Advertisement

Approximation Algorithms for Bin-Packing — An Updated Survey

  • E. G. CoffmanJr.
  • M. R. Garey
  • D. S. Johnson
Part of the International Centre for Mechanical Sciences book series (CISM, volume 284)

Abstract

This paper updates a survey [53] written about 3 years ago. All of the results mentioned there are covered here as well. However, as a major justification for this second edition we shall be presenting many new results, some of which represent important advances. As a measure of the impressive amount of research in just 3 years, the present reference list more than doubles the list in [53].

Keywords

Approximation Algorithm Assembly Line Balance Multiprocessor Schedule Item Size Strip Packing 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Albano, A. and Sapuppo, G., “Optimal allocation of two-dimensional irregular shapes using heuristic search methods,” IEEE Trans. Syste., Man, Cybern., SMC-10 (1980), 242–248.Google Scholar
  2. [2]
    Assmann, S. B., Doctoral Dissertation, Department of Mathematics, M.I.T., Cambridge, Mass. (1983).Google Scholar
  3. [3]
    Assmann, S. B., Johnson, D. J., Kleitman, D. J. and Leung, J. Y-T., “On a dual version of the one-dimensional bin packing problem,” J. of Algorithms (to appear).Google Scholar
  4. [4]
    Baker, B. S., “A new proof for the first-fit decreasing bin-packing algorithm,” Technical Memorandum (1983), Bell Laboratories, Murray Hill, N.J. 07974.Google Scholar
  5. [5]
    Baker, B. S., Brown, D. J., and Katseff, H. P., “A 5/4 algorithm for two-dimensional packing,” J. of Algorithms, 2 (1981), 348–368.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    Baker, B. S., Calderbank, A. R., Coffman, E. G., Jr., and Lagarias, J. C., “Approximation algorithms for maximizing the number of squares packing into a rectangle,” SIAM J. of Alg. Disc. Meth. (to appear).Google Scholar
  7. [7]
    Baker, B. S. and Coffman, E. G., Jr., “A tight asymptotic bound for next-fit-decreasing bin-packing,” SIAM J. Alg. Disc. Meth. 2 (1981), 147–152.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Baker, B. S., Coffman, E. G., Jr., and Rivest, R. L., “Orthogonal packings in two dimensions,” SIAM J. Comput. 9 (1980), 846–855.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    Baker, B. S. and Schwarz, J. S., “Shelf algorithms for two-dimensional packing problems,” SIAM J. Comput. (to appear).Google Scholar
  10. [10]
    Biro, M. and Boros, E., “A network flow approach to non-guillotine cutting problems,” Working Paper MO/30 (1982), Computer and Automation Institute, Hungarian Academy of Sciences, Budapest.Google Scholar
  11. [11]
    Brown, A. R., Optimum Packing and Depletion, American Elsevier, New York (1971).MATHGoogle Scholar
  12. [12]
    Brown, D. J., “A lower bound for on-line one-dimensional bin packing algorithms,” Technical Report R-864 (1979), Coordinated Science Laboratory, University of Illinois, Urbana, IL.Google Scholar
  13. [13]
    Brown, D. J., “An improved BL lower bound,” Inf. Proc. Letters 11 (1980) 37–39.CrossRefMATHGoogle Scholar
  14. [14]
    Brown, D. J., private communication (1980).Google Scholar
  15. [15]
    Brown, D. J. and Baker, B. S. and Katseff, H. P., “Lower bounds for the on-line two-dimensional packing algorithms,” Acta Informatica, 18 (1982), 207–225.CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    Bruno, J. L. and Downey, P. J., “Probbilistic bounds on the performance of list scheduling,” Tech. Rep. TR 82–19, Computer Science Dept., University of Arizona, Tucson, Ariz.Google Scholar
  17. [17]
    Chandra, A. K. and Wong, C. K., “Worst-case analysis of a placement algorithm related to storage allocation,” SIAM J. Comput. 4 (1975), 249–263.CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    Chandra, A. K., Hirschberg, D. S., and Wong, C. K., “Bin packing with geometric constraints in computer network design,” Computer Science Research Report RC 6895 (1977), IBM Research Center, Yorktown Heights, New York.Google Scholar
  19. [19]
    Christofides, N. and Whitlock, C., “An algorithm for two-dimensional cutting problems,” Oper. Res. 25 (1977), 30–44.CrossRefMATHGoogle Scholar
  20. [20]
    Chung, F. R. K., Garey, M. R. and Johnson, D. J., “On packing two-dimensional bins,” SIAM J. Alg. Disc. Meth. 3 (1982), 66–76.CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    Cody, R. A. and Coffman, E. G., Jr., “Record allocation for minimizing expected retrieval costs on drum-like storage devices,” Journal of the ACM 23 (1976), 103–115.CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    Coffman, E. G., Jr., “An introduction to proof techniques for packing and sequencing algorithms,” in Deterministic and Stochastic Scheduling, M.A.H. Dempster, et al. (eds.), (1982), 245–270, Reidel Publishing Co., Amsterdam.Google Scholar
  23. [23]
    Coffman, E. G., Jr., “An introduction to combinatorial models of dynamic storage allocation,” SIAM Review (to appear).Google Scholar
  24. [24]
    Coffman, E. G., Jr., Frederickson, G. and Lueker, G. S., “A note on expected makespans for largest-first sequences of independent tasks on two processors,” Math of OR (to appear).Google Scholar
  25. [25]
    Coffman, E. G., Jr., Frederickson, G. N. and Lueker, G. S., manuscript in preparation.Google Scholar
  26. [26]
    Coffman, E. G., Jr., Garey, M. R., and Johnson, D. S., “An application of bin-packing to multiprocessor scheduling,” SIAM J. Comput. 7 (1978), 1–17.CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    Coffman, E. G., Jr., Garey, M. R., and Johnson, D. S., “Dynamic bin packing,” SIAM J. Comput. 12 (1983), 227–258.CrossRefMATHMathSciNetGoogle Scholar
  28. [28]
    Coffman, E. G., Jr., Garey, M. R. and Johnson, D. S., “Performance of packing algorithms for divisible sequences of item sizes,” paper in preparation.Google Scholar
  29. [29]
    Coffman, E. G., Jr., Garey, M. R., Johnson, D. S., and Tarjan, R. E., “Performance bounds for level-oriented two-dimensional packing algorithms,” SIAM J. Comput. 9 (1980), 808–826.CrossRefMATHMathSciNetGoogle Scholar
  30. [30]
    Coffman, E. G., Jr. and Gilbert, E. N., “On the expected relative performance of list scheduling,” Technical Memorandum, Bell Laboratories, Murray Hill, N.J. 07974 (1983).Google Scholar
  31. [31]
    Coffman, E. G., Jr. and Gilbert, E. N., “Dynamic first-fit packings in two or more dimensions,” Technical Memorandum, Bell Laboratories, Murray Hill, N.J. 07974 (1983).Google Scholar
  32. [32]
    Coffman, E. G., Jr., Hofri, M., So, K., and Yao, A. C., “A stochastic model of bin packing,” Inf. and Control 44 (1980), 105–115.CrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    Coffman, E. G., Jr., and Leung, J. Y., “Combinatorial analysis of an efficient algorithm for processor and storage allocation,” SIAM J. Comput. 8 (1979), 202–217.CrossRefMATHMathSciNetGoogle Scholar
  34. [34]
    Coffman, E. G., Jr., Leung, J. Y., and Ting, D. W., “Bin packing: maximizing the number of pieces packed,” Acta Informatica 9 (1978), 263–271.CrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    Coffman, E. G., Jr. and Sethi, R., “A generalized bound on LPT sequencing,” RAIRO Informatique 10 (1976), 17–25.MATHMathSciNetGoogle Scholar
  36. [36]
    Dempster, M. A. H., Fisher, M. L., Jansen, L., Lageweg, B. J., Lenstra, J. K. and Rinnooy Kan, A. H. G., “Analysis of heuristics for stochastic programming: Results for Hierarchical scheduling problems,” Operations Res., 29 (1981), 707–716.CrossRefMATHGoogle Scholar
  37. [37]
    Deuermeyer, B. L., Friesen, D. K. and Langston, M. A., “Maximizing the minimum processor finish time in a multiprocessor system,” SIAM J. Alg. Disc. Meth. 3 (1982), 190–196.Google Scholar
  38. [38]
    Easton, M. C. and Wong, C. K., “The effect of a capacity constraint on the minimal cost of a partition,” J. Assoc. Comput. Mach. 22 (1975), 441–449.CrossRefMATHMathSciNetGoogle Scholar
  39. [39]
    Erdös, P. and Graham, R. L., “On packing squares with equal squares,” J. Combinatorial Theory Ser. A 19 (1975), 119–123.CrossRefMATHMathSciNetGoogle Scholar
  40. [40]
    Fernandez de la Vega, W. and Lueker, G. S., “Bin packing can be solved within 1+E in linear time,” Combinatorica 1 (1981), 349–355.CrossRefMATHMathSciNetGoogle Scholar
  41. [41]
    Finn, G. and Horowitz, E., “A linear time approximation algorithm for multiprocessor scheduling,” BIT 19 (1979), 312–320.CrossRefMATHMathSciNetGoogle Scholar
  42. [42]
    Frederickson, G. N., “Probabilistic analysis for simple one- and two-dimensional bin packing algorithms,” Inf. Proc. Letters 11 (1980), 156–161.CrossRefMATHMathSciNetGoogle Scholar
  43. [43]
    Frenk, H. and Rinnooy Kan, A. H. G., “The asymptotic optimality of the LPT heuristic,” Erasmus University, Rotterdam, The Netherlands, (to be published).Google Scholar
  44. [44]
    Friesen, D. K., “Sensitivity analysis for heuristic algorithms,” Technical Report UIUCDCS¬R-78–939 (1978), Dept. Comp. Sci., Univ. of Illinois, Urbana, IL. SIAM J. Comput., (to appear).Google Scholar
  45. [45]
    Friesen, D. K. and Langston, M. A., “Analysis of a compound bin-packing algorithm,” (to appear).Google Scholar
  46. [46]
    Galambos, G. and Turan, G., Laboratory of Cybernetics, Josef Attila University, Szeged, Hungary (private communication).Google Scholar
  47. [47]
    Gardner, M., “Some packing problems that cannot be solved by sitting on the suitcase,” in Mathematical Games column, Scientific American, Oct. 1979, 18–26.Google Scholar
  48. [48]
    Garey, M. R. and Graham, R. L., “Bounds on multiprocessor scheduling with resource constraints,” SIAM J. Comput. 4 (1974), 187–200.CrossRefMathSciNetGoogle Scholar
  49. [49]
    Garey, M. R., Graham, R. L., and Johnson, D. S., “On a number-theoretic bin packing conjecture,” Proc. 5th Hungarian Combinatorics Colloquium, North-Holland, Amsterdam (1978), 377–392.Google Scholar
  50. [50]
    Garey, M. R., Graham, R. L., Johnson, D. S. and Yao, A. C., “Resource constrained scheduling as generalized bin packing,” J. Combinatorial Theory Ser. A 21 (1976), 257–298.CrossRefMATHMathSciNetGoogle Scholar
  51. [51]
    Garey, M. R. and Johnson, D. S., “Approximation algorithms for combinatorial problems: an annotated bibliography,” in J. F. Traub (ed.), Algorithms and Complexity: New Directions and Recent Results, Academic Press, New York (1976), 41–52.Google Scholar
  52. [52]
    Garey, M. R. and Johnson, D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco (1979).MATHGoogle Scholar
  53. [53]
    Garey, M. R. and Johnson, D. S., “Approximation algorithms for bin-packing problems — A survey,” in Analysis and Design of Algorithms in Combinatorial Optimization, G. Ausiello and M. Lucertini (eds.), Springer-Verlag, New York, 1981, 147–172.CrossRefGoogle Scholar
  54. [54]
    Garey, M. R. and Johnson, D. S., paper in preparation.Google Scholar
  55. [55]
    Gilmore, P. C. and Gomory, R. E., “A linear programming approach to the cutting stock problem,” Operations Res. 9 (1961), 849–859.CrossRefMATHMathSciNetGoogle Scholar
  56. [56]
    Gilmore, P. C. and Gomory, R. E., “A linear programming approach to the cutting stock program — Part II,” Operations Res. 11 (1963), 863–888.CrossRefMATHGoogle Scholar
  57. [57]
    Gilmore, P. C. and Gomory, R. E., “Multistage cutting stock problems of two and more dimensions,” Operations Res. 13 (1965), 94–120.CrossRefMATHGoogle Scholar
  58. [58]
    Golan, I., “Two orthogonal oriented algorithms for packing in two dimension,” Report 1979/311/MHM, Computer Center M.O.D., P. O. Box 2250, Haifa, Israel (1979).Google Scholar
  59. [59]
    Golan, I., “Performance bounds for orthogonal, oriented two-dimensional packing algorithms,” SIAM J. Comput. 10 (1981), 571–582.CrossRefMATHMathSciNetGoogle Scholar
  60. [60]
    Graham, R. L., “Bounds for certain multiprocessing anomalies,” Bell System Tech. J. 45 (1966), 1563–1581.CrossRefGoogle Scholar
  61. [61]
    Graham, R. L., “Bounds for Dynamic Storage Allocation Strategies,” Technical Memorandum, Bell Laboratories, Murray Hill, N.J. (1968).Google Scholar
  62. [62]
    Graham, R. L., “Bounds on multiprocessing timing anomalies,” SIAM J. Appl. Math. 17 (1969), 263–269.Google Scholar
  63. [63]
    Graham, R. L., “Bounds on multiprocessing anomalies and related packing algorithms,” Proc. 1972 Spring Joint Computer Conference, AFIPS Press, Montvale, N.J. (1972), 205–217.Google Scholar
  64. [64]
    Graham, R. L., “Bounds on performnce of scheduling algorithms,” in E. G. Coffman, Jr. (ed.), Computer and Job-Shop Scheduling Theory, John Wiley & Sons, New York (1976), 165–227.Google Scholar
  65. [65]
    Graham, R. L., Lawler, E. L., Lenstra, J. K., and Rinnooy Kan, A. H. G., “Optimization and approximation in deterministic sequencing and scheduling: a survey,” Annals Disc. Math. 5 (1979), 287–326.CrossRefMATHGoogle Scholar
  66. [66]
    Hoffman, U., “A class of simple online bin packing algorithms,” Computing, 29 (1982), 227–239.Google Scholar
  67. [67]
    Hofri, M., “Two dimensional packing: expected performance of simple level algorithms,” Inf. and Control 45 (1980), 1–17.CrossRefMATHMathSciNetGoogle Scholar
  68. [68]
    Hofri, M., “Bin-packing: An analysis of the Next-Fit algorithm,” Tech. Rep. No. 242, Dept. of Computer Science, The Technion, Haifa, Israel (1982).Google Scholar
  69. [69]
    Johnson, D. S., “Near-optimal bin packing algorithms,” Technical Report MAC TR-109 (1973), Project MAC, Masschusetts Institute of Technology, Cambridge, Mass.Google Scholar
  70. [70]
    Johnson, D. S., “Fast algorithms for bin packing,” J. Comput. Syst. Sci. 8 (1974), 272–314.CrossRefMATHGoogle Scholar
  71. [71]
    Johnson, D. S., “The NP-completeness column: An ongoing guide, J. of Algorithms 2 (1981), 393–405 (and succeeding issues).Google Scholar
  72. [72]
    Johnson, D. S., Demers, A., Ullman, J. D., Garey, M. R., and Graham, R. L., “Worst-case performance bounds for simple one-dimensional packing algorithms,” SIAM J. Comput. 3 (1974), 299–325.CrossRefMathSciNetGoogle Scholar
  73. [73]
    Karmarkar, N., “Probabilistic analysis of some bin-packing problems,” Proc. 23rd Ann. Symp. on Foundations of Computer Science, IEEE Computer Soc., Nov. 1982 (full paper to appear elsewhere).Google Scholar
  74. [74]
    Karmarkar, N. and Karp, R. M., “An efficient approximation scheme for the one-dimensional bin packing problem,” Proc. 23rd Ann. Symp. on Foundations of Computer Science, IEEE Computer Soc., Nov. 1982 (full paper to appear elsewhere).Google Scholar
  75. [75]
    Karmarkar, N. and Karp, R. M., “The differencing method of set partitioning,” Computer Science Div., University of California, Berkeley, Calif., to be published.Google Scholar
  76. [76]
    Karmarkar, N., Karp, R. M., Lueker, G. S. and Murgolo, F., Computer Science Div., University of California, Berkeley, Calif., paper in preparation.Google Scholar
  77. [77]
    Karmarkar, N., Karp, R. M., Lueker, G. S., and Odlyzko, A., Bell Laboratories, Murray Hill, N.J., paper in preparation.Google Scholar
  78. [78]
    Karp, R. M., “Reducibility among combinatorial problems,” in R. E. Miller and J. W. Thatcher (eds.), Complexity of Computer Computations, Plenum Press, New York (1972), 85–103.CrossRefGoogle Scholar
  79. [79]
    Karp, R. M., Luby, M. G. and Spaccamela, A. M., “Probabilistic analysis of multidimensional bin packing problems,” Computer Science Div., University of California, Berkeley, Calif., paper in preparation.Google Scholar
  80. [80]
    Kaufman, M. T., “An almost-optimal algorithm for the assembly line scheduling problem,” IEEE Trans. Computers C-23 (1974), 1169–1174.Google Scholar
  81. [81]
    Kleitman, D. J. and Krieger, M. K., “An optimal bound for two dimensional bin packing,” Proc. 16th Ann. Symp. on Foundations of Computer Science, IEEE Computer Society, Long Beach, CA (1975), 163–168.Google Scholar
  82. [82]
    Knödel, W., “A bin-packing algorithm with complexity O (n log n) and performance 1 in the stochastic limit,” Proc., 10th Symp. on Math. Foundations in Comp. Sci. (1981). (to appear in Lecture Notes in Computer Science, Springer-Verlag).Google Scholar
  83. [83]
    Knuth, D. E., Fundamental Algorithms, Vol. 1, Second edition, Addison-Wesley (1973).Google Scholar
  84. [84]
    Kou, L. T. and Markowsky, G., “Multidimensional bin packing algorithms,” IBM J. Res. & Dev. 21 (1977), 443–448.CrossRefMATHMathSciNetGoogle Scholar
  85. [85]
    Krause, K. L., Shen, Y. Y., and Schwetman, H. D., “Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems,” J. Assoc. Comput. Mach. 22 (1975), 522–550.CrossRefMATHMathSciNetGoogle Scholar
  86. [86]
    Langston, M. A., Processor Scheduling with Improved Heuristic Algorithms, Doctoral dissertation, Texas A&M University, College Station, Texas (1981).Google Scholar
  87. [87]
    Langston, M. A., “Performance of bin-packing heuristics for maximizing the number of pieces packed into bins of different sizes,” Tech. Rep. No. CS-82–090, Computer Science Dept., Washington State University, Pullman, Wash. (1982).Google Scholar
  88. [88]
    Langston, M. A., “Improved LPT scheduling for identical processor systems,” RAIRO-Technique et Science Informatiques, 1 (1982), 69–75.MATHGoogle Scholar
  89. [89]
    Langston, M. A., Improved 0/1 interchange scheduling,“ BIT 22 (1982), 282–290.CrossRefMATHGoogle Scholar
  90. [90]
    Lee, C. C. and Lee, D. T., “A simple on-line packing algorithm,” Dept. of Electrical Engineering and Computer Science (1983), Northwestern Univ., Evanston, Ill. 60201 (to appear).Google Scholar
  91. [91]
    Liang, F. M., “A lower bound for on-line bin packing,” Information Processing Lett. 10 (1980), 76–79.CrossRefMATHGoogle Scholar
  92. [92]
    Loulou, R., “Tight bounds and probabilistic analysis of two heuristics for parallel processor scheduling,” Tech. Rep., Faculty of Management, McGill University, Montreal (1982).Google Scholar
  93. [93]
    Loulou, R., “Probabilistic behavior of optimal bin packing solutions,” Tech. Rep., Faculty of Management, McGill University, Montreal (1982).Google Scholar
  94. [94]
    Lueker, G. S., “An average-case analysis of bin packing with uniformly distributed item sizes,” Tech. Rep. No. 181 (1982), Dept. of Information and Computer Science, University of California, Irvine, CA 92717.Google Scholar
  95. [95]
    Lueker, G. S., “Bin packing with items uniformly distributed over intervals [a,b],” Dept. of Information and Computer Science (1983), University of California, Irvine, CA 92717 (to be published).Google Scholar
  96. [96]
    Maruyama, K., Chang, S. K., and Tang, D. T., “A general packing algorithm for multidimensional resource requirements,” Internat. J. Comput. Infor. Sci. 6 (1977), 131–149.CrossRefMathSciNetGoogle Scholar
  97. [97]
    Ong, H. L., Magazine, M. J., and Wee, T. S., “Probabilistic analysis of bin-packing heuristics,” Operations Res. (to appear).Google Scholar
  98. [98]
    Robson, J. M., “Bounds for some functions concerning dynamic storage allocation,” Journal of the ACM 21 (1974), 491–499.CrossRefMATHMathSciNetGoogle Scholar
  99. [99]
    Robson, J. M., “Worst-case fragmentation of first-fit and best-fit storage allocation strategies,” Computer J., 20 (1977), 242–244.CrossRefGoogle Scholar
  100. [100]
    Sahni, S., “Algorithms for scheduling independent tasks,” Journal of the ACM 23 (1976), 116–127.CrossRefMATHMathSciNetGoogle Scholar
  101. [101]
    Schrijver, A. (ed.), Packing and Covering in Combinatorics, published by Mathematical Centre, Tweede Boerhaavestraat 49, Amsterdam (1979).Google Scholar
  102. [102]
    Shapiro, S. D., “Performance of heuristic bin packing algorithms with segments of random length,” Information and Control 35 (1977), 146–148.MATHMathSciNetGoogle Scholar
  103. [103]
    Shearer, J. B., “A counterexample to a bin packing conjecture, SIAM J. Alg. Disc. Meth. 2 (1981), 309–310.CrossRefMATHMathSciNetGoogle Scholar
  104. [104]
    Sleator, D. K. D. B., “A 2.5 times optimal algorithm for bin packing in two dimensions,” Information Processing Lett. 10 (1980), 37–40.CrossRefMATHMathSciNetGoogle Scholar
  105. [105]
    Steele, M., Stanford University (private communication).Google Scholar
  106. [106]
    Taylor, D. B., “Container stacking: an application of mathematical programming,” Draft (1979).Google Scholar
  107. [107]
    Wee, T. S. and Magazine, M. J., “Assembly line balancing as generalized bin-packing,” Operation Res. Letters, 1 (1982), 56–58.CrossRefMATHGoogle Scholar
  108. [108]
    Wong, C. K. and Yao, A. C., “A combinatorial optimization problem related to data set allocation,” Rev. Francaise Automat. Informa t. Recherche Operationelle Ser. Bleue 10.5 (suppl.) (1976), 83–95.Google Scholar
  109. [109]
    Yao, A. C., “New algorithms for bin packing,” J. Assoc. Comput. Mach. 27 (1980), 207–227.Google Scholar
  110. [110]
    Yue, P. C. and Wong, C. K., “On the optimality of the probability ranking scheme in storage applications,” J. Assoc. Comput. Mach. 20 (1973), 624–633.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1984

Authors and Affiliations

  • E. G. CoffmanJr.
    • 1
  • M. R. Garey
    • 1
  • D. S. Johnson
    • 1
  1. 1.Bell LaboratoriesMurray HillUSA

Personalised recommendations