Bin packing can be solved within 1 + ε in linear time


For any listL ofn numbers in (0, 1) letL* denote the minimum number of unit capacity bins needed to pack the elements ofL. We prove that, for every positive ε, there exists anO(n)-time algorithmS such that, ifS(L) denotes the number of bins used byS forL, thenS(L)/L*≦1+ε for anyL providedL* is sufficiently large.

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  1. [1]

    M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest andR. E. Tarjan, Time bounds for selection,J. Comput. Sys. Sci.,7 (1973), 448–461.

    MATH  MathSciNet  Article  Google Scholar 

  2. [2]

    M. R. Garey, R. L. Graham, D. S. Johnson andA. C. Yao, Multiprocessor scheduling as generalized bin-packing,J. Combinatorial Theory A21 (1976), 257–298.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    M. R. Garey andD. S. Johnson,Computers and Intractability, Freeman, San Francisco, 1979.

    MATH  Google Scholar 

  4. [4]

    M. R. Garey andD. S. Johnson, Approximation algorithms for bin packing problems: a survey,preprint 1980.

  5. [5]

    D. S. Johnson,Near optimal bin packing algorithms, Ph. D. Th., MIT, Cambridge, Mass., June 1973.

    Google Scholar 

  6. [6]

    D. S. Johnson, Fast algorithms for bin packing,J. Comptr. Syst. Sci. 8 (1974), 272–314.

    MATH  Google Scholar 

  7. [7]

    D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey andR. L. Graham, Worst case bounds for simple one-dimensional packing algorithms,SIAM J. Comptg. 3 (1974), 299–325.

    Article  MathSciNet  Google Scholar 

  8. [8]

    R. M. Karp, Reducibility among combinatorial problems, in:Complexity of Computer calculations. (R. E. Miller and J. W. Thatcher, Eds.) Plenum Press, New York, 1972, 85–103.

    Google Scholar 

  9. [9]

    D. E. Knuth,The Art of Computer Programming, Vol. 3, Sorting and Searching, Addison-Wesley, Reading, Mass., 1973.

    Google Scholar 

  10. [10]

    A. Schönhage, M. S. Paterson andN. Pippenger, Finding the median,J. Comput. Sys. Sci. 13 (1976), 184–199.

    MATH  Google Scholar 

  11. [11]

    A. C. Yao, New algorithms for bin packing.J. ACM 27, 2 (Apr. 1980).

    Article  Google Scholar 

  12. [12]

    J. L. Bentley, Probabilistic analysis of algorithms,Applied Probability—Computer Science, the Interface, Boca Raton, Florida, January 1981.

    Google Scholar 

  13. [13]

    P. C. Gilmore andR. E. Gomory, A linear programming approach to the cutting-stock problem,Operations Research 9 (1961), 849–859.

    MATH  MathSciNet  Google Scholar 

  14. [14]

    O. H. Ibarra andC. E. Kim, Fast approximation algorithms for the knapsack and sum of subset problems,Journal of the ACM 22 (1975), 463–468.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    L. V. Kantorovtch, Mathematical methods of organizing and planning production,Management Science 6, 4 (July 1960), 366-422.

    MathSciNet  Google Scholar 

  16. [16]

    S. Sahni, General techniques for combinatorial approximation,Operations Research 25, 6 (1977), 920-936.

    MathSciNet  Article  Google Scholar 

  17. [17]

    B. W. Weide,Statistical Methods in Algorithm Design and Analysis, Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh, Pennsylvania (August 1978); appeared as CMU Computer Science Report CMU-CS-78-142.

    Google Scholar 

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The work of this author was supported by NSF Grant MCS 70-04997.

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Fernandez de la Vega, W., Lueker, G.S. Bin packing can be solved within 1 + ε in linear time. Combinatorica 1, 349–355 (1981).

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AMS subject classification (1980)

  • 68 C 25
  • 68 E 05
  • 90 C 10