Bin Packing Approximation Algorithms: Combinatorial Analysis

  • Edward G. Coffman
  • Gabor Galambos
  • Silvano Martello
  • Daniele Vigo


In the classical version of the bin packing problem one is given a list L = (a 1,...,a n ) of items (or elements) and an infinite supply of bins with capacity C. A function s(a i ) gives the size of item a i , and satisfies 0 < s(a i )≤C, 1 ≤ in. The problem is to pack the items into a minimum number of bins under the constraint that the sum of the sizes of the items in each bin is no greater than C. In simpler terms, a set of numbers is to be partitioned into a minimum number of blocks subject to a sum constraint common to each block. We use the bin packing terminology, as it eases considerably the problem of describing and analyzing algorithms.


Online Algorithm Item Size Longe Processing Time Current Item Closing Rule 
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]
    M. Adler, P. B. Gibbons, and Y. Matias. Scheduling space sharing for internet advertising. Technical report, Bell Labs, Lucent Technologies, Murray Hill, NJ 07974, 1997.Google Scholar
  2. [2]
    S. Albers. Better bounds for on-line scheduling. In Proc. 29th Annual ACM Symp. Theory of Comput., pages 130–139, 1997.Google Scholar
  3. [3]
    R. J. Anderson, E. W. Mayr, and M. K. Warmuth. Parallel approximation algorithms for bin packing. Inf. and Comput., 82: 262–277, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    S. F. Assmann Problems in Discrete Applied Mathematics. PhD thesis, Mathematics Department MIT, Cambridge, MA, 1983.Google Scholar
  5. [5]
    S. F. Assmann, D. S. Johnson, D. J. Kleitman, and J. Y.-T. Leung. On a dual version of the one-dimensional bin packing problem. J. Algorithms, 5: 502–525, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    B. S. Baker. A new proof for the first-fit decreasing bin-packing algorithm. J. Algorithms, 6: 49–70, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    B. S. Baker, D. J. Brown, and H. P. Katseff. A 5/4 algorithm for two-dimensional packing. J. Algorithms, 2: 348–368, 1981.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    B. S. Baker and E. G. Coffman, Jr. A tight asymptotic bound for nextfit-decreasing bin-packing. SIAM J. Algebraic Discr. Meth.,2(2):147152, 1981.Google Scholar
  9. [9]
    Y. Bartal, A. Fiat, H. Karloff, and R. Vohra. New algorithms for an ancient scheduling problem. In Proc. 24th Annual ACM Symp. Theory of Comput., pages 51–58, Victoria, Canada, 1992.Google Scholar
  10. [10]
    Y. Bartal, H. Karloff, and Y. Rabani. A better lower bound for on-line scheduling. Inform. Process. Lett., 50: 113–116, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    J. Bèkèsi, G. Galambos, and H. Kellerer. A 5/4 linear time bin packing algorithm. Technical Report OR-97–2, Teachers Trainer College, Szeged, Hungary, 1997.Google Scholar
  12. [12]
    J. Blazewicz and K. Ecker. A linear time algorithm for restricted bin packing and scheduling problems. Oper. Res. Lett., 2: 80–83, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    D. J. Brown. A lower bound for on-line one-dimensional bin-packing algorithms. Technical Report R-864, University of Illinois, Coordinated sc. lab., Urbana, 1979.Google Scholar
  14. [14]
    R. E. Burkard and G. Zhang. Bounded space on-line variable-sized bin packing. Acta Cybern., 13: 63–76, 1997.MathSciNetzbMATHGoogle Scholar
  15. [15]
    L. M. A. Chan, D. Simchi-Levi, and J. Bramel. Worst-case analyses, linear programming, and the bin-packing problem. Unpublished manuscript, 1994.Google Scholar
  16. [16]
    A. K. Chandra, D. S. Hirschler, and C. K. Wong. Bin packing with geometric constraints in computer network design. Oper. Res., 26: 760772, 1978.Google Scholar
  17. [17]
    B. Chandra. Does randomization help in on-line bin packing? Inform. Process. Lett., 43: 15–19, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    B. Chen, A. van Vliet, and G.J. Woeginger. New lower and upper bounds for on-line scheduling. Oper. Res. Lett., 16: 221–230, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    E. G. Coffman, Jr. An introduction to combinatorial models of dynamic storage allocation. SIAM Rev., 25 (3): 311–325, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    E. G. Coffman, Jr., M. Garey, and D. S. Johnson. Bin packing with divisible item sizes. J. Complexity, 3: 405–428, 1987.Google Scholar
  21. [21]
    E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson. An application of bin-packing to multiprocessor scheduling. SIAM J. Comput., 7 (1): 117, 1978.Google Scholar
  22. [22]
    E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson. Approximation algorithms for bin-packing: An updated survey. In G. Ausiello, M. Lucertini, and P. Serafini, editors, Algorithm Design for Computer System Design, pages 49–106. Springer Verlag, Wien, 1984.Google Scholar
  23. [23]
    E. G. Coffman, Jr., M. R. Carey, and D. S. Johnson. Approximation algorithms for bin packing. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, pages 46–93. PWS Publ. Company, 1997.Google Scholar
  24. [24]
    E. G. Coffman, Jr. and J. Y.-T. Leung. Combinatorial analysis of an efficient algorithm for processor and storage allocation. SIAM J. Comput., 8 (2): 202–217, 1979.MathSciNetzbMATHGoogle Scholar
  25. [25]
    E. G. Coffman, Jr., J. Y.-T. Leung, and D. W. Ting. Bin packing: Maximizing the number of pieces packed. Acta Inform., 9: 263–271, 1978.MathSciNetzbMATHGoogle Scholar
  26. [26]
    E. G. Coffman, Jr. and G. S. Lueker. Probabilistic analysis of packing and partitioning algorithms. John Wiley & Sons, New York, 1991.Google Scholar
  27. [27]
    J. Csirik. An on-line algorithm for variable-sized bin packing. Acta Inform., 26: 697–709, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    J. Csirik. The parametric behaviour of the first fit decreasing bin-packing algorithm. J. Algorithms, 15: 1–28, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    J. Csirik, G. Galambos, and G. Turan. Some results on bin-packing. In Proc. EURO VI Conf., page 52, Vienna, Austria, 1983.Google Scholar
  30. [30]
    J. Csirik and B. Imreh. On the worst-case performance of the next-k-fit bin-packing heuristic. Acta Cybern., 9: 89–105, 1989.MathSciNetzbMATHGoogle Scholar
  31. [31]
    J. Csirik and D. S. Johnson. Bounded space on-line bin-packing: best is better than first. In Proc. 2nd Annual ACM-SIAM Symp. Discr. Algorithms, pages 309–319, Philadelphia, 1991.Google Scholar
  32. [32]
    J. Csirik and V. Totik. Online algorithms for a dual version of bin paking. Discr. Appl. Math., 21: 163–167, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    J. Csirik and G. J. Woeginger. Online packing and covering problems. Technical Report No. 83, T.U. Graz (Austria), 1996.Google Scholar
  34. [34]
    M. Dror. Private communication.Google Scholar
  35. [35]
    U. Faigle, W. Kern, and G. Turan. On the performance of on-line algorithms for particular problems. Acta Cybern., 9: 107–119, 1989.MathSciNetzbMATHGoogle Scholar
  36. [36]
    W. Fernandez de la Vega and G. S. Lueker. Bin packing can be solved within 1 + e in linear time. Combinatorica, 1 (4): 349–355, 1981.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    D. C. Fisher. Next-fit packs a list and its reverse into the same number of bins. Oper. Res. Lett., 7 (6): 291–293, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    D. K. Friesen. Tighter bounds for the multifit processor scheduling algorithm. SIAM J. Comput., 13 (1): 170–181, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    D. K. Friesen and M. A. Langston. Variable sized bin packing. SIAM J. Comput., 15 (1): 222–230, 1986.zbMATHCrossRefGoogle Scholar
  40. [40]
    D. K. Friesen and M. A. Langston. Analysis of a compound bin packing algorithm. SIAM J. Discr. Math., 4 (1): 61–79, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    G. Galambos. A new heuristic for the classical bin-packing problem. Technical Report 82, Institute fuer Mathematik, Augsburg, 1985.Google Scholar
  42. [42]
    G. Galambos. Parametric lower bound for on-line bin-packing. SIAM J. Algebraic Discr. Meth., 7 (3): 362–367, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    G. Galambos and J. B. G. Frenk. A simple proof of Liang’s lower bound for on-line bin packing and the extension to the parametric case. Discr. Appl. Math., 41: 173–178, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    G. Galambos and G. J. Woeginger. An on-line scheduling heuristic with better worst case ratio than graham’s list scheduling. SIAM J. Comput., 22 (2): 345–355, 1993.MathSciNetCrossRefGoogle Scholar
  45. [45]
    G. Galambos and G. J. Woeginger. Repacking helps in bounded space on-line bin-packing. Computing, 49: 329–338, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    G. Galambos and G. J. Woeginger. On-line bin packing — a restricted survey. Z. Oper. Res., 42: 25–45, 1995.MathSciNetzbMATHGoogle Scholar
  47. [47]
    G. Gambosi, A. Postiglione, and M. Talamo. New algorithms for online bin packing. In R. Petreschi, G. Ausiello, and D. P. Bovet, editors, Algorithms and Complexity, pages 44–59. World Scientific, Singapore, 1990.Google Scholar
  48. [48]
    M. R. Garey, R. L. Graham, D. S. Johnson, and A. C.-C. Yao. Resource constrained scheduling as generalized bin packing. J. Combin. Theory, Ser. A, 21: 257–298, 1976.Google Scholar
  49. [49]
    M. R. Garey, R. L. Graham, and J. D. Ullmann. Worst-case analysis of memory allocation algorithms. In Proc. 4th Annual ACM Symp. Theory of Comput., pages 143–150, New York, 1972.Google Scholar
  50. [50]
    M. R. Garey and D. S. Johnson. Computers and intractability (A Guide to the theory of NP-Completeness. W. H. Freeman and Company, San Francisco, 1979.Google Scholar
  51. [51]
    M. R. Carey and D. S. Johnson. Approximation algorithm for bin-packing problems: a survey. In G. Ausiello and M. Lucertini, editors, Analysis and Design of Algorithm in Combinatorial Optimization, pages 147–172. Springer Verlag, New York, 1981.Google Scholar
  52. [52]
    M. R. Garey and D. S. Johnson. A 71/60 theorem for bin packing. J. Complexity, 1: 65–106, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    P. C. Gilmore and R. E. Gomory. A linear programming approach to the cutting-stock problem. Oper. Res., 9: 849–859, 1961.MathSciNetzbMATHCrossRefGoogle Scholar
  54. [54]
    P. C. Gilmore and R. E. Gomory. A linear programming approach to the cutting stock problem–(Part II). Oper. Res., 11: 863–888, 1963.zbMATHCrossRefGoogle Scholar
  55. [55]
    S. W. Golomb. On certain nonlinear recurring sequences. American Math. Monthly, 70: 403–405, 1963.MathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    R. L. Graham. Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J., 45 (45): 1563–1581, 1966.Google Scholar
  57. [57]
    R. L. Graham. Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math., 17 (2): 263–269, 1969.CrossRefGoogle Scholar
  58. [58]
    R. L. Graham. Bounds on multiprocessing anomalies and related packing algorithms. In Proc. 1972 Spring Joint Computer Conf.,pages 205–217, Montvale NJ, 1972. AFIPS Press.Google Scholar
  59. [59]
    E. F. Grove. Online bin packing with lookahead. Unpublished manuscript, 1994.Google Scholar
  60. [60]
    L. A. Hall. Approximation algorithms for scheduling. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, pages 1–45. PWS Publ. Company, 1997.Google Scholar
  61. [61]
    D. Hochbaum and D. Shmoys. Using dual approximation algorithms for scheduling problems: theoretical and practical results. J. ACM, 34: 144–162, 1987.MathSciNetCrossRefGoogle Scholar
  62. [62]
    D. S. Hochbaum. Approximation Algorithms for NP-Hard Problems. PWS Publ. Company, Boston, 1997.Google Scholar
  63. [63]
    D. S. Hochbaum. Various notions of approximation: Good, better, best, and more. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, pages 389–391. PWS Publ. Company, 1997.Google Scholar
  64. [64]
    M. Hofri. Analysis of Algorithms. Oxford University Press, New York, 1995.zbMATHGoogle Scholar
  65. [65]
    Z. Ivkovié and E. Lloyd. Fully dynamic algorithms for bin packing: being myopic helps. In Proc. 1st European Symp. on Algorithms, volume 726 of Lecture Notes in Computer Science, pages 224–235. Springer Verlag, New York, 1993.Google Scholar
  66. [66]
    Z. Ivkovié and E. Lloyd. A fundamental restriction on fully dynamic maintenance of bin packing. Inf. Proc. Lett., 59 (4): 229–232, 1996.CrossRefGoogle Scholar
  67. [67]
    Z. Ivkovié and E. Lloyd. Partially dynamic bin packing can be solved within 1 + e in (amortized) polylogarithmic time. Inf. Proc. Lett., 63 (1): 45–50, 1997.CrossRefGoogle Scholar
  68. [68]
    D. S. Johnson. Fast allocation algorithms. In Proc. 13th IEEE Symp. Switching and Automata Theory, pages 144–154, New York, 1972.Google Scholar
  69. [69]
    D. S. Johnson. Near-optimal bin packing algorithms. PhD thesis, MIT, Cambridge, MA, 1973.Google Scholar
  70. [70]
    D. S. Johnson. Fast algorithms for bin packing. J. Comput. Syst. Sci., 8: 272–314, 1974.zbMATHCrossRefGoogle Scholar
  71. [71]
    D. S. Johnson. The NP-completeness column: An ongoing guide. J. Algorithms, 3: 89–99, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  72. [72]
    D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey, and R. L. Graham. Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM J. Comput., 3: 256–278, 1974.MathSciNetCrossRefGoogle Scholar
  73. [73]
    D. R. Karger, S. J. Phillips, and E. Torng. A better algorithm for an ancient scheduling problem. J. Algorithms, 20: 400–430, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  74. [74]
    N. Karmarkar and R. M. Karp. An efficient approximation scheme for the one-dimensional bin-packing problem. In Proc. 23rd Annual IEEE Symp. Found. Comput. Sci., pages 312–320, 1982.Google Scholar
  75. [75]
    H. Kellerer and U. Pferschy. An on-line alorithm for cardinality constrained bin packing problem. Technical report, Universitaet Graz und TU Graz, 1997.Google Scholar
  76. [76]
    C. Kenyon. Best-fit bin-packing with random order. In Proc. 7th Annual ACM-SIAM Symp. Discr. Algorithms, pages 359–364, Philadelphia, 1996.Google Scholar
  77. [77]
    S. Khanna. Private communication.Google Scholar
  78. [78]
    N. G. Kinnersley and M. A. Langston. On-line variable sized bin packing. Discr. Appl. Math., 22: 143–148, 1988.MathSciNetCrossRefGoogle Scholar
  79. [79]
    K. L. Krause, Y. Y. Shen, and H. D. Schwetman. Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. J. ACM, 22 (4): 522–550, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [80]
    M. A. Langston. Improved 0/1 interchanged scheduling. BIT, 22: 28 2290, 1982.Google Scholar
  81. [81]
    E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys. Sequencing and scheduling: algorithms and complexity. In S. C. Graves, A. H. G. Rinnooy Kan, and P. H. Zipkin, editors, Logistics of production and inventory,volume 4 of Handbooks in operations research and management science,pages 445–522. North-Holland, Amsterdam, 1993.Google Scholar
  82. [82]
    C. C. Lee and D. T. Lee. A new algorithm for on-line bin-packing. Technical Report 83–03-FC-02, Department of Electrical Engineering and computer Science Northwestern University, Evanston, IL, 1983.Google Scholar
  83. [83]
    C. C. Lee and D. T. Lee. A simple on-line bin-packing algorithm. J. ACM, 32 (3): 562–572, 1985.zbMATHCrossRefGoogle Scholar
  84. [84]
    H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8 (4): 538–548, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  85. [85]
    F. M. Liang. A lower bound for on-line bin packing. Inform. Process. Lett., 10 (2): 76–79, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  86. [86]
    W. Mao. Best-k-fit bin packing. Computing, 50: 265–270, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  87. [87]
    W. Mao. Tight worst-case performance bounds for next-k-fit bin packing. SIAM J. Comput.,22(1):46–56, 1993Google Scholar
  88. [88]
    C. U. Martel. A linear time bin-packing algorithm. Oper. Res. Lett., 4 (4): 189–192, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  89. [89]
    F. D. Murgolo. An efficient approximation scheme for variable-sized bin packing. SIAM J. Comput.16(1):149–161 1987.Google Scholar
  90. [90]
    F. D. Murgolo. Anomalous behaviour in bin packing algorithms. Discr. Appl. Math. 21:229–243 1988.Google Scholar
  91. [91]
    P. Ramanan, D. J. Brown, C. C. Lee, and D. T. Lee. On-line bin packing in linear time. J. Algorithms, 10: 305–326, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  92. [92]
    M. B. Richey. Improved bounds for harmonic-based bin packing algorithms.Discr. Appl. Math. 34:203–227 1991.Google Scholar
  93. [93]
    S. Sahni. Algorithms for scheduling independent tasks. J. ACM 23:116–127 1976.Google Scholar
  94. [94]
    H. E. Salzer. The approximation of number as sums of reciprocals. American Math. Monthly 54:135–142 1947.Google Scholar
  95. [95]
    D. Simchi-Levi. New worst-case results for the bin packing problem. Naval Res. Log. Quart. 41:579–585 1994.Google Scholar
  96. [96]
    A. van Vliet. An improved lower bound for on-line bin packing algorithms. Inform. Process. Lett., 43: 277–284, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  97. [97]
    A. van Vliet. Lower and Upper Bounds for On-line Bin Packing and Scheduling Heuristic. PhD thesis, Erasmus University, Rotterdam, 1995.Google Scholar
  98. [98]
    A. van Vliet. On the asymptotic worst case behavoir of harmonic fit. J. Algorithms 20:113–136 1996.Google Scholar
  99. [99]
    T. S. Wee and M. J. Magazine. Assembly line balancing as generalized bin packing. Oper. Res. Lett.,1(2):56–58, 1982.Google Scholar
  100. [100]
    G. J. Woeginger. Improved space for bounded-space on-line bin-packing. SIAM J. Discr. Math. 6:575–581 1993.Google Scholar
  101. [101]
    K. Xu. A bin-packing problem with Item Sizes in the Interval (0, a] for G 2. PhD thesis, Chinese Academy of Sciences, Institute of Applied Mathematics, Beijing, China, 1993.Google Scholar
  102. [102]
    A. C.-C. Yao. New algorithms for bin packing. J. ACM 27(2):207–227 1980.Google Scholar
  103. [103]
    M. Yue. On the exact upper bound for the multifit processor scheduling algorithm. Ann. Oper. Res., 24: 233–259, 1991.CrossRefGoogle Scholar
  104. [104]
    M. Yue. A simple proof of the inequality FFD(L) OPT (L) + 1 dL for the FFD bin packing algorithm. Acta Math. App. Sinica,7(4):321331, 1991.Google Scholar
  105. [105]
    G. Zhang. Tight worst-case performance bound for AFBk. Technical Report 15, Inst. of Applied Mathematics. Academia Sinica, Beijng, China, 1994.Google Scholar
  106. [106]
    G. Zhang. Worst-case analysis of the FFH algorithm for on-line variable-sized bin paking. Computing, 56: 165–172, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  107. [107]
    G. Zhang. A new version of on-line variable-sized bin packing. Discr. Appl. Math., 72: 193–197, 1997.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Edward G. Coffman
    • 1
  • Gabor Galambos
    • 2
  • Silvano Martello
    • 3
  • Daniele Vigo
    • 4
  1. 1.Bell LabsLucent TechnologiesUSA
  2. 2.Computer Science DepartmentTeacher’s Training CollegeSzegedHungary
  3. 3.DEISUniversity of BolognaItaly
  4. 4.DEISUniversity of BolognaItaly

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