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The Pseudoflow Algorithm and the Pseudoflow-Based Simplex for the Maximum Flow Problem

  • Dorit S. Hochbaum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1412)

Abstract

We introduce an algorithm that solves the maximum flow problem without generating flows explicitly. The algorithm solves di- rectly a problem we call the maximum s-excess problem. That problem is equivalent to the minimum cut problem, and is a direct extension of the maximum closure problem. The concepts used also lead to a new parametric analysis algorithm generating all breakpoints in the amount of time of a single run.

The insights derived from the analysis of the new algorithm lead to a new simplex algorithm for the maximum flow problem — a pseudoflow-based simplex. We show that this simplex algorithm can perform a parametric analysis in the same amount of time as a single run. This is the first known simplex algorithm for maximum flow that generates all possible breakpoints of parameter values in the same complexity as required to solve a single maximum flow instance and the fastest one.

The complexities of our pseudoflow algorithm, the new simplex algo- rithm, and the parametric analysis for both algorithms are O(mnlog n) on a graph with n nodes and m arcs.

Keywords

Sink Node Extended Network Simplex Algorithm Balance Constraint Maximum Flow Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. C76.
    W. H. Cunningham. A network simplex method. Mathematical Programming, 1:105–116, 1976.CrossRefMathSciNetGoogle Scholar
  2. GGT89.
    G. Gallo, M. D. Grigoriadis and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal of Computing, 18(1):30–55, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  3. GT86.
    A. V. Goldberg and R. E. Tarjan. A new approach to the maximum flow problem. J. Assoc. Comput. Mach., 35:921–940, 1988.zbMATHMathSciNetGoogle Scholar
  4. GC96.
    D. Goldfarb and W. Chen. On strongly polynomial dual algorithms for the maximum flow problem. Special issue of Mathematical Programming B, 1996. To appear.Google Scholar
  5. GH90.
    D. Goldfarb and J. Hao. A primal simplex method that solves the Maximum flow problem in at most nm pivots and O(n2m) time. Mathematical Programming, 47:353–365, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Hoc96.
    D. S. Hochbaum. A new — old algorithm for minimum cut on closure graphs. Manuscript, June 1996.Google Scholar
  7. LG64.
    H. Lerchs, I. F. Grossmann. Optimum design of open-pit mines. Transactions, C.I.M., LXVIII:17–24, 1965.Google Scholar
  8. Pic76.
    J. C. Picard. Maximal closure of a graph and applications to combinatorial problems. Management Science, 22:1268–1272, 1976.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Dorit S. Hochbaum
    • 1
  1. 1.Department of Industrial Engineering and Operations Research, and Walter A. Haas School of BusinessUniversity of CaliforniaBerkeley

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