Abstract
We show how to approximate in NC the problem of scheduling unrelated parallel machines, for a fixed number of machines in which the makespan C max is the objective function to minimize. We develop an approximate NC algorithm which finds a schedule whose length is at most (1+o(1))(C* max + √3 C* maxln(2n(n-1)/ε)), where C*max denotes the optimal schedule, n the total number of jobs and √ a small positive constant. Our approach shows how to relate the linear program obtained by relaxing the integer programming formulation of the problem with a linear program formulation that is positive and in the packing/covering form. The established relationship enables us to transfer approximate fractional solutions from the later formulation that is known to be approximable in NC. Then, we show how to obtain an integer approximate solution, i.e. a schedule, from the fractional one, using the randomized rounding technique. We stress that our analysis assumes that the length of the schedule is Ω(ln n) and that the min p ij and max p ij values are not too disparate (where p ij is the time to run job j on machine i).
Finally, we show that the same technique can be applied to the general assignment problem with a fixed number of machines and makespan T.
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References
B. Berger, “Using randomness to design efficient deterministic algorithms,” PhD Thesis, Massachusetts Institute of Technology, 1990.
B. Berger and J. Rompel, “Simulating (logc n)-wise independence in NC,” Journal of the ACM, vol. 38, pp. 1026–1046, 1991.
J. Díaz, M. Serna, P. Spirakis, and J. Torán, Paradigms for Fast Parallel Approximability, Cambridge University Press: Cambridge, 1997.
D. Dobkin, J.R. Lipton, and S. Reiss, “Simulating (logc n)-wise independence in NC,” Journal of the ACM, vol. 38, pp. 1026–1046, 1991.
M.X. Goemans and D.P. Williamson, “New 3/4-approximation algorithms for the maximum satisfability problem,” SIAM Journal on Discrete Mathematics, vol. 7, pp. 656–666, 1994.
L.A. Hall, “Approximability of flow shop scheduling,” in Proc. of 36th IEEE Symp. on Found. of Comp. Science, 1995, pp. 82–91.
D.S. Hochbaum, “Approximation algorithms for the set cover and the vertex cover,” SIAM Journal of Computing, vol. 11, no. 3, pp. 555–556, 1982.
E. Horowitz and S. Sahni, “Exact and approximate algorithms for scheduling nonidentical processors,” Journal of the ACM, vol. 23, pp. 317–327, 1976.
K. Jansen and L. Parkolab, “Improved approximation schemes for scheduling unrelated parallel machines,” Technical Report MPI-I-98-1-026, 1998.
D.R. Karger and D. Koller, “(De)randomized construction of small sample spaces inNC,” in Proc. of the 35th Annual Symp. on Found. of Comp. Science, 1994.
R.M. Karp, Reducibility among combinatorial problems,” in R.E. Miller and J.W. Thatcher (Eds.), Complexity of Computer Computations, Plenum Press: New York, 1972, pp. 85–103.
L.G. Khachyan, “Apolynomial algorithm in linear programming,” Translated in Soviet Mathematics Doklady, vol. 20, pp. 191–194, 1979.
Y. Kopidakis, D. Fayard, and V. Zissimopoulos, “Linear time approximation schemes for parallel processor scheduling,” in Proc. of 8th IEEE Symp. on Parallel and Distributed Processing, 1996, pp. 482–485.
J.K. Lenstra, D.B. Shmoys, and È. Tardos, “Approximation algorithms for scheduling unrelated parallel machines,” Mathematical Programming, vol. 46, pp. 259–271, 1990.
M. Luby, “Asimple parallel algorithm for the maximal independent set problem,” SIAMJournal of Computing, vol. 15, pp. 1036–1053, 1986.
M. Luby and N. Nisan, “A parallel approximation algorithm for positive linear programming,” in Proc. of 25th ACM Symposium on Theory of Computing, 1993, pp. 448–457.
N. Megiddo, “A note on approximate linear programming,” Information Processing Letters, vol. 42, p. 53, 1992.
R. Motwani, J.S. Naor, and M. Naor, “The probabilistic method yields deterministic parallel algorithms,” Journal of Computer and System Sciences, vol. 49, pp. 478–516, 1994.
A.L. Peresini, F.E. Sullivan, and J.J. Uhl, Jr., The Mathematics of Nonlinear Programming, Springer Verlag: Berlin, 1988.
C.N. Potts, “Analysis of a linear programming heuristic for scheduling unrelated parallel machines,” Discrete Applied Mathematics, vol. 10, pp. 155–164, 1985.
P. Raghavan and C. Thompson, “Randomized rounding: A technique for provably good algorithms and algorithmic proofs,” Combinatorica, vol. 7, pp. 365–374, 1987.
M. Serna, “Approximating linear programming is logspace complete for P,” Information Processing Letters, vol. 37, pp. 233–236, 1991.
D.B. Shmoys and E. Tardos, “An approximation algorithms for the generalized assignment problem,” Mathematical Programming, vol. 62, pp. 461–474, 1993.
L. Trevisan, “Parallel approximation algorithms by positive linear programming,” Algorithmica, vol. 21, pp. 72–88, 1998.
L. Trevisan and F. Xhafa, “The parallel complexity of positive linear programming,” Parallel Processing Letters, vol. 8, no. 4, pp. 527–533, 1998.
S. Walukiewicz, Integer Programming, Kluwer Academic Publishers: Dordrecht, 1990.
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Serna, M., Xhafa, F. Approximating Scheduling Unrelated Parallel Machines in Parallel. Computational Optimization and Applications 21, 325–338 (2002). https://doi.org/10.1023/A:1013781304506
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DOI: https://doi.org/10.1023/A:1013781304506