Abstract
In this paper, we discuss scheduling problems with m identical machines and n jobs where each job has to be assigned to some machine. The objective is to minimize the weighted makespan of jobs, i.e., the maximum weighted completion time of jobs. This scheduling problem is a generalization of minimizing the makespan on parallel machine scheduling problem. We present a (\(2-\frac{1}{m}\))-approximation algorithm and a randomized efficient polynomial time approximation scheme (EPTAS) for the problem. We also design a randomized fully polynomial time approximation scheme (FPTAS) for the special case when the number of machines is fixed.
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References
Afrati FN, Bampis E, Chekuri C, Karger DR, Kenyon C, Khanna S, Milis I, Queyranne M, Skutella M, Stein C, Sviridenko M (1999) Approximation schemes for minimizing average weighted completion time with release dates. In: Proceedings of the 40th annual IEEE symposium on foundations of computer science. IEEE Computer Society Press, Los Alamitos, CA, pp 32–43
Alon N, Azar Y, Woeginger GJ, Yadid T (1998) Approximation schemes for scheduling on parallel machines. J Sched 1(1):55–66
Bansal N, Srinivasan A, Svensson O (2016) Lift-and-round to improve weighted completion time on unrelated machines. In: Proceedings of 48th annual ACM symposium theory computing (STOC). ACM, pp 156–167
Baveja A, Qu X, Srinivasan A (2023) Approximating weighted completion time via stronger negative correlation. J Sched 1–10
Bleuse R, Kedad-Sidhoum S, Monna F, Mounié G, Trystram D (2015) Scheduling independent tasks on multi-cores with gpu accelerators. Concurr Comput Pract Exp 27(6):1625–1638
Chai X, Lu LF, Li WH, Zhang LQ (2018) Best-possible online algorithms for single machine scheduling to minimize the maximum weighted completion time. Asia-Pac J Oper Res 35:1850048
Feng Q, Yuan JJ (2007) NP-hardness of a multicriteria scheduling on two families of jobs. OR Trans 11:121–126
Gehrke JC, Jansen K, Kraft SE, Schikowski J (2018) A PTAS for scheduling unrelated machines of few different types. Int J Found Comput Sci 29(4):591–621
Graham RL (1966) Bounds for certain multiprocessing anomalies. Bell Syst Tech J 45(9):1563–1581
Graham RL (1969) Bounds on multiprocessing timing anomalies. SIAM J Appl Math 17(2):416–429
Hochbaum DS (1997) Various notions of approximations: good, better, best and more. In: Hochbaum DS (ed) Approximation algorithms. PWS Publishing Company, Boston
Hochbaum DS, Shmoys DB (1987) Using dual approximation algorithms for scheduling problems theoretical and practical results. J ACM 34(1):144–162
Imreh C (2003) Scheduling problems on two sets of identical machines. Computing 70(4):277–294
Im S, Shadloo M (2020) Weighted completion time minimization for unrelated machines via iterative fair contention resolution. In: Symposium on discrete algorithms, SODA, pp 2790–2809
Jansen K, Maack M (2019) An EPTAS for scheduling on unrelated machines of few different types. Algorithmica 81:4134–4164
Jansen K, Klein KM, Verschae J (2020) Closing the gap for makespan scheduling via sparsification techniques. Math Oper Res 45(4):1371–1392
Kawaguchi T, Kyan S (1986) Worst case bound of an LRF schedule for the mean weighted flow-time problem. SIAM J Comput 15(4):1119–1129
Kones I, Levin A (2019) A unified framework for designing EPTAS for load balancing on parallel machines. Algorithmica 81(7):3025–3046
Lenstra HW Jr (1983) Integer programming with a fixed number of variables. Math Oper Res 8(4):538–548
Li WJ (2015) A best possible online algorithm for the parallel-machine scheduling to minimize the maximum weighted completed time. Asia-Pac J Oper Res 32:1550030
Li S (2020) Scheduling to minimize total weighted completion time via time-indexed linear programming relaxations. SIAM J Comput 49(4):FOCS 17-409
Lu L, Zhang L, Ou J (2021) Single machine scheduling with rejection to minimize the weighted makespan. In: Wu W, Du H (eds) AAIM 2021, vol 13153. LNCS. Springer, Cham, pp 96–110
Phillips C, Stein C, Wein J (1998) Minimizing average completion time in the presence of release dates. Math Program 82:199–223
Raravi G, Nélis V (2012) A PTAS for assigning sporadic tasks on two-type heterogeneous multiprocessors. In: 2012 IEEE 33rd real-time systems symposium (RTSS). IEEE, pp 117–126
Sahni SK (1976) Algorithms for scheduling independent tasks. J ACM 23(1):116–127
Skutella M (2001) Convex quadratic and semidefinite programming relaxations in scheduling. J ACM 48(2):206–242
Skutella M, Woeginger GJ (2000) A PTAS for minimizing the total weighted completion time on identical parallel machines. Math Oper Res 25(1):63–75
Smith WE (1956) Various optimizers for single-stage production. Nav Res Logist Quart 3:59–66
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The work is supported in part by the National Natural Science Foundation of China [No. 12071417].
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Sun, R. Randomized approximation schemes for minimizing the weighted makespan on identical parallel machines. J Comb Optim 47, 34 (2024). https://doi.org/10.1007/s10878-024-01118-w
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DOI: https://doi.org/10.1007/s10878-024-01118-w