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Approximation Algorithm for the Single Machine Scheduling Problem with Release Dates and Submodular Rejection Penalty

Author

Listed:
  • Xiaofei Liu

    (School of Electronic Engineering and Computer Science, Peking University, Beijing 100871, China)

  • Weidong Li

    (School of Mathematics and Statistics, Yunnan University, Kunming 650504, China)

Abstract

In this paper, we consider the single machine scheduling problem with release dates and nonmonotone submodular rejection penalty. We are given a single machine and multiple jobs with probably different release dates and processing times. For each job, it is either accepted and processed on the machine or rejected. The objective is to minimize the sum of the makespan of the accepted jobs and the rejection penalty of the rejected jobs which is determined by a nonmonotone submodular function. We design a combinatorial algorithm based on the primal-dual framework to deal with the problem, and study its property under two cases. For the general case where the release dates can be different, the proposed algorithm have an approximation ratio of 2. When all the jobs release at the same time, the proposed algorithm becomes a polynomial-time exact algorithm.

Suggested Citation

  • Xiaofei Liu & Weidong Li, 2020. "Approximation Algorithm for the Single Machine Scheduling Problem with Release Dates and Submodular Rejection Penalty," Mathematics, MDPI, vol. 8(1), pages 1-11, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:133-:d:309327
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    References listed on IDEAS

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    1. E. L. Lawler, 1973. "Optimal Sequencing of a Single Machine Subject to Precedence Constraints," Management Science, INFORMS, vol. 19(5), pages 544-546, January.
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    3. Cheng He & Joseph Y.-T. Leung & Kangbok Lee & Michael L. Pinedo, 2016. "Improved algorithms for single machine scheduling with release dates and rejections," 4OR, Springer, vol. 14(1), pages 41-55, March.
    4. Xueling Zhong & Jinwen Ou, 2017. "Improved approximation algorithms for parallel machine scheduling with release dates and job rejection," 4OR, Springer, vol. 15(4), pages 387-406, December.
    5. Dvir Shabtay & Nufar Gaspar & Liron Yedidsion, 2012. "A bicriteria approach to scheduling a single machine with job rejection and positional penalties," Journal of Combinatorial Optimization, Springer, vol. 23(4), pages 395-424, May.
    6. Akiyoshi Shioura & Natalia V. Shakhlevich & Vitaly A. Strusevich, 2016. "Application of Submodular Optimization to Single Machine Scheduling with Controllable Processing Times Subject to Release Dates and Deadlines," INFORMS Journal on Computing, INFORMS, vol. 28(1), pages 148-161, February.
    7. Liqi Zhang & Lingfa Lu, 2016. "Parallel-machine scheduling with release dates and rejection," 4OR, Springer, vol. 14(2), pages 165-172, June.
    8. Ou, Jinwen & Zhong, Xueling & Wang, Guoqing, 2015. "An improved heuristic for parallel machine scheduling with rejection," European Journal of Operational Research, Elsevier, vol. 241(3), pages 653-661.
    9. Weidong Li & Qianna Cui, 2018. "Vector scheduling with rejection on a single machine," 4OR, Springer, vol. 16(1), pages 95-104, March.
    10. Zhang, Liqi & Lu, Lingfa & Yuan, Jinjiang, 2009. "Single machine scheduling with release dates and rejection," European Journal of Operational Research, Elsevier, vol. 198(3), pages 975-978, November.
    11. Xueling Zhong & Zhangming Pan & Dakui Jiang, 2017. "Scheduling with release times and rejection on two parallel machines," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 934-944, April.
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    Cited by:

    1. Hongye Zheng & Suogang Gao & Wen Liu & Weili Wu & Ding-Zhu Du & Bo Hou, 2022. "Approximation algorithm for the parallel-machine scheduling problem with release dates and submodular rejection penalties," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 343-353, August.
    2. Xiaofei Liu & Man Xiao & Weidong Li & Yaoyu Zhu & Lei Ma, 2023. "Algorithms for single machine scheduling problem with release dates and submodular penalties," Journal of Combinatorial Optimization, Springer, vol. 45(4), pages 1-18, May.

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