Advertisement

Job Shop Scheduling with Unit Length Tasks: Bounds and Algorithms

  • Juraj Hromkovič
  • Kathleen Steinhöfel
  • Peter Widmayer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2202)

Abstract

We consider the job shop scheduling problem unit-J m, where each job is processed once on each of m given machines. The execution of anyt ask on its corresponding machine takes exactly one time unit. The objective is to minimize the overall completion time, called makespan. The contribution of this paper are the following results: (i) For anyi nput instance of unit-J m with d jobs, the makespan of an optimum schedule is at most m+o(m), for d = o(m 1/2). For d = o(m 1/2), this improves on the upper bound O(m+d) given in [LMR99] with a constant equal to two as shown in [S98]. For d = 2 the upper bound is improved to \( m + \left\lceil {\sqrt m } \right\rceil \). (ii) There exist input instances of unit-J m with d = 2 such that the makespan of an optimum schedule is at least \( m + \left\lceil {\sqrt m } \right\rceil \), i.e., the result (i) cannot be improved for d = 2. (iii) We present a randomized on-line approximation algorithm for unit-J m with the best known approximation ratio for d = o(m 1/2). (iv) A deterministic approximation algorizhm for unit-J m is described that works in quadratic time for constant d and has an approximation ratio of \( 1 + 2^d /\left\lfloor {\sqrt m } \right\rfloor \) for d ≤ 2 log2 m.

Keywords

Optimum Schedule Approximation Ratio Competitive Ratio Additional Delay Relative Delay 
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. [B88]
    Brucker, P.: An Efficient Algorithm for the Job Shop Problem with Two Jobs. Computing, 40:353–359, 1988.MATHCrossRefMathSciNetGoogle Scholar
  2. [FS98]
    Feige U., Scheideler, C.: Improved Bounds for Acyclic Job Shop Scheduling. Proc. 28th ACM Symposium on Theory of Computing, pp. 624–233, 1998.Google Scholar
  3. [GPSS97]
    Goldberg, L. A., Paterson, M., Srinivasan, A., Sweedyk, E.: Better Approximation Guarantees for Job-shop Scheduling. Proc. 8th ACM-SIAM Symposium on Discrete Algorithms, pp. 599–608, 1997.Google Scholar
  4. [LMR94]
    Leighton, F. T., Maggs, B. M., Rao, S. B.: Packet Routing and Job-Shop Scheduling in O(Congestion+Dilation) steps. Combinatorica, 14:167–186, 1994.MATHCrossRefMathSciNetGoogle Scholar
  5. [LMR99]
    Leighton, F. T., Maggs, B. M., Richa, A. W.: Fast Algorithms for Finding O(Congestion+Dilation) Packet Routing Schedules. Combinatorica, 19:375–401, 1999.MATHCrossRefMathSciNetGoogle Scholar
  6. [LR79]
    Lenstra, J. K., Rinnooy Kan A. H. G.: Computational Complexity of Discrete Optimization Problems. Annals of Discrete Mathematics, 4:121–140, 1979.MATHCrossRefMathSciNetGoogle Scholar
  7. [S98]
    Scheideler, C.: Universal Routing Strategies for Interconnection Networks. Lecture Notes in Computer Science 1390, Springer Verlag, 1998.Google Scholar
  8. [SSW94]
    Shmoys, D. B., Stein, C., Wein, J.: Improved Approximation Algorithms for Shop Scheduling Problems. SIAM J. on Computing, 23:617–632, 1994.MATHCrossRefMathSciNetGoogle Scholar
  9. [WHH97]
    Williamson, D. P., Hall, L. A., Hoogeveen, J. A., Hurkens, C. A. J., Lenstra, J. K., Sevast’janov, S. V., Shmoys, D. B.: Short Shop Schedules. Operations Research, 45:288–294, 1997.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Juraj Hromkovič
    • 1
  • Kathleen Steinhöfel
    • 2
  • Peter Widmayer
    • 3
  1. 1.Lehrstuhl für Informatik IRWTH AachenAachenGermany
  2. 2.GMD - Forschungszentrum Informationstechnik GmbHBerlinGermany
  3. 3.Institut für Theoretische InformatikETH ZürichZürichSwitzerland

Personalised recommendations