Advertisement

Scheduling fully parallel jobs

  • Kai Wang
  • Vincent Chau
  • Minming Li
Article
  • 109 Downloads

Abstract

We consider the following scheduling problem. We have m identical machines, where each machine can accomplish one unit of work at each time unit. We have a set of n fully parallel jobs, where each job j has \(s_j\) units of workload, and each unit workload can be executed on any machine at any time unit. A job is considered complete when its entire workload has been executed. The objective is to find a schedule that minimizes the total weighted completion time \(\sum w_j C_j\), where \(w_j\) is the weight of job j and \(C_j\) is the completion time of job j. We provide theoretical results for this problem. First, we give a PTAS of this problem with fixed m. We then consider the special case where \(w_j = s_j\) for each job j, and we show that it is polynomial solvable with fixed m. Finally, we study the approximation ratio of a greedy algorithm, the Largest-Ratio-First algorithm. For the special case, we show that the approximation ratio depends on the instance size, i.e. n and m, while for the general case where jobs have arbitrary weights, we prove that the upper bound of the approximation ratio is \(1 + \frac{m-1}{m+2}\).

Keywords

Approximation ratio Total weighted completion time Parallel jobs Integer parallel units 

Notes

Acknowledgements

We are grateful to Gruia Cǎlinescu for helpful discussions introducing the idea of the PTAS.

The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project Nos. CityU 11268616), by NSFC (Nos. 61433012, U1435215), and by a Shenzhen basic Research Grant JCYJ20160229195940462.

References

  1. Afrati, F. N., Bampis, E., Chekuri, C., Karger, D. R., Kenyon, C., Khanna, S., Milis, I., Queyranne, M., Skutella, M., Stein, C., & Sviridenko, M. (1999). Approximation schemes for minimizing average weighted completion time with release dates. In 40th IEEE computer society annual symposium on foundations of computer science, FOCS (pp. 32–44). http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6604.
  2. Brucker, P. (2010). Scheduling algorithms (5th ed.). Berlin: Springer.Google Scholar
  3. Bruno, J., Coffman, E. G, Jr., & Sethi, R. (1974). Scheduling independent tasks to reduce mean finishing time. Communications of the ACM, 17(7), 382–387.CrossRefGoogle Scholar
  4. Fishkin, A. V., Jansen, K., & Porkolab, L. (2001). On minimizing average weighted completion time: A PTAS for scheduling general multiprocessor tasks. In 13th FCT 2001, Springer, LNCS, (vol. 2138, pp. 495–507).Google Scholar
  5. Garg, N., Kumar, A., & Pandit, V. (2007). Order scheduling models: Hardness and algorithms. In FSTTCS 2007, Springer, LNCS, (vol. 4855, pp 96–107).Google Scholar
  6. Hendel, Y., Kubiak, W., & Trystram, D. (2015). Scheduling semi-malleable jobs to minimize mean flow time. Journal of Scheduling, 18(4), 335–343.CrossRefGoogle Scholar
  7. Kalaitzis, C., Svensson, O., & Tarnawski, J. (2017). Unrelated machine scheduling of jobs with uniform smith ratios. In Proceedings of the twenty-eighth annual ACM-SIAM symposium on discrete algorithms, society for industrial and applied mathematics, Philadelphia, PA, USA, SODA ’17 (pp. 2654–2669) http://dl.acm.org/citation.cfm?id=3039686.3039861.
  8. Kawaguchi, T., & Kyan, S. (1986). Worst case bound of an LRF schedule for the mean weighted flow-time problem. SIAM Journal on Computing, 15(4), 1119–1129.CrossRefGoogle Scholar
  9. Leung, J. Y., Li, H., & Pinedo, M. (2005). Order scheduling in an environment with dedicated resources in parallel. Journal of Scheduling, 8(5), 355–386.CrossRefGoogle Scholar
  10. Mastrolilli, M., Queyranne, M., Schulz, A. S., Svensson, O., & Uhan, N. A. (2010). Minimizing the sum of weighted completion times in a concurrent open shop. Operations Research Letters, 38(5), 390–395.CrossRefGoogle Scholar
  11. Roemer, T. A. (2006). A note on the complexity of the concurrent open shop problem. Journal Scheduling, 9(4), 389–396.CrossRefGoogle Scholar
  12. Sahni, S. (1976). Algorithms for scheduling independent tasks. Journal of the ACM (JACM), 23(1), 116–127.CrossRefGoogle Scholar
  13. Schulz, A.S., & Skutella, M. (1997). Scheduling-LPs bear probabilities: Randomized approximations for min-sum criteria. In 5th Annual European symposium algorithms - ESA ’97, Springer, LNCS (vol. 1284, pp 416–429).Google Scholar
  14. Skutella, M., & Woeginger, G.J. (1999). A PTAS for minimizing the weighted sum of job completion times on parallel machines. In Proceedings of the thirty-first annual ACM STOC, ACM (pp. 400–407).Google Scholar
  15. Smith, W. E. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3(1–2), 59–66.CrossRefGoogle Scholar
  16. Sung, C. S., & Yoon, S. H. (1998). Minimizing total weighted completion time at a pre-assembly stage composed of two feeding machines. International Journal of Production Economics, 54(3), 247–255.CrossRefGoogle Scholar
  17. Zhang, Q., Wu, W., & Li, M. (2013). Minimizing the total weighted completion time of fully parallel jobs with integer parallel units. Theoretical Computer Science, 507, 34–40.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceCity University of Hong KongHong KongChina
  2. 2.Shenzhen Institutes of Advanced TechnologyShenzhenChina

Personalised recommendations