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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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