A Dense Hierarchy of Sublinear Time Approximation Schemes for Bin Packing

  • Richard Beigel
  • Bin Fu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7285)


The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes a 1,…, a n in (0,1]. Using uniform sampling, which selects a random element from the input list each time, we develop a randomized \(O({n(\log\log n)\over \sum_{i=1}^n a_i}+({1\over \epsilon})^{O({1\over\epsilon})})\) time (1 + ε)-approximation scheme for the bin packing problem. We show that every randomized algorithm with uniform random sampling needs \(\Omega({n\over \sum_{i=1}^n a_i})\) time to give an (1 + ε)-approximation. For each function s(n): N → N, define ∑ (s(n)) to be the set of all bin packing problems with the sum of item sizes equal to s(n). We show that ∑ (n b ) is NP-hard for every b ∈ (0,1]. This implies a dense sublinear time hierarchy of approximation schemes for a class of NP-hard problems, which are derived from the bin packing problem. We also show a randomized streaming approximation scheme for the bin packing problem such that it needs only constant updating time and constant space, and outputs an (1 + ε)-approximation in \(({1\over \epsilon})^{O({1\over\epsilon})}\) time. Let S(δ)-bin packing be the class of bin packing problems with each input item of size at least δ. This research also gives a natural example of NP-hard problem (S(δ)-bin packing) that has a constant time approximation scheme, and a constant time and space sliding window streaming approximation scheme, where δ is a positive constant.


Approximation Scheme Minimum Span Tree Uniform Sampling Large Item Small Item 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. In: Proceedings of the Symposium on Theory of Computing, pp. 20–29 (1996)Google Scholar
  2. 2.
    Applegate, D., Buriol, L., Dillard, B., Johnson, D., Shore, P.: The cutting-stock approach to bin packing: Theory and experiments. In: Proceedings of Algorithm Engineering and Experimentation (ALENEX), pp. 1–15 (2003)Google Scholar
  3. 3.
    Batu, T., Berenbrink, P., Sohler, C.: A sublinear-time approximation scheme for bin packing. Theoretical Computer Science 410, 5082–5092 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brown, D.: A lower bound for on-line one-dimensional bin packing problem. Technical Report 864, University of Illinois, Urbana, IL (1979)Google Scholar
  5. 5.
    Chazelle, B., Liu, D., Magen, A.: Sublinear geometric algorithms. SIAM Journal on Computing 35, 627–646 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chazelle, B., Rubfinfeld, R., Trevisan, L.: Approximating the minimum spanning tree weight in sublinear time. SIAM Journal on Computing 34, 1370–1379 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Csirik, J.A., Johnson, D.S., Kenyon, C., Shor, P.W., Weber, R.R.: A Self Organizing Bin Packing Heuristic. In: Goodrich, M.T., McGeoch, C.C. (eds.) ALENEX 1999. LNCS, vol. 1619, pp. 246–265. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. 8.
    Csirik, J., Johnson, D., Kenyon, C., Orlin, J., Shore, P., Weber, R.: On the sum-of-squares algorithm for bin-packing. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC), pp. 208–217 (2000)Google Scholar
  9. 9.
    Czumaj, A., Ergun, F., Fortnow, L., Magen, I.N.A., Rubinfeld, R., Sohler, C.: Sublinear approximation of euclidean minimum spanning tree. SIAM Journal on Computing 35, 91–109 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Czumaj, A., Sohler, C.: Estimating the weight of metric minimum spanning trees in sublinear-time. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 175–183 (2004)Google Scholar
  11. 11.
    Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within 1+epsilon in linear time. Combinatorica 1(4), 349–355 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Flajolet, P., Martin, G.: Probabilistic counting algorithms for data base application. Journal of Computer and System Sciences 31, 182–209 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fu, B., Chen, Z.: Sublinear-time algorithms for width-bounded geometric separators and their applications to protein side-chain packing problems. Journal of Combinatorial Optimization 15, 387–407 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Gilmore, M., Gomory, R.: A linear programming approach to the cutting-stock problem - part ii. Operations ResearchGoogle Scholar
  16. 16.
    Gilmore, M., Johnson, D.: A linear programming approach to the cutting-stock problem. Operations ResearchGoogle Scholar
  17. 17.
    Goldreich, O., Ron, D.: On testing expansion in bounded-degree graphs. Technical Report 00-20, Electronic Colloquium on Computational Complexity (2000),
  18. 18.
    Li, M., Ma, B., Wang, L.: On the closest string and substring problems. Journal of the ACM 49(2), 157–171 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Liang, F.: A lower bound for on-line bin packing. Information Processing Letters 10, 76–79 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press (2000)Google Scholar
  21. 21.
    Munro, J.I., Paterson, M.S.: Selection and sorting with limited storage. Theoretical Computer Science 12, 315–323 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Goldreich, S.G.O., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45, 653–750 (1998)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Richard Beigel
    • 1
  • Bin Fu
    • 2
  1. 1.CIS DepartmentTemple UniversityPhiladelphiaUSA
  2. 2.Department of Computer ScienceUniversity of Texas-Pan AmericanEdinburgUSA

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