Abstract
In this paper, we demonstrate the existence of a polynomial time approximation scheme for makespan minimization in the open shop scheduling problem with an arbitrary fixed numberm of machines. For the variant of the problem where the number of machines is part of the input, it is known that the existence of an approximation scheme would implyP = NP. Hence, our result draws a precise separating line between approximable cases (i.e., withm fixed) and non-approximable cases (i.e., withm part of the input) of this shop problem. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
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References
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, Sequencing and scheduling: algorithms and complexity, in Handbooks in Operations Research and Management Science, vol. 4, North Holland, Amsterdam, 1993, pp. 445–522.
T. Gonzalez, S. Sahni, Open shop scheduling to minimize finish time, Journal of the Association of Computing Machinery 23 (1976) 665–679.
I. Bárány, T. Fiala, Nearly optimum solution of multimachine scheduling problems, Szigma Mathematika Közgazdasági Folyóirat 15 (1982) 177–191 (in Hungarian).
V.A. Aksjonov, A polynomial-time algorithm for an approximate solution of a scheduling problem, Upravlyaemye Sistemy 28 (1988) 8–11 (in Russian).
B. Chen, V.A. Strusevich, Approximation algorithms for three-machine open shop scheduling, ORSA Journal on Computing 5 (1993) 321–326.
D.P. Williamson, L.A. Hall, J.A. Hoogeveen, C.A.J. Hurkens, J.K. Lenstra, S.V. Sevastianov, D.B. Shmoys, Short shop schedules, Operations Research, 45 (1997) 288–294.
D.B. Shmoys, C. Stein, J. Wein, Improved approximation algorithms for shop scheduling problems, SIAM Journal on Computing 23 (1994) 617–632.
J.P. Schmidt, A. Siegel, A. Srinivasan, Chernoff-Hoeffding bounds for applications with limited independence, SIAM Journal on Discrete Mathematics 8 (1995) 223–250.
L.A. Hall, Approximability of flow shop scheduling, Mathematical Programming (this issue).
D.S. Hochbaum, D.B. Shmoys, Using dual approximation algorithms for scheduling problems: theoretical and practical results, Journal of the Association of Computing Machinery 34 (1987) 144–162.
L.A. Hall, D.B. Shmoys, Approximation algorithms for constrained scheduling problems, in: Proceedings of 30th IEEE Symposium on Foundations of Computer Science, 1989, pp. 134–139.
M.R. Garey, D.S. Johnson, Computers and Intractability, Freeman, San Francisco, CA, 1979.
T. Fiala, An algorithm for the open-shop problem, Mathematics of Operations Research 8 (1983) 100–109.
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Supported by the DIMANET/PECO Program of the European Union.
Supported by a research fellowship of the Euler Institute for Discrete Mathematics and its Applications. This research was done while Gerhard Woeginger was with the Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands.
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Sevastianov, S.V., Woeginger, G.J. Makespan minimization in open shops: A polynomial time approximation scheme. Mathematical Programming 82, 191–198 (1998). https://doi.org/10.1007/BF01585871
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DOI: https://doi.org/10.1007/BF01585871