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Tight lower bounds for semi-online scheduling on two uniform machines with known optimum

Author

Listed:
  • György Dósa

    (University of Pannonia)

  • Armin Fügenschuh

    (Brandenburgische Technische Universität Cottbus-Senftenberg)

  • Zhiyi Tan

    (Zhejiang University)

  • Zsolt Tuza

    (University of Pannonia
    Hungarian Academy of Sciences)

  • Krzysztof Węsek

    (Helmut Schmidt University/University of the Federal Armed Forces Hamburg
    Warsaw University of Technology)

Abstract

This problem is about scheduling a number of jobs on two uniform machines with given speeds 1 and $$s\ge 1$$ s ≥ 1 , so that the overall finishing time, i.e., the makespan, is earliest possible. We consider a semi-online variant (introduced for equal speeds) by Azar and Regev, where the jobs are arriving one after the other, while the scheduling algorithm knows the optimum value of the corresponding offline problem. One can ask how close any possible algorithm could get to the optimum value, that is, to give a lower bound on the competitive ratio: the supremum over ratios between the value of the solution given by the algorithm and the optimal offline solution. We contribute to this question by constructing tight lower bounds for all values of s in the intervals $$[\frac{1+\sqrt{21}}{4},\frac{3+\sqrt{73}}{8}]\approx [1.3956,1.443]$$ [ 1 + 21 4 , 3 + 73 8 ] ≈ [ 1.3956 , 1.443 ] and $$[\frac{5}{3},\frac{4+\sqrt{133}}{9}]\approx [\frac{5}{3},1.7258]$$ [ 5 3 , 4 + 133 9 ] ≈ [ 5 3 , 1.7258 ] , except a very narrow interval.

Suggested Citation

  • György Dósa & Armin Fügenschuh & Zhiyi Tan & Zsolt Tuza & Krzysztof Węsek, 2019. "Tight lower bounds for semi-online scheduling on two uniform machines with known optimum," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(4), pages 1107-1130, December.
  • Handle: RePEc:spr:cejnor:v:27:y:2019:i:4:d:10.1007_s10100-018-0536-9
    DOI: 10.1007/s10100-018-0536-9
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    References listed on IDEAS

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    1. Martin Böhm & Jiří Sgall & Rob Stee & Pavel Veselý, 2017. "A two-phase algorithm for bin stretching with stretching factor 1.5," Journal of Combinatorial Optimization, Springer, vol. 34(3), pages 810-828, October.
    2. Martin Böhm & Jiří Sgall & Rob Stee & Pavel Veselý, 2017. "Online bin stretching with three bins," Journal of Scheduling, Springer, vol. 20(6), pages 601-621, December.
    3. György Dósa & M. Grazia Speranza & Zsolt Tuza, 2011. "Two uniform machines with nearly equal speeds: unified approach to known sum and known optimum in semi on-line scheduling," Journal of Combinatorial Optimization, Springer, vol. 21(4), pages 458-480, May.
    4. György Dósa & Armin Fügenschuh & Zhiyi Tan & Zsolt Tuza & Krzysztof Węsek, 2018. "Tight upper bounds for semi-online scheduling on two uniform machines with known optimum," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(1), pages 161-180, March.
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    Cited by:

    1. Feifeng Zheng & Yuhong Chen & Ming Liu & Yinfeng Xu, 2022. "Competitive analysis of online machine rental and online parallel machine scheduling problems with workload fence," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 1060-1076, September.

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