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Minimizing Sum of Completion Times and Makespan in Master-Slave Systems
August 2006 (vol. 55 no. 8)
pp. 985-999
We consider scheduling problems in the master-slave model. In this model, each job has to be processed sequentially in three stages. In the first stage, a preprocessing task runs on a master machine, in the second stage, a slave task runs on a dedicated slave machine, and, in the last stage, a postprocessing task again runs on a master machine, possibly different from the master machine in the first stage. It has been shown that the problem of minimizing the makespan or the sum of completion times is NP-hard in the strong sense even if preemption is allowed. In this paper, we design efficient approximation algorithms to minimize the sum of completion times in various settings. These are the first general results for the minsum problem in the master-slave model. We also show that these algorithms generate schedules with small makespan as well.

[1] M. Dell'Amico, “Shop Problems with Two Machines and Time Lags,” Operations Research, vol. 44, no. 5, pp. 777-787, 1996.
[2] R.E. Buten and V.Y. Shen, “A Scheduling Model for Computer Systems with Two Classes of Processors,” Proc. Sagamore Computer Conf. Parallel Processing, pp. 130-138, 1973.
[3] C. Chekuri, R. Motwani, B. Natarajan, and C. Stein, “Approximation Techniques for Average Completion Time Scheduling,” SIAM J. Computing, vol. 31, no. 1, pp. 146-166, 2001.
[4] S. Chakrabarti, C. Phillips, A. Schulz, D.B. Shmoys, C. Stein, and J. Wein, “Improved Scheduling Algorithms for Minsum Criteria,” Proc. 23rd Int'l Colloquium Automata, Languages and Programming, pp. 646-657, 1996.
[5] J. Du and J.Y-T. Leung, “Minimizing Mean Flow Time in Two-Machine Open Shops and Flow Shops,” J. Algorithms, vol. 14, pp.24-44, 1993.
[6] M.X. Goemans, “Improved Approximation Algorithms for Scheduling with Release Dates,” Proc. Eighth ACM-SIAM Symp. Discrete Algorithms, pp. 591-598, 1997.
[7] J.N. D. Gupta, “Two-Stage, Hybrid Flowshop Scheduling Problem,” J. Operational Research Soc., vol. 38, pp. 359-364, 1988.
[8] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: Freeman, 1979.
[9] T.F. Gonzalez and S. Sahni, “Flowshop and Jobshop Schedules: Complexity and Approximation,” Operations Research, vol. 26, pp.26-52, 1978.
[10] L.A. Hall, A.S. Schulz, D.B. Shmoys, and J. Wein, “Scheduling to Minimize Average Completion Time: Offine and Online Algorithms,” Math. Operations Research, vol. 22, pp. 513-544, 1997.
[11] J.A. Hoogeveen and T. Kawaguchi, “Minimizing Sum of the Completion Times in a Two Machine Flowshop: Analysis of Special Cases,” Math. Operations Research, vol. 24, no. 4, pp. 887-910, 1999.
[12] L.A. Hall, “Approximability of Flow Shop Scheduling,” Math. Programming, vol. 82, pp. 175-190, 1998.
[13] S.M. Johnson, “Optimal Two and Three-Stage Production Schedules with Setup Times Included,” Naval Research Logistics Quarterly, vol. 1, pp. 61-68, 1954.
[14] W. Kern and W. Nawijn, “Scheduling Multioperation Jobs with Time Lags on a Single Machine,” technical report, Univ. of Twente, 1993.
[15] M.A. Langston, “Interstage Transportation Planning in the Deterministic Flow-Shop Environment,” Operations Research, vol. 35, no. 4, pp. 556-564, 1987.
[16] C.-Y. Lee and G.L. Vairaktarakis, “Minimizing Makespan in Hybrid Flowshops,” Operations Research Letters, vol. 16, pp. 149-158, 1994.
[17] J.Y-T. Leung and H. Zhao, “Minimizing Mean Flowtime and Makespan on Master-Slave Systems,” J. Parallel and Distributed Computing, vol. 65, pp. 843-856, 2005.
[18] C. Phillips, C. Stein, and J. Wein, “Minimizing Average Completion Time in the Presence of Release Dates,” Math. Programming, vol. 82, pp. 199-223, 1998.
[19] M. Queyranne, “Structure of a Simple Scheduing Polyhedron,” Math. Programming, vol. 58, pp. 263-285, 1993.
[20] M. Queyranne, personal communication, 1995.
[21] S. Sahni, “Scheduling Master-Slave Multiprocessor Systems,” IEEE Trans. Computers, vol. 45, no. 10, pp. 1195-1199, Oct. 1996.
[22] A.S. Schulz, “Scheduling to Minimize Total Weighted Completion Time: Performance Guarantees of LP-Based Heuristics and Lower Bounds,” Proc. Fifth Integer Programming and Combinatorial Optimaization (IPCO), pp. 301-315, 1996.
[23] L. Schrage, “A Proof of the Optimality of the Shortest Remaining Processing Time Discipline,” Operations Research, vol. 16, pp. 687-690, 1968.
[24] D. Smith, “A New Proof of the Optimality of the Shortest Remaining Processing Time Discipline,” Operations Research, vol. 26, no. 1, pp. 197-199, 1976.
[25] W. Szwarc, “Flow-Shop Problems with Time Lags,” Management Science, vol. 29, pp. 447-491, 1983.
[26] C. Sriskandarajah and S.P. Sethi, “Scheduling Algorithms for Flexible Flowshops: Worst and Average Case Performance,” European J. Operational Research, vol. 43, pp. 143-160, 1989.
[27] A.S. Schulz and M. Skutella, “Scheduling-LPs Bear Probabilities: Randomized Approximations for Min-Sum Criteria,” Proc. Fifth Ann. European Symp. Algorithms, pp. 416-429, 1997.
[28] S. Sahni and G. Vairaktarakis, “The Master-Slave Paradigm in Parallel Computer and Industrial Settings,” J. Global Optimization, vol. 9, pp. 357-377, 1996.
[29] S. Sahni and G. Vairaktarakis, “The Master-Slave Scheduling Model,” Handbook of Scheduling: Algorithms, Models, and Performance Analysis, 2004.
[30] G. Vairaktarakis, “Analysis of Algorithms for Master-Slave System,” IIE Trans., vol. 29, no. 11, pp. 939-949, 1997.
[31] L.A. Wolsey, “Mixed Integer Programming Formulations for Production Planning and Scheduling Problems,” Invited talk at the 12th Int'l Symp. Math. Programming, 1985.
[32] W. Yu, H. Hoogeveen, and J.K. Lenstra, “Minimizing Makespan in a Two-Machine Flowshop with Delays and Unit-Time Operations is NP-Hard,” J. Scheduling, vol. 7, no. 5, pp. 333-348, 2004.

Index Terms:
Sequence and scheduling, approximation algorithms, makespan, linear programming.
Citation:
Joseph Y.-T. Leung, Hairong Zhao, "Minimizing Sum of Completion Times and Makespan in Master-Slave Systems," IEEE Transactions on Computers, vol. 55, no. 8, pp. 985-999, Aug. 2006, doi:10.1109/TC.2006.128
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