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Scheduling Fully Parallel Jobs with Integer Parallel Units

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Theory and Applications of Models of Computation (TAMC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10185))

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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 jobs, where each job j has \(s_j\) units of workload, and each unit workload could be executed on any machine at any time unit. A job is said completed when its whole 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 first give a PTAS of this problem when m is constant. Then we study the approximation ratio of a greedy algorithm, Largest-Ratio-First algorithm. Any permutation is a possible outcome of this algorithm when \(w_j = s_j\) for each job j, and for this special case we show that the approximation ratio depends on the instance size, i.e. n and m. Finally, when jobs have arbitrary weights, we prove that the upper bound of the approximation ratio is \(1 + \frac{m-1}{m+2}\).

The work described in this paper was fully supported by a grant from Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11268616).

The original version of this chapter was revised. The erratum to this chapter is available at https://doi.org/10.1007/978-3-319-55911-7_50

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Notes

  1. 1.

    Without preemption and without idle time in order to preserve the order of jobs.

  2. 2.

    One can always manipulate the order returned by LRF algorithm by changing the weight a little bit.

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Acknowledgement

We gratefully thank Gruia CÇŽlinescu for our helpful discussions introducing the idea of the PTAS.

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Authors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Hong Kong SAR, China

    Vincent Chau, Minming Li & Kai Wang

Authors
  1. Vincent Chau
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  2. Minming Li
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  3. Kai Wang
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Corresponding author

Correspondence to Kai Wang .

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Editors and Affiliations

  1. Anna University, Chennai, India

    T.V. Gopal

  2. Universität Bern , Bern, Switzerland

    Gerhard Jäger

  3. Universität Bern , Bern, Switzerland

    Silvia Steila

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Chau, V., Li, M., Wang, K. (2017). Scheduling Fully Parallel Jobs with Integer Parallel Units. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_11

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  • DOI: https://doi.org/10.1007/978-3-319-55911-7_11

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  • Online ISBN: 978-3-319-55911-7

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