Skip to main content
SpringerLink
Log in
Book cover

International Computing and Combinatorics Conference

COCOON 2015: Computing and Combinatorics pp 264–276Cite as

Packing Cubes into a Cube in (D>3)-Dimensions

  • Conference paper
  • First Online:

  • 1292 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9198))

Abstract

The problem of determining whether a set of cubes can be orthogonally packed into a cube has been studied in 2-diminson, 3-dimension, and (d>3)-dimensions. Open questions were asked on whether this problem is NP-complete in three articles in 1989, 2005, and 2009, respectively. In 1990, the problem of packing squares into a square was shown to be NP-complete by Leung et al. Recently, the problem of packing cubes into a cube in 3-D was shown to be NP-complete by Lu et al. In this paper, we show that the problem in (d>3)-dimensions is NP-complete in the strong sense, thus settling the related open question posed by previous researchers.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Caprara, A., Lodi, A., Monaci, M.: Fast Approximation Schemes for Two-stage, Two-dimensional Bin Packing. Mathematics of Operations Research 30, 136–156 (2005)

    Article  MathSciNet  Google Scholar 

  3. 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 

  4. Correa, J.R., Kenyon, C.: Approximation schemes for multidimensional packing. In: Proc. 15th ACM–SIAM Symposium on Discrete Algorithms, pp. 179–188 (2004)

    Google Scholar 

  5. Epstein, L., van Stee, R.: Online Square and Cube Packing. Acta Informatica 41(9), 595–606 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Garey, M., Johnson, D.: Computer and Intractability – A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    Google Scholar 

  7. Harren, R.: Approximation Algorithms for Orthogonal Packing Problems for Hypercubes. Theoretical Computer Science 410(44), 4504–4532 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kohayakawa, Y., Miyazawa, F.K., Raghavan, P., Wakabayashi, Y.: Multidimensional Cube Packing. Algorithmica 40, 173–187 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Leung, J.Y.-T., Tam, W.T., Wong, C.S., Chin, F.Y.L.: Packing Squares into a Square. Journal of Parallel and Distributed Computing 10, 271–275 (1990)

    Article  MathSciNet  Google Scholar 

  10. Li, K., Cheng, K.H.: Complexity of resource allocation and job scheduling problems in partitionable mesh connected systems. In: Proc. of 1st Annual IEEE Symposium of Parallel and Distributed Processing, pp. 358–365. Silver Spring, MD (1989)

    Google Scholar 

  11. Lu, Y., Chen, D.Z., Cha, J.: Packing Cubes into a Cube is NP-complete in the Strong Sense. Journal of Combinatorial Optimization 29(1), 197–215 (2015)

    Article  MathSciNet  Google Scholar 

  12. Miyazawa, F.K., Wakabayashi, Y.: Cube Packing. Theoretical Computer Science 297, 355–366 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, 100044, China

    Yiping Lu & Jianzhong Cha

  2. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Danny Z. Chen

Authors
  1. Yiping Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Danny Z. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianzhong Cha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yiping Lu .

Editor information

Editors and Affiliations

  1. Beijing University of Technology, Beijing, China

    Dachuan Xu

  2. University of New Brunswick, Fredericton, Canada

    Donglei Du

  3. University of Texas at Dallas, Richardson, USA

    Dingzhu Du

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Lu, Y., Chen, D.Z., Cha, J. (2015). Packing Cubes into a Cube in (D>3)-Dimensions. In: Xu, D., Du, D., Du, D. (eds) Computing and Combinatorics. COCOON 2015. Lecture Notes in Computer Science(), vol 9198. Springer, Cham. https://doi.org/10.1007/978-3-319-21398-9_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21398-9_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21397-2

  • Online ISBN: 978-3-319-21398-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation