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Approximation algorithms for the multiprocessor scheduling with submodular penalties

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Abstract

In this paper, we consider multiprocessor scheduling with submodular penalties to extend multiprocessor scheduling with rejection to submodular function. An instance of the problem is given by n jobs and m machines with each job having a certain processing time on a machine. We aim to find a subset R of rejected jobs, and assign each of other jobs to one of the m machines. The objective is to minimize the sum of the makespan of the m machines and the rejection penalty R, where the rejection penalty is determined by a submodular function. For this problem, we design a non-combinatorial Lovász rounding algorithm that achieves a worst-case guarantee of \(\frac{3+\sqrt{5}}{2}\). Then, we consider a special case of this problem in which all the machines are identical, i.e. each job has the same processing time on any machine, and we design a combinatorial \((2-\frac{1}{m})\)-approximation algorithm based on the greedy method and list scheduling (LS) algorithm.

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Acknowledgements

The work is supported in part by the Program for Excellent Young Talents of Yunnan University, Training Program of National Science Fund for Distinguished Young Scholars, IRTSTYN, and Key Joint Project of the Science and Technology Department of Yunnan Province and Yunnan University [No. 2018FY001(-014)].

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Authors and Affiliations

  1. School of Electronic Engineering and Computer Science, Peking University, Beijing, China

    Xiaofei Liu

  2. School of Mathematics and Statistics, Yunnan University, Kunming, China

    Weidong Li

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  1. Xiaofei Liu
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  2. Weidong Li
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Correspondence to Xiaofei Liu.

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Liu, X., Li, W. Approximation algorithms for the multiprocessor scheduling with submodular penalties. Optim Lett 15, 2165–2180 (2021). https://doi.org/10.1007/s11590-021-01724-1

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