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Resource Scheduling with Supply Constraint and Linear Cost

  • Qiang Zhang
  • Weiwei Wu
  • Minming Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7402)

Abstract

We consider the following resource scheduling problem to minimize the total weighted completion time. There are m resources available at each time unit, and n jobs, each requiring an arbitrary number s i of resources. Each resource can only be assigned to one job. The objective is to find a schedule that minimizes ∑ w i c i , where w i is the weight/importance of job J i and c i is the time that job J i receives all resources it requires. We show this problem is NP-hard when m is the input of the problem. We then give a simple greedy algorithm with 2-approximation ratio. Finally, we present a polynomial time algorithm with complexity O(n d + 1) to solve this problem when the number of different resources requirements that are not multiples of m is at most d.

Keywords

Algorithms Machine scheduling Parallel tasks Supply allocation 

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References

  1. 1.
    Afrati, F., Bampis, E., Chekuri, C., Karger, D., Kenyon, C., Khanna, S., Milis, I., Queyranne, M., Skutella, M., Stein, C., Sviridenkom, M.: Approximation schemes for minimizing average weighted completion time with release dates. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–43 (1999)Google Scholar
  2. 2.
    Brucker, P., Drexl, A., Mohring, R., Neumann, K., Pesch, E.: Resource-constrained project scheduling: Notation, classification, models, and methods. European Journal of Operational Research 112(1), 3–41 (1999)zbMATHCrossRefGoogle Scholar
  3. 3.
    Conway, R., Maxwell, W., Miller, L.: Theory of scheduling. Dover Publications (2003)Google Scholar
  4. 4.
    Dobzinski, S., Schapira, M.: An improved approximation algorithm for combinatorial auctions with submodular bidders. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm, pp. 1064–1073 (2006)Google Scholar
  5. 5.
    Feige, U., Vondrak, J.: Approximation algorithms for allocation problems: Improving the factor of 1-1/e. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 667–676 (2006)Google Scholar
  6. 6.
    Fishkin, A., Jansen, K., Porkolab, L.: On minimizing average weighted completion time: A ptas for scheduling general multiprocessor tasks. In: Proceedings of the 13th International Symposium on Fundamentals of Computation Theory, pp. 495–507 (2001)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Complexity results for multiprocessor scheduling under resource constraints. SIAM Journal on Computing 4, 397 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  9. 9.
    Gonçalves, J., Mendes, J., Resende, M.: A genetic algorithm for the resource constrained multi-project scheduling problem. European Journal of Operational Research 189(3), 1171–1190 (2008)zbMATHCrossRefGoogle Scholar
  10. 10.
    Graham, R., Lawler, E., Lenstra, J., Kan, A.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Mathematics 5(2), 287–326 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Graves, S.C.: A review of production scheduling. Operations Research, 646–675 (1981)Google Scholar
  12. 12.
    Hartmann, S., Briskorn, D.: A survey of variants and extensions of the resource-constrained project scheduling problem. European Journal of Operational Research 207(1), 1–14 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Herroelen, W., Leus, R.: Project scheduling under uncertainty: Survey and research potentials. European Journal of Operational Research 165(2), 289–306 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kolisch, R.: Serial and parallel resource-constrained project scheduling methods revisited: Theory and computation. European Journal of Operational Research 90(2), 320–333 (1996)zbMATHCrossRefGoogle Scholar
  15. 15.
    Kolisch, R., Hartmann, S.: Experimental investigation of heuristics for resource-constrained project scheduling: An update. European Journal of Operational Research 174(1), 23–37 (2006)zbMATHCrossRefGoogle Scholar
  16. 16.
    Leung, J.Y.T.: Handbook of Scheduling: Algorithms, Models, and Performance Analysis. CRC Press (2004)Google Scholar
  17. 17.
    Mirrokni, V., Schapira, M., Vondrák, J.: Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions. In: Proceedings of the 9th ACM Conference on Electronic Commerce, pp. 70–77 (2008)Google Scholar
  18. 18.
    Nisan, N., Segal, I.: The communication requirements of efficient allocations and supporting prices. Journal of Economic Theory 129(1), 192–224 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Skutella, M., Woeginger, G.: A ptas for minimizing the weighted sum of job completion times on parallel machines. In: Proceedings of the 31th Annual ACM Symposium on Theory of Computing, pp. 400–407. ACM (1999)Google Scholar
  20. 20.
    Smith, W.E.: Various optimizers for single-stage production. Naval Research Logistics Quarterly 3(1-2), 59–66 (1956)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Qiang Zhang
    • 1
  • Weiwei Wu
    • 2
  • Minming Li
    • 1
  1. 1.Department of Computer ScienceCity University of Hong KongHong Kong SARChina
  2. 2.Division of Mathematical SciencesNanyang Technological UniversitySingapore

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