Abstract
This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least \(1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}\), where \(x=\sqrt{2m+4}\). We then present a greedy algorithm with tight competitive ratio of \(1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}\). The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when \(m=2\), we propose an optimal online algorithm with competitive ratio of \(16\) \(/\) \(13\), which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., \(n\rightarrow \infty \).
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Acknowledgments
This work is supported by the National Natural Science Foundation of China (11571013, 11471286, 11571252) and Zhejiang Province Natural Science Foundation of China (LY14A010031, LY16A010015).
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Hu, J., Jiang, Y., Zhou, P. et al. Total completion time minimization in online hierarchical scheduling of unit-size jobs. J Comb Optim 33, 866–881 (2017). https://doi.org/10.1007/s10878-016-0011-2
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DOI: https://doi.org/10.1007/s10878-016-0011-2