Abstract
We consider the well known problem of scheduling jobs with release dates to minimize their average weighted completion time. When multiple machines are available, the machine environment may range from identical machines (the processing time required by a job is invariant across the machines) at one end of the spectrum to unrelated machines (the processing time required by a job on each machine is specified by an arbitrary vector) at the other end. While the problem is strongly NP-hard even in the case of a single machine, constant factor approximation algorithms are known for even the most general machine environment of unrelated machines. Recently a PTAS was discovered for the case of identical parallel machines [1]. In contrast, the problem is MAX SNP-hard for unrelated machines [11]. An important open problem was to determine the approximability of the intermediate case of uniformly related machines where each machine has a speed and it takes p=s time to process a job of size p on a machine with speed s. We resolve the complexity of this problem by obtaining a PTAS. This improves the earlier known approximation ratio of (2 + ∈).
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F. Afrati, E. Bampis, C. Chekuri, D. Karger, C. Kenyon, S. Khanna, I. Milis, M. Queyranne, M. Skutella, C. Stein, and M. Sviridenko. Approximation Schemes for Minimizing Average Weighted Completion Time with Release Dates. FOCS’ 99.
N. Alon, Y. Azar, G. J. Woeginger, and T. Yadid. Approximation schemes for scheduling on parallel machines. Journal of Scheduling, 1:55–66, 1998.
S. Chakrabarti, C. A. Phillips, A. S. Schulz, D. B. Shmoys, C. Stein, and J. Wein. Improved scheduling algorithms for minsum criteria. ICALP’ 96.
C. Chekuri and S. Khanna. A PTAS for the Multiple Knapsack Problem. SODA’ 00.
C. Chekuri, R. Motwani, B. Natarajan, and C. Stein. Approximation techniques for average completion time scheduling. SODA’ 97.
M. X. Goemans. Improved approximation algorithms for scheduling with release dates. SODA’ 97.
R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math., 5:287–326, 1979.
L. A. Hall, A. S. Schulz, D. B. Shmoys, and J. Wein. Scheduling to minimize average completion time: Offline and online algorithms. Math. of OR, 513-44,’ 97.
D. S. Hochbaum and D. B. Shmoys. Using dual approximation algorithms for scheduling problems: theoretical and practical results. JACM, 34:144–162, 1987.
D. S. Hochbaum and D. B. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM Journal on Computing, 17:539–551, 1988.
J. A. Hoogeveen, P. Schuurman, and G. J. Woeginger. Non-approximability results for scheduling problems with minsum criteria. IPCO’ 98.
J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343–362, 1977.
A. Munier, M. Queyranne, and A. S. Schulz. Approximation bounds for a general class of precedence constrained parallel machine scheduling problems. IPCO’ 98.
C. Phillips, C. Stein, and J. Wein. Minimizing average completion time in the presence of release dates. Mathematical Programming B, 82:199–223, 1998.
A. S. Schulz and M. Skutella. Scheduling-LPs bear probabilities: Randomized approximations for min-sum criteria. ESA’ 97.
M. Skutella and G. J. Woeginger. A PTAS for minimizing the weighted sum of job completion times on parallel machines. STOC’ 99.
W. E. Smith. Various optimizers for single-stage production. Naval Res. Logist. Quart., 3:59–66, 1956.
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Chekuri, C., Khanna, S. (2001). A PTAS for Minimizing Weighted Completion Time on Uniformly Related Machines. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48224-5_69
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DOI: https://doi.org/10.1007/3-540-48224-5_69
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