Abstract
A two-agent scheduling problem on parallel machines is considered. Our objective is to minimize the makespan for agent A, subject to an upper bound on the makespan for agent B. When the number of machines, denoted by \(m\), is chosen arbitrarily, we provide an \(O(n)\) algorithm with performance ratio \(2-\frac{1}{m}\), i.e., the makespan for agent A given by the algorithm is no more than \(2-\frac{1}{m}\) times the optimal value, while the makespan for agent B is no more than \(2-\frac{1}{m}\) times the threshold value. This ratio is proved to be tight. Moreover, when \(m=2\), we present an \(O(nlogn)\) algorithm with performance ratio \(\frac{1+\sqrt{17}}{4}\approx 1.28\) which is smaller than \(\frac{3}{2}\). The ratio is weakly tight.
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Acknowledgments
The authors would like to thank anonymous referees for their helpful comments and suggestions, which significantly improve the results and presentation of this paper. This research is supported by NSFC (11371137) and the Fund for the Doctoral Program of China (20120074110021).
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Zhao, K., Lu, X. Two approximation algorithms for two-agent scheduling on parallel machines to minimize makespan. J Comb Optim 31, 260–278 (2016). https://doi.org/10.1007/s10878-014-9744-y
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DOI: https://doi.org/10.1007/s10878-014-9744-y