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Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times

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Abstract

We revisit the classic problem of preemptive scheduling on m uniformly related machines. In this problem, jobs can be arbitrarily split into parts, under the constraint that every job is processed completely, and that the parts of a job are not assigned to run in parallel on different machines. We study a new objective which is motivated by fairness, where the goal is to minimize the sum of the two maximal job completion times. We design a polynomial time algorithm for computing an optimal solution. The algorithm can act on any set of machine speeds and any set of input jobs. The algorithm has several cases, many of which are very different from algorithms for makespan minimization (algorithms that minimize the maximum completion time of any job), and from algorithms that minimize the total completion time of all jobs.

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Notes

  1. The number of parts for each job must be finite, and the parts must have positive sizes.

  2. A short summary of the action of the algorithm of Ebenlendr and Sgall (2009) is as follows. It keeps an infinite number of machines that can be used during the time [0, T). The machines have time-dependent speeds. The actual machines are initialized with their actual speeds, while dummy machines are initialized with zero speeds. Each job is assigned to run during [0, T), possibly on several machines, with the convention that running on a machine of speed zero during some time means not running at all at that time. By assigning a job, two machines are merged into what will be seen as one machine (this merged machine usually will not have a constant speed).

  3. The action of the function can be seen as solving a linear program, but since the program is simple, we solve it directly. We do not use this property (that a linear program is being solved) in the proof, and thus we do not describe the function as such.

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Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Haifa, Israel

    Leah Epstein & Ido Yatsiv

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  1. Leah Epstein
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  2. Ido Yatsiv
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Correspondence to Leah Epstein.

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Epstein, L., Yatsiv, I. Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times. J Sched 20, 115–127 (2017). https://doi.org/10.1007/s10951-016-0476-y

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