Probabilistic Analysis of the Dual Next-Fit Algorithm for Bin Covering

  • Carsten FischerEmail author
  • Heiko Röglin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


In the bin covering problem, the goal is to fill as many bins as possible up to a certain minimal level with a given set of items of different sizes. Online variants, in which the items arrive one after another and have to be packed immediately on their arrival without knowledge about the future items, have been studied extensively in the literature. We study the simplest possible online algorithm Dual Next-Fit, which packs all arriving items into the same bin until it is filled and then proceeds with the next bin in the same manner. The competitive ratio of this and any other reasonable online algorithm is 1 / 2.

We study Dual Next-Fit in a probabilistic setting where the item sizes are chosen i.i.d. according to a discrete distribution and we prove that, for every distribution, its expected competitive ratio is at least \(1/2 + \epsilon \) for a constant \(\epsilon >0\) independent of the distribution. We also prove an upper bound of 2 / 3 and better lower bounds for certain restricted classes of distributions. Finally, we prove that the expected competitive ratio equals, for a large class of distributions, the random-order ratio, which is the expected competitive ratio when adversarially chosen items arrive in uniformly random order.


Markov Chain Competitive Ratio Online Algorithm Discrete Distribution Large 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.


  1. 1.
    Asgeirsson, E.I., Stein, C.: Bounded-space online bin cover. J. Sched. 12(5), 461–474 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Assmann, S.F., Johnson, D.S., Kleitman, D.J., Leung, J.Y.-T.: On a dual version of the one-dimensional bin packing problem. J. Algorithms 5(4), 502–525 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Christ, M.G., Favrholdt, L.M., Larsen, K.S.: Online bin covering: expectations vs. guarantees. Theor. Comput. Sci. 556, 71–84 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Courcoubetis, C., Weber, R.R.: Stability of on-line bin packing with random arrivals and long-run average constraints. Probab. Eng. Informational Sci. 4(4), 447–460 (1990)CrossRefzbMATHGoogle Scholar
  5. 5.
    Csirik, J., Frenk, J.B.G., Galambos, G., Kan, A.H.G.R.: Probabilistic analysis of algorithms for dual bin packing problems. J. Algorithms 12(2), 189–203 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Csirik, J., Johnson, D.S., Kenyon, C.: Better approximation algorithms for bin covering. In: Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 557–566 (2001)Google Scholar
  7. 7.
    Csirik, J., Totik, V.: Online algorithms for a dual version of bin packing. Discrete Appl. Math. 21(2), 163–167 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coffman Jr., E.G., Csirik, J., Rónyai, L., Zsbán, A.: Random-order bin packing. Discrete Appl. Math. 156(6), 2810–2816 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fischer, C., Röglin, H.: Probabilistic analysis of the dual next-fit algorithm for bin covering, December 2015.
  10. 10.
    Jansen, K., Solis-Oba, R.: An asymptotic fully polynomial time approximation scheme for bin covering. Theor. Comput. Sci. 306(1–3), 543–551 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Coffman Jr., E.G., Csirik, J., Rónyai, L., Zsbán, A.: Random-order bin packing. Discrete Appl. Math. 156(14), 2810–2816 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kenyon, C.: Best-fit bin-packing with random order. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 359–364 (1996)Google Scholar
  13. 13.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. AMS (2009)Google Scholar
  14. 14.
    Lorden, G.: On excess over the boundary. Ann. Math. Stat. 41(2), 520–527 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Naaman, N., Rom, R.: Average case analysis of bounded space bin packing algorithms. Algorithmica 50, 72–97 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany

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