Scheduling With Release Times and Deadlines on A Minimum Number of Machines

  • Mark Cieliebak
  • Thomas Erlebach
  • Fabian Hennecke
  • Birgitta Weber
  • Peter Widmayer
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 155)


In this paper we study the SRDM problem motivated by a variety of practical applications. We are given n jobs with integer release times,deadlines, and processing times. The goal is to find a non-preemptive schedule such that all jobs meet their deadlines and the number of machines needed to process all jobs is minimum. If all jobs have equal release times and equal deadlines, SRDM is the classical bin packing problem, which is NP-complete. The slack of a job is the difference between its release time and the last possible time it may be started while still meeting its deadline. We show that instances consisting of jobs with slack at most one can be solved efficiently. We close the resulting gap by showing that the problem already becomes NP-complete if slacks up to 2 are allowed. Additionally, we consider several variants of the SRDM problem and provide exact and approximation algorithms.


Schedule Problem Approximation Algorithm Release Time Interval Graph Arbitrary Placement 
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]
    A. Bar-Noy, R. Bar-Yehuda, A. Freund, J.S. Naor, and B. Schieber. A unified approach to approximating resource allocation and scheduling. Journal of the ACM, 48(5): 1069–1090, 2001.CrossRefMathSciNetGoogle Scholar
  2. [2]
    A. Bar-Noy, S. Guha, J.S. Naor, and B. Schieber. Approximating the throughput of multiple machines in real-time scheduling. SIAM Journal on Computing, 31(2):331–352, 2001.MathSciNetGoogle Scholar
  3. [3]
    P. Berman and B. DasGupta. Multi-phase algorithms for throughput maximization for real-time scheduling. Journal of Combinatorial Optimization, 4(3):307–323, 2000.CrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Brandstädt, V.B. Le, and J.P. Spinrad. Graph Classes: a Survey. SIAM Monographs on Discrete Mathematics and Applications, 1999.Google Scholar
  5. [5]
    J. Chuzhoy and S. Naor. New hardness results for congestion minimization and machine scheduling. accepted for STOC’04, 2004.Google Scholar
  6. [6]
    J. Chuzhoy, R. Ostrovsky, and Y. Rabani. Approximation algorithms for the job interval selection problem and related scheduling problems. In IEEE Symposium on Foundations of Computer Science, pages 348–356, 2001.Google Scholar
  7. [7]
    M. Cieliebak, T. Erlebach, F. Hennecke, B. Weber, and P. Widmayer. Scheduling jobs on a minimum number of machines. Technical Report 419, Institute of Theoretical Computer Science, ETH Zürich, 2003.Google Scholar
  8. [8]
    E.G. Coffman Jr., M.R. Garey, and D.S. Johnson. Approximation algorithms for bin packing: A survey. In D. Hochbaum, editor, Approximation Algorithms for NP-hard Problems. PWS, 1996.Google Scholar
  9. [9]
    T. Erlebach and F.C.R. Spieksma. Interval selection: Applications, algorithms, and lower bounds. Journal of Algorithms, 46(1):27–53, 2003.CrossRefMathSciNetGoogle Scholar
  10. [10]
    M.R. Garey and D.S. Johnson. Computers and Intractability. W.H. Freeman and Company, New York, 1979.Google Scholar
  11. [11]
    E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys. Sequencing and scheduling: Algorithms and complexity. In S.C. Graves, A.H.G. Rinnooy Kan, and P. Zip kin, editors, Handbooks in Operations Research and Management Science, volume 4, pages 445–522. North-Holland, 1993.Google Scholar
  12. [12]
    F. Malucelli and S. Nicoloso. Shiftable interval graphs. In Proc. 6th International Conference on Graph Theory, 2000.Google Scholar
  13. [13]
    D.D. Sleator and R.E. Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences, 26(3):362–391, 1983.CrossRefMathSciNetGoogle Scholar
  14. [14]
    F.C.R. Spieksma. Approximating an interval scheduling problem. In International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, volume 1444, pages 169–180. Springer-Verlag LNCS, 1998.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2004

Authors and Affiliations

  • Mark Cieliebak
    • 1
  • Thomas Erlebach
    • 2
  • Fabian Hennecke
    • 1
  • Birgitta Weber
    • 1
  • Peter Widmayer
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland
  2. 2.Computer Engineering and Networks Laboratory (TIK)ETH ZurichZurichSwitzerland

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