On the Value of Job Migration in Online Makespan Minimization
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 α m -competitive, for any m ≥ 2, where α m is the solution of a certain equation. For m = 2, α 2 = 4/3 and lim m → ∞ α m = W − 1( − 1/e 2)/(1 + W − 1( − 1/e 2)) ≈ 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 α m . 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.
KeywordsCompetitive Ratio Online Algorithm Online Schedule Makespan Minimization Loaded Machine
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- 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
- 14.Graham, R.L.: Bounds for certain multi-processing anomalies. Bell System Technical Journal 45, 1563–1581 (1966)Google Scholar
- 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
- 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