Abstract
In this paper, we study a combination problem of parallel machine scheduling and the s–t path problem, which is to find a s–t path \(P_{st}\) of the given directed graph, and to schedule the jobs corresponding to the arcs of the path \(P_{st}\) on m parallel machines, such that the makespan is minimized. It has been proved that this problem is NP-hard and admits 2-approximation algorithm. We present a polynomial-time algorithm with approximation ratio 1.5. By modifying the dynamic programming method for the restricted shortest path problem, we also give a polynomial time approximation scheme.
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Acknowledgements
We are grateful to the anonymous referees for numerous helpful comments and suggestions which helped to improve the presentation of our work. The work is supported in part by the National Natural Science Foundation of China [Nos. 61662088, 11761078, 11861075], Program for Excellent Young Talents of Yunnan University, Training Program of National Science Fund for Distinguished Young Scholars, IRTSTYN, and Key Joint Project of Yunnan Province Science and Technology Department and Yunnan University [No. 2018FY001(-014)].
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Guan, L., Li, J., Li, W. et al. Improved approximation algorithms for the combination problem of parallel machine scheduling and path. J Comb Optim 38, 689–697 (2019). https://doi.org/10.1007/s10878-019-00406-0
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DOI: https://doi.org/10.1007/s10878-019-00406-0