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A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: few machines, small looseness, and small slack

Author

Listed:
  • René Bevern

    (Novosibirsk State University)

  • Rolf Niedermeier

    (TU Berlin)

  • Ondřej Suchý

    (Czech Technical University in Prague)

Abstract

We study the problem of non-preemptively scheduling n jobs, each job j with a release time $$t_j$$ t j , a deadline $$d_j$$ d j , and a processing time $$p_j$$ p j , on m parallel identical machines. Cieliebak et al. (2004) considered the two constraints $$|d_j-t_j|\le \lambda {}p_j$$ | d j - t j | ≤ λ p j and $$|d_j-t_j|\le p_j +\sigma $$ | d j - t j | ≤ p j + σ and showed the problem to be NP-hard for any $$\lambda >1$$ λ > 1 and for any $$\sigma \ge 2$$ σ ≥ 2 . We complement their results by parameterized complexity studies: we show that, for any $$\lambda >1$$ λ > 1 , the problem remains weakly NP-hard even for $$m=2$$ m = 2 and strongly W[1]-hard parameterized by m. We present a pseudo-polynomial-time algorithm for constant m and $$\lambda $$ λ and a fixed-parameter tractability result for the parameter m combined with $$\sigma $$ σ .

Suggested Citation

  • René Bevern & Rolf Niedermeier & Ondřej Suchý, 2017. "A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: few machines, small looseness, and small slack," Journal of Scheduling, Springer, vol. 20(3), pages 255-265, June.
    • Handle: RePEc:spr:jsched:v:20:y:2017:i:3:d:10.1007_s10951-016-0478-9
    DOI: 10.1007/s10951-016-0478-9
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    References listed on IDEAS

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    1. Sevastianov, S. V., 2005. "An introduction to multi-parameter complexity analysis of discrete problems," European Journal of Operational Research, Elsevier, vol. 165(2), pages 387-397, September.
    2. Federico Malucelli & Sara Nicoloso, 2007. "Shiftable intervals," Annals of Operations Research, Springer, vol. 150(1), pages 137-157, March.
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    Citations

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    Cited by:

    1. Matthias Bentert & Robert Bredereck & Péter Györgyi & Andrzej Kaczmarczyk & Rolf Niedermeier, 2023. "A multivariate complexity analysis of the material consumption scheduling problem," Journal of Scheduling, Springer, vol. 26(4), pages 369-382, August.
    2. Danny Hermelin & Shlomo Karhi & Michael Pinedo & Dvir Shabtay, 2021. "New algorithms for minimizing the weighted number of tardy jobs on a single machine," Annals of Operations Research, Springer, vol. 298(1), pages 271-287, March.
    3. de Weerdt, Mathijs & Baart, Robert & He, Lei, 2021. "Single-machine scheduling with release times, deadlines, setup times, and rejection," European Journal of Operational Research, Elsevier, vol. 291(2), pages 629-639.
    4. Hermelin, Danny & Kubitza, Judith-Madeleine & Shabtay, Dvir & Talmon, Nimrod & Woeginger, Gerhard J., 2019. "Scheduling two agents on a single machine: A parameterized analysis of NP-hard problems," Omega, Elsevier, vol. 83(C), pages 275-286.
    5. Danny Hermelin & Dvir Shabtay & Chen Zelig & Michael Pinedo, 2022. "A general scheme for solving a large set of scheduling problems with rejection in FPT time," Journal of Scheduling, Springer, vol. 25(2), pages 229-255, April.
    6. Klaus Heeger & Danny Hermelin & George B. Mertzios & Hendrik Molter & Rolf Niedermeier & Dvir Shabtay, 2023. "Equitable scheduling on a single machine," Journal of Scheduling, Springer, vol. 26(2), pages 209-225, April.
    7. Hermelin, Danny & Pinedo, Michael & Shabtay, Dvir & Talmon, Nimrod, 2019. "On the parameterized tractability of single machine scheduling with rejection," European Journal of Operational Research, Elsevier, vol. 273(1), pages 67-73.
    8. S. Bessy & R. Giroudeau, 2019. "Parameterized complexity of a coupled-task scheduling problem," Journal of Scheduling, Springer, vol. 22(3), pages 305-313, June.

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