Theory of Computing Systems

, Volume 46, Issue 2, pp 246–260 | Cite as

Class Constrained Bin Covering

Article

Abstract

We study the following variant of the bin covering problem. We are given a set of unit sized items, where each item has a color associated with it. We are given an integer parameter k≥1 and an integer bin size Bk. The goal is to assign the items (or a subset of the items) into a maximum number of subsets of at least B items each, such that in each such subset the total number of distinct colors of items is at least k. We study both the offline and the online variants of this problem. We first design an optimal polynomial time algorithm for the offline problem. For the online problem we give a lower bound of 1+Hk−1 (where Hk−1 denotes the (k−1)-th harmonic number), and an O(k)-competitive algorithm. Finally, we analyze the performance of the natural heuristic First fit.

Keywords

Bin covering Online algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Assmann, S.F.: Problems in discrete applied mathematics. Ph.D.Tthesis, Mathematics Department, Massachusetts Institute of Technology, Cambridge, MA (1983) Google 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, 502–525 (1984) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Babel, L., Chen, B., Kellerer, H., Kotov, V.: Algorithms for on-line bin-packing problems with cardinality constraints. Discrete Appl. Math. 143(1–3), 238–251 (2004) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998) MATHGoogle Scholar
  5. 5.
    Caprara, A., Kellerer, H., Pferschy, U.: Approximation schemes for ordered vector packing problems. Nav. Res. Logist. 92, 58–69 (2003) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Coffman, E.G., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: A survey. In: Hochbaum, D. (ed.) Approximation Algorithms. PWS-Kent, Boston (1997) Google Scholar
  7. 7.
    Coffman, E.G. Jr., Csirik, J.: Performance guarantees for one-dimensional bin packing. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics, Chap. 32, pp. 32-1–32-18. Chapman & Hall/CRC, Boca Raton (2007) Google Scholar
  8. 8.
    Csirik, J., Johnson, D.S., Kenyon, C.: Better approximation algorithms for bin covering. In: Proc. of the 12th Annual Symposium on Discrete Algorithms (SODA2001), pp. 557–566 (2001) Google Scholar
  9. 9.
    Csirik, J., Leung, J.Y.-T.: Variable-sized bin packing and bin covering. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics, Chap. 34, pp. 34-1–34-11. Chapman & Hall/CRC, Boca Raton (2007) Google Scholar
  10. 10.
    Csirik, J., Leung, J.Y.-T.: Variants of classical one-dimensional bin packing. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics, Chap. 33, pp. 33-1–33-13. Chapman & Hall/CRC, Boca Raton (2007) Google Scholar
  11. 11.
    Csirik, J., Totik, V.: On-line algorithms for a dual version of bin packing. Discrete Appl. Math. 21, 163–167 (1988) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Csirik, J., Woeginger, G.J.: On-line packing and covering problems. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms: The State of the Art, Chap. 7, pp. 147–177. Springer, Berlin (1998) CrossRefGoogle Scholar
  13. 13.
    Epstein, L., Imreh, C., Levin, A.: Bin covering with cardinality constraints. Manuscript (2007) Google Scholar
  14. 14.
    Golubchik, L., Khanna, S., Khuller, S., Thurimella, R., Zhu, A.: Approximation algorithms for data placement on parallel disks. In: Proc. of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2000), pp. 223–232 (2000) Google Scholar
  15. 15.
    Jansen, K., Solis-Oba, R.: An asymptotic fully polynomial time approximation scheme for bin covering. Theor. Comput. Sci. 306(1–3), 543–551 (2003) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Krumke, S.O., de Paepe, W., Rambau, J., Stougie, L.: Online bin coloring. In: Proc. of the 9th Annual European Symposium on Algorithms (ESA2001), pp. 74–85 (2001) Google Scholar
  17. 17.
    Shachnai, H., Tamir, T.: On two class-constrained versions of the multiple knapsack problem. Algorithmica 29(3), 442–467 (2001) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Shachnai, H., Tamir, T.: Polynomial time approximation schemes for class-constrained packing problems. J. Sched. 4(6), 313–338 (2001) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Shachnai, H., Tamir, T.: Tight bounds for online class-constrained packing. Theor. Comput. Sci. 321(1), 103–123 (2004) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Xavier, E.C., Miyazawa, F.K.: The class constrained bin packing problem with applications to video-on-demand. In: Proc. of the 12th Annual International Conference on Computing and Combinatorics (COCOON 2006), pp. 439–448 (2006) Google Scholar
  21. 21.
    Yao, A.C.C.: Probabilistic computations: towards a unified measure of complexity. In: Proc. of the 18th Symposium on Foundations of Computer Science (FOCS’77), pp. 222–227 (1977) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Department of InformaticsUniversity of SzegedSzegedHungary
  3. 3.Department of StatisticsThe Hebrew UniversityJerusalemIsrael

Personalised recommendations