## 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

- 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. - Brucker, P. (2010).
*Scheduling algorithms*(5th ed.). Berlin: Springer.Google Scholar - 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 - 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 - 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 - 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 - 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. - 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 - 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 - 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 - Roemer, T. A. (2006). A note on the complexity of the concurrent open shop problem.
*Journal Scheduling*,*9*(4), 389–396.CrossRefGoogle Scholar - Sahni, S. (1976). Algorithms for scheduling independent tasks.
*Journal of the ACM (JACM)*,*23*(1), 116–127.CrossRefGoogle Scholar - 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 - 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 - Smith, W. E. (1956). Various optimizers for single-stage production.
*Naval Research Logistics Quarterly*,*3*(1–2), 59–66.CrossRefGoogle Scholar - 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 - 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