Abstract
In this paper, we investigate an open problem by Györgyi and Kis for a single-machine scheduling problem with a non-renewable resource (NR-SSP) and total weighted completion time criterion. The problem is NP-hard, even if every job has the same processing time, and each weight is equal to its required amount of the resource. Györgyi and Kis prove that a simple list scheduling algorithm for this special case is a 3-approximation algorithm and conjecture that the algorithm for the case is a 2-approximation algorithm. We prove their conjecture. More strongly, we show that the approximation ratio is strictly less than 2 for any instance, and for any \( \epsilon > 0 \), there exists an instance for which the approximation ratio is more than \( 2-\epsilon \).
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References
Bérczi, K., Király, T., & Omlor, S. (2020). Scheduling with non-renewable resources: Minimizing the sum of completion times. In International symposium on combinatorial optimization (pp. 167–178). Springer.
Briskorn, D., Jaehn, F., & Pesch, E. (2013). Exact algorithms for inventory constrained scheduling on a single machine. Journal of Scheduling, 16(1), 105–115.
Carlier, J., & Rinnooy Kan, A. H. G. (1982). Scheduling subject to nonrenewable-resource constraints. Operations Research Letters, 1(2), 52–55.
Gafarov, E. R., Lazarev, A. A., & Werner, F. (2011). Single machine scheduling problems with financial resource constraints: Some complexity results and properties. Mathematical Social Sciences, 62(1), 7–13.
Graham, R. L., Lawler, E. L., Lenstra, J. K., & Kan, A. R. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 287–326.
Grigoriev, A., Holthuijsen, M., & Van De Klundert, J. (2005). Basic scheduling problems with raw material constraints. Naval Research Logistics, 52(6), 527–535.
Györgyi, P., & Kis, T. (2015). Approximability of scheduling problems with resource consuming jobs. Annals of Operations Research, 235(1), 319–336.
Györgyi, P., & Kis, T. (2019). Minimizing total weighted completion time on a single machine subject to non-renewable resource constraints. Journal of Scheduling, 22(6), 623–634.
Györgyi, P., & Kis, T. (2020). New complexity and approximability results for minimizing the total weighted completion time on a single machine subject to non-renewable resource constraints. arXiv e-prints arxiv:1608.02792
Kis, T. (2015). Approximability of total weighted completion time with resource consuming jobs. Operations Research Letters, 43(6), 595–598.
Morsy, E., & Pesch, E. (2015). Approximation algorithms for inventory constrained scheduling on a single machine. Journal of Scheduling, 18(6), 645–653.
Slowiński, R. (1984). Preemptive scheduling of independent jobs on parallel machines subject to financial constraints. European Journal of Operational Research, 15(3), 366–373.
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This research is supported in part by Grant-in-Aid for Science Research (A) 19H00808 of Japan Society for the Promotion of Science.
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Hashimoto, S., Mizuno, S. A tight approximation ratio of a list scheduling algorithm for a single-machine scheduling problem with a non-renewable resource. J Sched 24, 259–267 (2021). https://doi.org/10.1007/s10951-021-00681-y
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DOI: https://doi.org/10.1007/s10951-021-00681-y