Abstract
We consider the following scheduling problem. We have m identical machines, where each machine can accomplish one unit of work at each time unit. We have a set of n fully parallel jobs, where each job j has \(s_j\) units of workload, and each unit workload can be executed on any machine at any time unit. A job is considered complete when its entire workload has been executed. The objective is to find a schedule that minimizes the total weighted completion time \(\sum w_j C_j\), where \(w_j\) is the weight of job j and \(C_j\) is the completion time of job j. We provide theoretical results for this problem. First, we give a PTAS of this problem with fixed m. We then consider the special case where \(w_j = s_j\) for each job j, and we show that it is polynomial solvable with fixed m. Finally, we study the approximation ratio of a greedy algorithm, the Largest-Ratio-First algorithm. For the special case, we show that the approximation ratio depends on the instance size, i.e. n and m, while for the general case where jobs have arbitrary weights, we prove that the upper bound of the approximation ratio is \(1 + \frac{m-1}{m+2}\).
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Notes
One can always manipulate the order returned by the LRF algorithm by changing the weight slightly.
Without preemption and without idle time in order to preserve the order of jobs.
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Acknowledgements
We are grateful to Gruia Cǎlinescu for helpful discussions introducing the idea of the PTAS.
The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project Nos. CityU 11268616), by NSFC (Nos. 61433012, U1435215), and by a Shenzhen basic Research Grant JCYJ20160229195940462.
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Vincent Chau: Work done in City University of Hong Kong, Hong Kong, China.
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Wang, K., Chau, V. & Li, M. Scheduling fully parallel jobs. J Sched 21, 619–631 (2018). https://doi.org/10.1007/s10951-018-0563-3
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DOI: https://doi.org/10.1007/s10951-018-0563-3