Abstract
In this paper we discuss the parallel asynchronous implementation of the classical primaldual method for solving the linear minimum cost network flow problem. Multiple augmentations and price rises are simultaneously attempted starting from several nodes with possibly outdated price and flow information. The results are then merged asynchronously subject to rather weak compatibility conditions. We show that this algorithm is valid, terminating finitely to an optimal solution. We also present computational results using an Encore MULTIMAX that illustrate the speedup that can be obtained by parallel implementation.
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R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, “Network flows,” Handbooks in Operations Research and Management Science, vol. 1, Optimization, G. L. Nemhauser, A. H. G. Rinnooy-Kan, and M. J. Todd, (eds.), North-Holland: amsterdam, 1989, pp. 211–369.
E. Balas, D. Miller, J. Pekny, and P. Toth, “A parallel shortest part algorithm for the assign problem,” J. ACM, Vol. 34, pp. 985–1004, 1991.
D. P. Bertsekas, “A unified framework for minimum cost network flow problems,” Math. Programming, pp. 125–145, 1985.
D. P. Bertsekas, Linear Network Optimization: Algorithms and Codes, MIT Press: Cambridge, MA, 1991.
D. P. Bertsekas, and D. A. Castañon, “Parallel asynchronous Hungarian methods for the assignment problem,” Alphatech Report, Feb. 1990, (to appear in ORSA J. of Comput).
D. P. Bertsekas, and D. A. Castañon, “Parallel synchronous and asynchronous implementations of the auction algorithm,” Parallel Computing, vol. 17, pp. 707–732, 1991.
D. P. Bertsekas, D. A. Castañon, J. Eckstein, and S. Zenios, “A survey of parallel algorithms for network optimization,” Lab. for Information and Decision Systems Report P-1606, Massachusetts Institute of Technology, November 1991; to appear in Handbooks in Operations Research and Management Science, vol. 4, North-Holland: Amsterdam.
D. P. Bertsekas, and P. Tseng, “Relaxation methods for minimum cost ordinary and generalized network flow problems,” Oper. Res. J., vol. 36, pp. 93–114, 1988.
D. P. Bertsekas, and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall: Englewood Cliffs, NJ, 1989.
J. Boyle, R. Butler, T. Disz, B. Glickfield, E. Lusk, R. Overbeek, J. Patterson, and R. Stevens, Portable Programs for Parallel Processors, Holt, Rinehart, and Winston: New York, NY, 1987.
R. G. Busaker, and P. J. Gowen, “A procedure for determining a family of minimal cost network flow patterns” O.R.O. Technical Report No. 15, Operational Research Office, Johns Hopkins University, Baltimore, MD, 1961.
L. R. Ford, Jr. and D. R. Fulkerson, “A primal-dual algorithm for the capacitated Hitchcock problem,” Naval Res. Log. Q. vol. 4, pp. 47–54, 1957.
L. R. Ford, Jr. and D. R. Fulkerson, Flows in Networks, Princeton Univ. Press: Princeton, NJ, 1962.
D. Klingman, A. Napier, and J. Stutz, “NETGEN — A program for generating large scale (un) capacitated assignment, transportation, and minimum cost flow network problems,” Mgmt. Sci., vol. 20, pp. 814–822, 1974.
C. H. Papadimitriou, and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall: Englewood Cliffs, NJ, 1982.
R. T. Rockafellar, Network Flows and Monotropic Programming, Wiley-Interscience: NY, New York, 1984.
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This work supported in part by the BM/C3 Technology branch of the United States Army Strategic Defense Command.
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Bertsekas, D.P., Castañon, D.A. Parallel primal-dual methods for the minimum cost flow problem. Comput Optim Applic 2, 317–336 (1993). https://doi.org/10.1007/BF01299544
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DOI: https://doi.org/10.1007/BF01299544