Lower Bounds for Several Online Variants of Bin Packing

  • János Balogh
  • József Békési
  • György Dósa
  • Leah Epstein
  • Asaf LevinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10787)


We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.


  1. 1.
    Angelopoulos, S., Dürr, C., Kamali, S., Renault, M., Rosén, A.: Online bin packing with advice of small size. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 40–53. Springer, Cham (2015). Scholar
  2. 2.
    Babel, L., Chen, B., Kellerer, H., Kotov, V.: Algorithms for on-line bin-packing problems with cardinality constraints. Discret. Appl. Math. 143(1–3), 238–251 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balogh, J., Békési, J.: Semi-on-line bin packing: a short overview and a new lower bound. CEJOR 21(4), 685–698 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balogh, J., Békési, J., Dósa, G., Epstein, L., Levin, A.: Online bin packing with cardinality constraints resolved. The Computing Research Repository (CoRR) (2016). Also in ESA 2017 (to appear)
  5. 5.
    Balogh, J., Békési, J., Dósa, G., Epstein, L., Levin, A.: A new and improved algorithm for online bin packing. The Computing Research Repository (CoRR) (2017).
  6. 6.
    Balogh, J., Békési, J., Dósa, G., Epstein, L., Levin, A.: Lower bounds for several online variants of bin packing. The Computing Research Repository (CoRR) (2017).
  7. 7.
    Balogh, J., Békési, J., Galambos, G.: New lower bounds for certain classes of bin packing algorithms. Theoret. Comput. Sci. 440–441, 1–13 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bansal, N., Correa, J., Kenyon, C., Sviridenko, M.: Bin packing in multiple dimensions: inapproximability results and approximation schemes. Math. Oper. Res. 31(1), 31–49 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Békési, J., Dósa, G., Epstein, L.: Bounds for online bin packing with cardinality constraints. Inf. Comput. 249, 190–204 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Blitz, D.: Lower bounds on the asymptotic worst-case ratios of on-line bin packing algorithms. M.Sc. thesis, University of Rotterdam, Number 114682 (1996)Google Scholar
  11. 11.
    Boyar, J., Kamali, S., Larsen, K.S., López-Ortiz, A.: Online bin packing with advice. Algorithmica 74(1), 507–527 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coppersmith, D., Raghavan, P.: Multidimensional online bin packing: algorithms and worst case analysis. Oper. Res. Lett. 8(1), 17–20 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Epstein, L.: Online bin packing with cardinality constraints. SIAM J. Discret. Math. 20(4), 1015–1030 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Epstein, L., Imreh, C., Levin, A.: Class constrained bin packing revisited. Theoret. Comput. Sci. 411(34–36), 3073–3089 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Epstein, L., Levin, A.: On bin packing with conflicts. SIAM J. Optim. 19(3), 1270–1298 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Epstein, L., Levin, A.: Robust approximation schemes for cube packing. SIAM J. Optim. 23(2), 1310–1343 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Epstein, L., van Stee, R.: Online square and cube packing. Acta Inform. 41(9), 595–606 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fujiwara, H., Kobayashi, K.: Improved lower bounds for the online bin packing problem with cardinality constraints. J. Comb. Optim. 29(1), 67–87 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Heydrich, S., van Stee, R.: Improved lower bounds for online hypercube packing. The Computing Research Repository (CoRR) (2016).
  20. 20.
    Johnson, D.S.: Fast algorithms for bin packing. J. Comput. Syst. Sci. 8, 272–314 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L.: Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM J. Comput. 3, 256–278 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kellerer, H., Pferschy, U.: Cardinality constrained bin-packing problems. Ann. Oper. Res. 92, 335–348 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Krause, K.L., Shen, V.Y., Schwetman, H.D.: Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. J. ACM 22(4), 522–550 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liang, F.M.: A lower bound for on-line bin packing. Inf. Process. Lett. 10(2), 76–79 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Seiden, S.S.: On the online bin packing problem. J. ACM 49(5), 640–671 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Seiden, S.S., van Stee, R.: New bounds for multi-dimensional packing. Algorithmica 36(3), 261–293 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Shachnai, H., Tamir, T.: Tight bounds for online class-constrained packing. Theoret. Comput. Sci. 321(1), 103–123 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shachnai, H., Tamir, T.: Polynomial time approximation schemes for class-constrained packing problems. J. Sched. 4(6), 313–338 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ullman, J.D.: The performance of a memory allocation algorithm. Technical report 100, Princeton University, Princeton, NJ (1971)Google Scholar
  30. 30.
    van Vliet, A.: An improved lower bound for online bin packing algorithms. Inf. Process. Lett. 43(5), 277–284 (1992)CrossRefzbMATHGoogle Scholar
  31. 31.
    Xavier, E.C., Miyazawa, F.K.: The class constrained bin packing problem with applications to video-on-demand. Theoret. Comput. Sci. 393(1–3), 240–259 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yao, A.C.C.: New algorithms for bin packing. J. ACM 27, 207–227 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • János Balogh
    • 1
  • József Békési
    • 1
  • György Dósa
    • 2
  • Leah Epstein
    • 3
  • Asaf Levin
    • 4
    Email author
  1. 1.Department of Applied Informatics, Gyula Juhász Faculty of EducationUniversity of SzegedSzegedHungary
  2. 2.Department of MathematicsUniversity of PannoniaVeszpremHungary
  3. 3.Department of MathematicsUniversity of HaifaHaifaIsrael
  4. 4.Faculty of Industrial Engineering and ManagementThe TechnionHaifaIsrael

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