Online Makespan Minimization with Parallel Schedules

Abstract

Online makespan minimization is a classical problem in which a sequence of jobs \(\sigma = J_1, \ldots , J_n\) has to be scheduled on m identical parallel machines so as to minimize the maximum completion time of any job. In this paper we investigate the problem with an essentially new model of resource augmentation. More specifically, an online algorithm is allowed to build several schedules in parallel while processing \(\sigma \). At the end of the scheduling process the best schedule is selected. This model can be viewed as providing an online algorithm with extra space, which is invested to maintain multiple solutions. The setting is of particular interest in parallel processing environments where each processor can maintain a single or a small set of solutions. As a main result we develop a \((4/3+\varepsilon )\)-competitive algorithm, for any \(0<\varepsilon \le 1\), that uses a constant number of schedules. The constant is equal to \(1/\varepsilon ^{O(\log (1/\varepsilon ))}\). We also give a \((1+\varepsilon )\)-competitive algorithm, for any \(0<\varepsilon \le 1\), that builds a polynomial number of \((m/\varepsilon )^{O(\log (1/\varepsilon ) / \varepsilon )}\) schedules. This value depends on m but is independent of the input \(\sigma \). The performance guarantees are nearly best possible. We show that any algorithm that achieves a competitiveness smaller than 4 / 3 must construct \(\Omega (m)\) schedules. Our algorithms make use of novel guessing schemes that (1) predict the optimum makespan of a job sequence \(\sigma \) to within a factor of \(1+\varepsilon \) and (2) guess the job processing times and their frequencies in \(\sigma \). In (2) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant. We remark that the competitive ratios achieved using parallel schedules are considerably smaller than those in the standard problem without resource augmentation. Furthermore they are at least as good and in most cases better than the ratios obtained with other means of resource augmentation for makespan minimization.

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Acknowledgments

We thank two anonymous referees for helpful comments improving the presentation of this paper.

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Correspondence to Susanne Albers.

Additional information

A preliminary version of this paper has appeared in Proc. 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), 2014.

S. Albers: Work supported by the German Research Foundation, project Al 464/7-1.

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Albers, S., Hellwig, M. Online Makespan Minimization with Parallel Schedules. Algorithmica 78, 492–520 (2017). https://doi.org/10.1007/s00453-016-0172-5

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Keywords

  • Scheduling
  • Makespan minimization
  • Online algorithms
  • Competitive analysis