Abstract
We propose a related machine scheduling problem in which the processing times of jobs are given and known, but the speeds of machines are variables and must satisfy a system of linear constraints. The objective is to decide the speeds of machines and minimize the makespan of the schedule among all the feasible choices. The problem is motivated by some practical application scenarios. This problem is strongly NP-hard in general, and we discuss various cases of it. In particular, we obtain polynomial time algorithms for two special cases. If the number of constraints is more than one and the number of machines is a fixed constant, then we give a \((2+\epsilon )\)-approximation algorithm. For the case where the number of machines is an input of the problem instance, we propose several approximation algorithms, and obtain a polynomial time approximation scheme when the number of distinct machine speeds is a fixed constant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burer S, Letchford AN (2012) Non-convex mixed-integer nonlinear programming: a survey. Surv Oper Res Manag Sci 17:97–106
Chen B, Potts CN, Woeginger GJ (1998) A review of machine scheduling: complexity, algorithms and approximability. In: Du D, Pardalos PM (eds) Handbook of combinatorial optimization. Springer, Boston, pp 1493–1641
Cho Y, Sahni S (1980) Bounds for list schedules on uniform processors. SIAM J Comput 9:91–103
Daniels RL, Kouvelis P (1995) Robust scheduling to hedge against processing time uncertainty in single-stage production. Manag Sci 41:363–376
Dobson G (1984) Scheduling independent tasks on uniform processors. SIAM J Comput 13:705–716
Friesen DK (1987) Tighter bounds for LPT scheduling on uniform processors. SIAM J Comput 16:554–560
Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Co., San Francisco
Gonzalez T, Ibarra OH, Sahni S (1977) Bounds for LPT schedules on uniform processors. SIAM J Comput 6:155–166
Hochbaum DS, Shmoys DB (1987) Using dual approximation algorithms for scheduling problems: theoretical and practical results. J ACM 34:144–162
Hochbaum DS, Shmoys DB (1988) A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM J Comput 17:539–551
Köppe M (2012) On the complexity of nonlinear mixed-integer optimization. In: Lee J, Leyffer S (eds) Mixed integer nonlinear programming. The IMA volumes in mathematics and its applications, vol 154. Springer, New York, pp 533–557
Kovács A (2006) Tighter approximation bounds for LPT scheduling in two special cases. In Calamoneri T, Finocchi I, Italiano G F (eds) Algorithms and complexity: 6th Italian conference, CIAC 2006. Lecture Notes in Computer Science, vol 3998. Springer, Berlin, pp 187–198
Kovács A (2010) New approximation bounds for LPT scheduling. Algorithmica 57:413–433
Kunde M (1982) A multifit algorithm for uniform multiprocessor scheduling. In: Cremers AB, Kriegel HP (eds) Theoretical computer science. Lecture Notes in Computer Science, vol 145. Springer, Berlin, pp 175–185
Lenstra JK, Shmoys DB, Tardos É (1990) Approximation algorithms for scheduling unrelated parallel machines. Math Program 46:259–271
Möhring RH, Radermacher FJ, Weiss G (1984) Stochastic scheduling problems I—general strategies. Zeitschrift für Oper Res 28:193–260
Nip K, Wang Z (2019) Two-machine flow shop scheduling problem under linear constraints. In: Li Y, Cardei M, Huang Y (eds) The 13th annual international conference on combinatorial optimization and applications (COCOA 2019). Lecture Notes in Computer Science, vol 11949. Springer Nature, Switzerland AG, pp 400–411
Nip K, Wang Z, Wang Z (2016) Scheduling under linear constraints. Eur J Oper Res 253:290–297
Nip K, Wang Z, Wang Z (2017) Knapsack with variable weights satisfying linear constraints. J Global Optim 69:713–725
Nip K, Wang Z, Shi T (2019) Some graph optimization problems with weights satisfying linear constraints. In: Li Y, Cardei M, Huang Y (eds) The 13th annual international conference on combinatorial optimization and applications (COCOA 2019). Lecture Notes in Computer Science, vol 11949. Springer Nature, Switzerland AG, pp 412–424
Wang Z, Nip K (2017) Bin packing under linear constraints. J Comb Optim 34:1198–1209
Acknowledgements
This work has been supported by National Natural Science Foundation of China Nos. 11801589, 11771245 and 11371216. We also thank Tianning Shi for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A preliminary version of this paper has appeared in D. Kim et al. (Eds.): COCOA 2018, LNCS 11346, pp. 314–328, 2018.
Rights and permissions
About this article
Cite this article
Zhang, S., Nip, K. & Wang, Z. Related machine scheduling with machine speeds satisfying linear constraints. J Comb Optim 44, 1724–1740 (2022). https://doi.org/10.1007/s10878-020-00523-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-020-00523-1