Abstract
We consider the problem of scheduling n jobs withrelease dates on m identical parallel machines to minimize the average completion time of the jobs. We prove that the ratio of the average completion time of the optimal nonpreemptive schedule to that of the optimal preemptive schedule is at most 7/3, improving a bound of \((3 - \frac{1}{m})\)Shmoys and Wein.
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S. Chakrabarti, C. Phillips, A.S. Schulz, D.B. Shmoys, C. Stein, and J. Wein, “Improved Approximation Algorithms for Minsum Criteria,” in Proc. of the 1996 International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 1099, Springer-Verlag, Berlin, 1996, pp. 646–657.
C. Chekuri, R. Motwani, B. Natarajan, and C. Stein, “Approximation Techniques for Average Completion Time Scheduling,” in Proc. of the 8th ACM-SIAM Symposium on Discrete Algorithms, 1997, pp. 609–618.
F.A. Chudak and D.B. Shmoys, “Approximation Algorithms for Precedence-Constrained Scheduling Problems on Parallel Machines that Run at Different Speeds,” in Proc. of the 8th ACM-SIAM Symposium on Discrete Algorithms, 1997, pp. 581–590.
E.G. Coffman, Jr., and M.R. Garey, “Proof of the 4/3 Conjecture for Preemptive vs. Nonpreemptive Two-Processor Scheduling,” in Proc. of the 23rd Annual ACM Symposium on Theory of Computing, 1991, pp. 241–248.
M. Goemans, “A Supermodular Relaxation for Scheduling with Release Dates,” in Proc. of the 5th MPS Conference on Integer Programming and Combinatorial Optimization, 1996, pp. 288–300. (Published as Lecture Notes in Computer Science 1084, Springer-Verlag.)
M. Goemans, “Improved Approximation Algorithms for Scheduling with Release Dates,” in Proc. of the 8th ACM-SIAM Symposium on Discrete Algorithms, 1997, pp. 591–598.
D.K. Goyal, “Nonpreemptive Scheduling of Unequal Execution Time Tasks on Two Identical Processors,” Technical Report CS-77-039, Washington State University, Pullman, WA, 1977.
L.A. Hall, D.B. Shmoys, and J. Wein, “Scheduling to Minimize Average Completion Time: Off-line and On-line Algorithms,” in Proc. of the 7th ACM-SIAM Symposium on Discrete Algorithms, 1996, pp. 142–151.
L.A. Hall, A.S. Schulz, D.B. Shmoys, and J. Wein, “Scheduling to minimize average completion time: Off-line and on-line approximation algorithms,” Mathematics of Operations Research, vol. 22, pp. 513–544, 1997.
K.S. Hong and J.Y. Leung, “Some results on Liu's conjecture,” SIAM Journal on Discrete Mathematics, vol. 5, pp. 500–523, 1992.
T.C. Lai, Personal communication, 1995.
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys, “Sequencing and Scheduling: Algorithms and Complexity,” in Handbooks in Operations Research and Management Science, Vol. 4., Logistics of Production and Inventory, S.C. Graves, A.H.G. Rinnooy Kan, and P.H. Zipkin (Eds.), North-Holland, 1993, pp. 445–522.
J.K. Lenstra, A.H.G. Rinnooy Kan, and P. Brucker, “Complexity of machine scheduling problems,” Annals of Discrete Mathematics, vol. 1, pp. 343–362, 1977.
J.H. Lin and J.S. Vitter, “ε-Approximation with Minimum Packing Constraint Violation,” in Proc. of the 24th Annual ACM Symposium on Theory of Computing, 1992, pp. 771–782.
C.L. Liu, “Optimal Scheduling on Multiprocessor Computing Systems,” in Proc. of the 13th Annual IEEE Symposium on Switching and Automata Theory, 1972, pp. 155–160.
R. McNaughton, “Scheduling with deadlines and loss functions,” Management Science, vol. 6, pp. 1–12, 1959.
R.H. Möhring, M.W. Schäffter, and A.S. Schulz, “Scheduling Jobs with Communication Delays: Using Infeasible Solutions for Approximation,” in Algorithms—ESA’96, J. Diaz and M. Serna (Eds.), Vol. 1136 of Lecture Notes in Computer Science, pp. 76–90. Springer, Berlin, 1996, Proceedings of the 4th Annual European Symposium on Algorithm.
C. Phillips, C. Stein, and J. Wein, “Scheduling Jobs that Arrive Over Time,” in Proc. of Fourth Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science, 955, Springer-Verlag, Berlin, 1995, pp. 86–97, Journal version to appear in Mathematical Programming B.
M. Queyranne, “Structure of a simple scheduling polyhedron,” Mathematical Programming, vol. 58, pp. 263–285, 1993.
M. Queyranne, Private communication, 1995.
M. Queyranne and A.S. Schulz, “Polyhedral Approaches to Machine Scheduling,” Technical Report 408/1994, Technical University of Berlin, 1994.
A.S. Schulz, “Scheduling to Minimize Total Weighted Completion Time: Performance Guarantees of lp Based Heuristics and Lower Bounds,” in Proc. of the 5th MPS Conference on Integer Programming and Combinatorial Optimization, 1996, pp. 301–315. Published as Lecture Notes in Computer Science 1084, Springer-Verlag.
A. Schulz and M. Skutella, “Randomization Strikes in lP-based Scheduling: Improved Approximations for Minsum Criteria,” Manuscript, 1996.
Y. Wang, “Bicriteria Job Scheduling with Release Dates,” Technical Report, Max-Planck-Institut für Informatik, Saarbrücken, Germany, 1996.
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Phillips, C.A., Schulz, A.S., Shmoys, D.B. et al. Improved Bounds on Relaxations of a Parallel Machine Scheduling Problem. Journal of Combinatorial Optimization 1, 413–426 (1998). https://doi.org/10.1023/A:1009750913529
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DOI: https://doi.org/10.1023/A:1009750913529