Scheduling Fully Parallel Jobs with Integer Parallel Units

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10185)


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 jobs, where each job j has \(s_j\) units of workload, and each unit workload could be executed on any machine at any time unit. A job is said completed when its whole 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 first give a PTAS of this problem when m is constant. Then we study the approximation ratio of a greedy algorithm, Largest-Ratio-First algorithm. Any permutation is a possible outcome of this algorithm when \(w_j = s_j\) for each job j, and for this special case we show that the approximation ratio depends on the instance size, i.e. n and m. Finally, when jobs have arbitrary weights, we prove that the upper bound of the approximation ratio is \(1 + \frac{m-1}{m+2}\).



We gratefully thank Gruia Cǎlinescu for our helpful discussions introducing the idea of the PTAS.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceCity University of Hong KongHong Kong SARChina

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