ESA 2012: Algorithms – ESA 2012 pp 84-95

# On the Value of Job Migration in Online Makespan Minimization

• Susanne Albers
• Matthias Hellwig
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)

## Abstract

Makespan minimization on identical parallel machines is a classical scheduling problem. We consider the online scenario where a sequence of n jobs has to be scheduled non-preemptively on m machines so as to minimize the maximum completion time of any job. The best competitive ratio that can be achieved by deterministic online algorithms is in the range [1.88,1.9201]. Currently no randomized online algorithm with a smaller competitiveness is known, for general m.

In this paper we explore the power of job migration, i.e. an online scheduler is allowed to perform a limited number of job reassignments. Migration is a common technique used in theory and practice to balance load in parallel processing environments. As our main result we settle the performance that can be achieved by deterministic online algorithms. We develop an algorithm that is αm-competitive, for any m ≥ 2, where αm is the solution of a certain equation. For m = 2, α2 = 4/3 and limm → ∞ αm = W− 1( − 1/e2)/(1 + W− 1( − 1/e2)) ≈ 1.4659. Here W− 1 is the lower branch of the Lambert W function. For m ≥ 11, the algorithm uses at most 7m migration operations. For smaller m, 8m to 10m operations may be performed. We complement this result by a matching lower bound: No online algorithm that uses o(n) job migrations can achieve a competitive ratio smaller than αm. We finally trade performance for migrations. We give a family of algorithms that is c-competitive, for any 5/3 ≤ c ≤ 2. For c = 5/3, the strategy uses at most 4m job migrations. For c = 1.75, at most 2.5m migrations are used.

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### References

1. 1.
Aggarwal, G., Motwani, R., Zhu, A.: The load rebalancing problem. Journal of Algorithms 60(1), 42–59 (2006)
2. 2.
Albers, S.: Better bounds for online scheduling. SIAM J. Comput. 29, 459–473 (1999)
3. 3.
Bartal, Y., Karloff, H., Rabani, Y.: A better lower bound for on-line scheduling. Infomation Processing Letters 50, 113–116 (1994)
4. 4.
Bartal, Y., Fiat, A., Karloff, H., Vohra, R.: New algorithms for an ancient scheduling problem. Journal of Computer and System Sciences 51, 359–366 (1995)
5. 5.
Cao, Q., Liu, Z.: Online scheduling with reassignment on two uniform machines. Theoretical Computer Science 411(31-33), 2890–2898 (2010)
6. 6.
Chen, X., Lan, Y., Benko, A., Dósa, G., Han, X.: Optimal algorithms for online scheduling with bounded rearrangement at the end. Theoretical Computer Science 412(45), 6269–6278 (2011)
7. 7.
Chen, B., van Vliet, A., Woeginger, G.J.: A lower bound for randomized on-line scheduling algorithms. Information Processing Letters 51, 219–222 (1994)
8. 8.
Chen, B., van Vliet, A., Woeginger, G.J.: A optimal algorithm for preemptive online scheduling. Operations Research Letters 18, 127–131 (1995)
9. 9.
Dósa, G., Epstein, L.: Preemptive Online Scheduling with Reordering. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 456–467. Springer, Heidelberg (2009)
10. 10.
Englert, M., Özmen, D., Westermann, M.: The power of reordering for online minimum makespan scheduling. In: Proc. 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 603–612 (2008)Google Scholar
11. 11.
Faigle, U., Kern, W., Turan, G.: On the performance of on-line algorithms for partition problems. Acta Cybernetica 9, 107–119 (1989)
12. 12.
Fleischer, R., Wahl, M.: Online scheduling revisited. Journal of Scheduling 3, 343–353 (2000)
13. 13.
Galambos, G., Woeginger, G.: An on-line scheduling heuristic with better worst case ratio than Graham’s list scheduling. SIAM J. on Computing 22, 349–355 (1993)
14. 14.
Graham, R.L.: Bounds for certain multi-processing anomalies. Bell System Technical Journal 45, 1563–1581 (1966)Google Scholar
15. 15.
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM Journal of Applied Mathematics 17(2), 416–429 (1969)
16. 16.
Gormley, T., Reingold, N., Torng, E., Westbrook, J.: Generating adversaries for request-answer games. In: Proc. 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 564–565 (2000)Google Scholar
17. 17.
Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: Theoretical and practical results. Journal of the ACM 34, 144–162 (1987)
18. 18.
Karger, D.R., Phillips, S.J., Torng, E.: A better algorithm for an ancient scheduling problem. Journal of Algorithms 20, 400–430 (1996)
19. 19.
Min, X., Liu, J., Wang, Y.: Optimal semi-online algorithms for scheduling problems with reassignment on two identical machines. Information Processing Letters 111(9), 423–428 (2011)
20. 20.
Rudin III, J.F.: Improved bounds for the on-line scheduling problem. Ph.D. Thesis. The University of Texas at Dallas (May 2001)Google Scholar
21. 21.
Sanders, P., Sivadasan, N., Skutella, M.: Online scheduling with bounded migration. Mathematics of Operations Research 34(2), 481–498 (2009)
22. 22.
Sgall, J.: A lower bound for randomized on-line multiprocessor scheduling. Information Processing Letters 63, 51–55 (1997)
23. 23.
Tan, Z., Yu, S.: Online scheduling with reassignment. Operations Research Letters 36(2), 250–254 (2008)
24. 24.
Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Communications of the ACM 28, 202–208 (1985)