Advertisement

Online Interval Coloring with Packing Constraints

  • Leah Epstein
  • Meital Levy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)

Abstract

We study online interval coloring problems with bandwidth. We are interested in some variants motivated by bin packing problems. Specifically we consider open-end coloring, cardinality constrained coloring, coloring with vector constraints and finally a combination of both the cardinality and the vector constraints. We construct competitive algorithms for each of the variants. Additionally, we present a lower bound of 24/7 for interval coloring with bandwidth, which holds for all the above models, and improves the current lower bound for the standard interval coloring with bandwidth.

Keywords

Competitive Ratio Online Algorithm Interval Graph Cardinality Constraint Full Color 
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.
    Adamy, U., Erlebach, T.: Online coloring of intervals with bandwidth. In: Solis-Oba, R., Jansen, K. (eds.) WAOA 2003. LNCS, vol. 2909, pp. 1–12. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Babel, L., Chen, B., Kellerer, H., Kotov, V.: Algorithms for on-line bin-packing problems with cardinality constraints. Discrete Applied Mathematics 143(1-3), 238–251 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blitz, D., van Vliet, A., Woeginger, G.J.: Lower bounds on the asymptotic worst-case ratio of online bin packing algorithms. Unpublished manuscript (1996)Google Scholar
  4. 4.
    Caprara, A., Kellerer, H., Pferschy, U.: Approximation schemes for ordered vector packing problems. Naval Research Logistics 92, 58–69 (2003)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chrobak, M., Ślusarek, M.: On some packing problems relating to dynamical storage allocation. RAIRO Journal on Information Theory and Applications 22, 487–499 (1988)zbMATHGoogle Scholar
  6. 6.
    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, pp. 147–177 (1998)Google Scholar
  7. 7.
    Epstein, L., Levy, M.: Online interval coloring and variants. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 602–613. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Galambos, G., Kellerer, H., Woeginger, G.J.: A lower bound for online vector packing algorithms. Acta Cybernetica 10, 23–34 (1994)MathSciNetGoogle Scholar
  9. 9.
    Garey, M.R., Graham, R.L., Johnson, D.S., Yao, A.C.C.: Resource constrained scheduling as generalized bin packing. Journal of Combinatorial Theory (Series A) 21, 257–298 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jensen, T.R., Toft, B.: Graph coloring problems. Wiley, Chichester (1995)zbMATHGoogle Scholar
  11. 11.
    Johnson, D.S.: Near-optimal bin packing algorithms. PhD thesis, MIT, Cambridge, MA (1973)Google Scholar
  12. 12.
    Kierstead, H.A.: The linearity of first-fit coloring of interval graphs. SIAM Journal on Discrete Mathematics 1(4), 526–530 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kierstead, H.A., Qin, J.: Coloring interval graphs with First-Fit. SIAM Journal on Discrete Mathematics 8, 47–57 (1995)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Kierstead, H.A., Trotter, W.T.: An extremal problem in recursive combinatorics. Congressus Numerantium 33, 143–153 (1981)MathSciNetGoogle Scholar
  15. 15.
    Kou, L.T., Markowsky, G.: Multidimensional bin packing algorithms. IBM Journal on Research and Development 21, 443–448 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Krause, K.L., Shen, V.Y., Schwetman, H.D.: Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. Journal of the ACM 22(4), 522–550 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Krause, K.L., Shen, V.Y., Schwetman, H.D.: Errata: Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. Journal of the ACM 24(3), 527–527 (1977)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Narayanaswamy, N.S.: Dynamic storage allocation and on-line colouring interval graphs. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 329–338. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Pemmaraju, S., Raman, R., Varadarajan, K.: Buffer minimization using max-coloring. In: Proc. of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), pp. 562–571 (2004)Google Scholar
  20. 20.
    Ullman, J.D.: The performance of a memory allocation algorithm. Technical Report 100, Princeton University, Princeton, NJ (1971)Google Scholar
  21. 21.
    Yang, J., Leung, J.Y.-T.: The ordered open-end bin-packing problem. Oper. Res. 51(5), 759–770 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Leah Epstein
    • 1
  • Meital Levy
    • 2
  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.School of Computer ScienceTel-Aviv UniversityIsrael

Personalised recommendations