Abstract
In this paper, we consider the single machine scheduling problem with release dates and submodular penalties, in which each job can be either assigned to the machine or rejected. The objective is to minimize the sum of the makespan of the processed jobs and the penalty of the rejected jobs which is determined by a submodular function. First, we present a simple algorithm for the off-line problem. Second, for the on-line problem, we prove that there is no on-line algorithm with a constant competitive ratio if the penalty submodular function is not monotone, and present an on-line algorithm with a competitive ratio of 3 if the penalty submodular function is monotone. Finally, we consider a special case of the on-line problem in which all jobs have the same release date. We prove that there is no on-line algorithm with a competitive ratio of \(\frac{\sqrt{5}+1}{2}\approx 1.618\), and the competitive ratio of the on-line algorithm we presented is 2.
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Acknowledgements
The work is supported in part by the National Natural Science Foundation of China [grant No. 12071417], National Key R &D Program of China [grant No. 2020AAA 0105200], the Yunnan Provincial Research Foundation for Basic Research, China. [grant No. 202301AU070197], and the Project of Yunnan Provincial Department of Education Science Research Fund [grant No. 2023J0014].
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Liu, X., Xiao, M., Li, W. et al. Algorithms for single machine scheduling problem with release dates and submodular penalties. J Comb Optim 45, 105 (2023). https://doi.org/10.1007/s10878-023-01032-7
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DOI: https://doi.org/10.1007/s10878-023-01032-7