Abstract
This paper presents dual network simplex algorithms that require at most 2nm pivots and O(n 2 m) time for solving a maximum flow problem on a network ofn nodes andm arcs. Refined implementations of these algorithms and a related simplex variant that is not strictly speaking a dual simplex algorithm are shown to have a complexity of O(n 3). The algorithms are based on the concept of apreflow and depend upon the use of node labels that are underestimates of the distances from the nodes to the sink node in the extended residual graph associated with the current flow. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
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References
R.K. Ahuja, T.L. Magnanti and J.B. Orlin,Network Flows: Theory, Algorithms, and Applications (Prentice Hall, Englewood Cliffs, NJ, 1993).
R.K. Ahuja and J.B. Orlin, A fast and simple algorithm for the maximum flow problem,Operations Research 37 (1989) 748–759.
R.D. Armstrong and Z. Jin, A strongly polynomial dual (simplex) method for the max flow problem, Manuscript, Graduate School of Management, Rutgers University, Newark, NJ (1992).
A.V. Goldberg, A new max-flow algorithm, Technical Report MIT/LCS/ TM 291, Laboratory for Computer Science, MIT, Cambridge, MA (1985).
A.V. Goldberg, M.D. Grigoriadis and R.E. Tarjan, Use of dynamic trees in a network simplex algorithm for the maximum flow problem,Mathematical Programming 50 (1991) 277–290.
A.V. Goldberg and R.E. Tarjan, A new approach to the maximum flow problem,Journal of the Association of Computing Machinery 35 (1988) 921–940.
D. Goldfarb and W. Chen, An O(n 3) dual simplex method for the maximum flow problem, Manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, NY (1992).
D. Goldfarb and J. Hao, A primal simplex algorithm that solves the maximum flow problem in at mostnm pivots and O(n 2 m) time,Mathematical Programming 47 (1990) 353–365.
D. Goldfarb and J. Hao, On strongly polynomial variants of the network simplex algorithm for the maximum flow problem,Operations Research Letters 10 (1991) 383–387.
A.V. Karzanov, Determining the maximal flow in a network by the method of preflow,Soviet Math. Dokl. 15 (1974) 434–437.
J.B. Orlin, S.A. Plotkin and E. Tardos, Polynomial dual network simplex algorithms,Mathematical Programming 60 (1993) 255–276.
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Research was supported by NSF Grants DMS 91-06195, DMS 94-14438 and CDR 84-21402 and DOE Grant DE-FG02-92ER25126.
Research was supported by NSF Grant CDR 84-21402 at Columbia University.
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Armstrong, R.D., Chen, W., Goldfarb, D. et al. Strongly polynomial dual simplex methods for the maximum flow problem. Mathematical Programming 80, 17–33 (1998). https://doi.org/10.1007/BF01582129
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DOI: https://doi.org/10.1007/BF01582129