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An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities

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Abstract

We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \).

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Acknowledgements

We would like to thank the associate editor and an anonymous referee for their constructive comments and kind suggestions. This work was supported by NSFC(11301528), NSF-Jiangsu(BK20130169), and NSFC(61573362). Li was also supported by NSFC(11401605) and NSFC(11501279), and Yuan was also supported by NSFC(11671368) and NSF-Henan(15IRTSTHN006).

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Authors and Affiliations

  1. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Ruyan Fu & Ji Tian

  2. Department of Information and Computation Science, Zhongyuan University of Technology, Zhengzhou, 450007, Henan, People’s Republic of China

    Shisheng Li

  3. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, Henan, People’s Republic of China

    Jinjiang Yuan

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  1. Ruyan Fu
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  2. Ji Tian
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  3. Shisheng Li
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  4. Jinjiang Yuan
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Correspondence to Jinjiang Yuan.

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Fu, R., Tian, J., Li, S. et al. An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities. J Comb Optim 34, 1187–1197 (2017). https://doi.org/10.1007/s10878-017-0139-8

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  • DOI: https://doi.org/10.1007/s10878-017-0139-8

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