• DOI:10.1007/s10107-022-01829-0
• Corpus ID: 233875282

Speed-robust scheduling: sand, bricks, and rocks

@article{Eberle2020SpeedrobustSS,
  title={Speed-robust scheduling: sand, bricks, and rocks},
  author={Franziska Eberle and Ruben Hoeksma and Nicole Megow and Lukas N{\"o}lke and Kevin Schewior and Bertrand Simon},
  journal={Mathematical Programming},
  year={2020},
  volume={197},
  pages={1009 - 1048}
}

• Franziska Eberle, R. Hoeksma, +3 authors B. Simon
• Published 10 November 2020
• Geology
• Mathematical Programming

The speed-robust scheduling problem is a two-stage problem where, given m machines, jobs must be grouped into at most m bags while the processing speeds of the machines are unknown. After the speeds are revealed, the grouped jobs must be assigned to the machines without being separated. To evaluate the performance of algorithms, we determine upper bounds on the worst-case ratio of the algorithm’s makespan and the optimal makespan given full information. We refer to this ratio as the robustness factor. We give an algorithm with a robustness factor 2-1m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2-\frac{1}{m}$$\end{document} for the most general setting and improve this to 1.8 for equal-size jobs. For the special case of infinitesimal jobs, we give an algorithm with an optimal robustness factor equal to ee-1≈1.58\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{e}{e-1}\approx 1.58$$\end{document}. The particular machine environment in which all machines have either speed 0 or 1 was studied before by Stein and Zhong (ACM Trans Algorithms 16(1):1-20, 2020. https://doi.org/10.1145/3340320). For this setting, we provide an algorithm for scheduling infinitesimal jobs with an optimal robustness factor of 1+22≈1.207\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1+\sqrt{2}}{2}\approx 1.207$$\end{document}. It lays the foundation for an algorithm matching the lower bound of 43\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{4}{3}$$\end{document} for equal-size jobs.