Online Splitting Interval Scheduling on m Identical Machines

  • Feifeng Zheng
  • Bo Liu
  • Yinfeng Xu
  • E. Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6124)


This paper investigates online scheduling on m identical machines with splitting intervals, i.e., intervals can be split into pieces arbitrarily and processed simultaneously on different machines. The objective is to maximize the throughput, i.e., the total length of satisfied intervals. Intervals arrive over time and the knowledge of them becomes known upon their arrivals. The decision on splitting and assignment for each interval is made irrecoverably upon its arrival. We first show that any non-split online algorithms cannot have bounded competitive ratios if the ratio of longest to shortest interval length is unbounded. Our main result is giving an online algorithm ES (for Equivalent Split) which has competitive ratio of 2 and \(\frac{2m-1}{m-1}\) for m = 2 and m ≥ 3, respectively. We further present a lower bound of \(\frac{m}{m-1}\), implying that ES is optimal as m = 2.


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  1. 1.
    Kolen, A.W.J., Lenstra, J.K., Papadimitriou, C.H., Spieksma, F.C.R.: Interval Scheduling: A survey. Naval Research Logistics 54, 530–543 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Lipton, R.J., Tomkins, A.: Online interval scheduling. In: Proceedings of Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1994), pp. 302–311 (1994)Google Scholar
  3. 3.
    Faigle, U., Garbe, R., Kern, W.: Randomized online algorithms for maximizing busy time interval scheduling. Computing, 95–104 (1996)Google Scholar
  4. 4.
    Faigle, U., Nawijn, W.M.: Note on scheduling intervals on-line. Discrete Applied Mathmatics 58, 13–17 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Carlisle, M.C., Lloyd, E.L.: On the k-coloring of intervals. Discrete Applied Mathmatics 59, 225–235 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Thibault, N., Laforest, C.: On-line time-constrained scheduling problem for the size on k machines. In: The 8th International Symposium on Parallel Architectures, Algorithms and Networks, pp. 20–24 (2005)Google Scholar
  7. 7.
    Woeginger, G.J.: On-line scheduling of jobs with fixed start and end times. Theoretical Computer Science 130, 5–16 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Canetti, R., Irani, S.: Bounding the power of preemption in randomized scheduling. SIAM Journal of Computing 27, 993–1015 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Seiden, S.S.: Randomized online interval scheduling. Operations Research Letters, 171–177 (1998)Google Scholar
  10. 10.
    Fung, S.P.Y., Poon, C.K., Zheng, F.F.: Online Interval Scheduling: Randomized and Multiprocessor Cases. Journal of Combinatorial Optimization 16, 248–262 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fung, S.P.Y., Poon, C.K., Zheng, F.F.: Improved Randomized Online Scheduling of Unit Length Intervals and Jobs. In: Bampis, E., Skutella, M. (eds.) WAOA 2008. LNCS, vol. 5426, pp. 53–66. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Xing, W., Zhang, J.: Parallel machine scheduling with splitting jobs. Discrete Applied Mathematics 103, 259–269 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kim, Y.D., Shim, S.O., Kim, S.B., Choi, Y.C., Yoon, H.: Parallel machine scheduling considering a job splitting property. International Journal of Production Research 42, 4531–4546 (2004)zbMATHCrossRefGoogle Scholar
  14. 14.
    Shim, S.O., Kim, Y.D.: A branch and bound algorithm for an identical parallel machine scheduling problem with a job splitting property. Computers and Operations Research 35, 863–875 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics 5, 287–326 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Borodin, A., El-yaniv, R.: Online computation and competitive analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Feifeng Zheng
    • 1
    • 2
    • 3
    • 4
  • Bo Liu
    • 1
  • Yinfeng Xu
    • 1
    • 2
    • 3
    • 4
  • E. Zhang
    • 5
  1. 1.School of ManagementXi’an JiaoTong UniversityXi’anChina
  2. 2.Ministry of Education Key Lab for Intelligent Networks and Network SecurityXi’anChina
  3. 3.Ministry of Education Key Lab for Process Control and Efficiency EngineeringXi’anChina
  4. 4.State Key Lab for Manufacturing Systems EngineeringXi’anChina
  5. 5.School of Information Management and EngineeringShanghai University of Finance and EconomicsShanghaiChina

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