Fast Divide-and-Conquer Algorithms for Preemptive Scheduling Problems with Controllable Processing Times – A Polymatroid Optimization Approach

  • Natalia V. Shakhlevich
  • Akiyoshi Shioura
  • Vitaly A. Strusevich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5193)


We consider a variety of preemptive scheduling problems with controllable processing times on a single machine and on identical/uniform parallel machines, where the objective is to minimize the total compression cost. In this paper, we propose fast divide-and-conquer algorithms for these scheduling problems. Our approach is based on the observation that each scheduling problem we discuss can be formulated as a polymatroid optimization problem. We develop a novel divide-and-conquer technique for the polymatroid optimization problem and then apply it to each scheduling problem. We show that each scheduling problem can be solved in \( \O({\rm T}_{\rm feas}(n) \times\log n)\) time by using our divide-and-conquer technique, where n is the number of jobs and Tfeas(n) denotes the time complexity of the corresponding feasible scheduling problem with n jobs. This approach yields faster algorithms for most of the scheduling problems discussed in this paper.


Stein Sorting 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Orlin, J.B., Stein, C., Tarjan, R.E.: Improved algorithms for bipartite network flow. SIAM J. Comput. 23, 906–933 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chung, J.Y., Shih, W.-K., Liu, J.W.S., Gillies, D.W.: Scheduling imprecise computations to minimize total error. Microprocessing and Microprogramming 27, 767–774 (1989)CrossRefGoogle Scholar
  3. 3.
    Federgruen, A., Groenevelt, H.: Preemptive scheduling of uniform machines by ordinary network flow techniques. Management Sci. 32, 341–349 (1986)MATHMathSciNetGoogle Scholar
  4. 4.
    Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Elsevier, Amsterdam (2005)MATHGoogle Scholar
  5. 5.
    Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18, 30–55 (1989)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gonzales, T.F., Sahni, S.: Preemptive scheduling of uniform processor systems. J. ACM 25, 92–101 (1978)CrossRefGoogle Scholar
  7. 7.
    Hochbaum, D.S., Shamir, R.: Minimizing the number of tardy job unit under release time constraints. Discrete Appl. Math. 28, 45–57 (1990)MATHCrossRefGoogle Scholar
  8. 8.
    Horn, W.: Some simple scheduling algorithms. Naval Res. Logist. Quat. 21, 177–185 (1974)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Janiak, A., Kovalyov, M.Y.: Single machine scheduling with deadlines and resource dependent processing times. European J. Oper. Res. 94, 284–291 (1996)MATHCrossRefGoogle Scholar
  10. 10.
    Leung, J.Y.-T.: Minimizing total weighted error for imprecise computation tasks and related problems. In: Leung, J.Y.-T. (ed.) Handbook of Scheduling, ch. 34. Chapman & Hall, Boca Raton (2004)Google Scholar
  11. 11.
    Leung, J.Y.-T., Yu, V.K.M., Wei, W.-D.: Minimizing the weighted number of tardy task units. Discrete Appl. Math. 51, 307–316 (1994)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    McCormick, S.T.: Fast algorithms for parametric scheduling come from extensions to parametric maximum flow. Oper. Res. 47, 744–756 (1999)MATHMathSciNetGoogle Scholar
  13. 13.
    McNaughton, R.: Scheduling with deadlines and loss functions. Management Sci. 12, 1–12 (1959)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nowicki, E., Zdrzałka, S.: A bicriterion approach to preemptive scheduling of parallel machines with controllable job processing times. Discrete Appl. Math. 63, 237–256 (1995)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Sahni, S.: Preemptive scheduling with due dates. Oper. Res. 27, 925–934 (1979)MATHMathSciNetGoogle Scholar
  16. 16.
    Sahni, S., Cho, Y.: Scheduling independent tasks with due times on a uniform processor system. J. ACM 27, 550–563 (1980)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Shakhlevich, N.V., Strusevich, V.A.: Preemptive scheduling problems with controllable processing times. J. Sched. 8, 233–253 (2005)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Shakhlevich, N.V., Strusevich, V.A.: Preemptive scheduling on uniform parallel machines with controllable job processing times. Algorithmica 51, 451–473 (2008)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Shih, W.-K., Lee, C.-R., Tang, C.-H.: A fast algorithm for scheduling imprecise computations with timing constraints to minimize weighted error. In: 21th IEEE Real-Time Syst. Symp., pp. 305–310. IEEE Computer Society, Los Alamitos (2000)CrossRefGoogle Scholar
  20. 20.
    Shih, W.-K., Liu, J.W.S., Chung, J.-Y.: Algorithms for scheduling imprecise computations with timing constraints. SIAM J. Comput. 20, 537–552 (1991)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Shih, W.-K., Liu, J.W.S., Chung, J.-Y., Gillies, D.W.: Scheduling tasks with ready times and deadlines to minimize average error. ACM SIGOPS Oper. Syst., Rev. 23, 14–28 (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Natalia V. Shakhlevich
    • 1
  • Akiyoshi Shioura
    • 2
  • Vitaly A. Strusevich
    • 3
  1. 1.School of ComputingUniversity of LeedsLeedsU.K.
  2. 2.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  3. 3.Department of Mathematical SciencesUniversity of GreenwichLondonU.K.

Personalised recommendations