu Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen, Lars Prädel & Ulrich M. Schwarz

Authors
  1. Klaus Jansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lars Prädel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ulrich M. Schwarz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina Gavrilova

  3. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, P.O. Box, K1S 5B6, Ontario, Canada

    Jörg-Rüdiger Sack

  4. University of Calgary, Calgary, AB, Canada

    Csaba D. Tóth

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jansen, K., Prädel, L., Schwarz, U.M. (2009). Two for One: Tight Approximation of 2D Bin Packing . In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_35

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-03367-4_35

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03366-7

  • Online ISBN: 978-3-642-03367-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation