# Generalized max flows and augmenting paths

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## Abstract

It is well-known that if augmenting paths are used to solve a max flow problem then two problems can arise: 1) the number of augmenting paths needed can depend on the size of the capacities; and 2) if the capacities are irrational then the algorithm need not even converge to an optimal solution. Edmonds and Karp [4] and Dinic [3] were the first to show that these problems can be overcome using minimum length augmenting paths. In this paper we examine to what extent augmenting paths and the results of Edmonds and Karp can be generalized. In particular, we consider these ideas for the following generalized max flow problem (first studied by Fulkerson [7]): max*x*_{1}:*Ax*=0,0≤*x*≤*c*.

## Keywords

Feasible Solution Flow Problem Polynomial Algorithm Independent Column Proportional Flow
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## References

- 1.M.S. Bazaraa, J.J. Jarvis, H.D. Sherali.
*Linear Programming and Network Flows*. John Wiley & Sons, New York, 1990.Google Scholar - 2.W.H. Cunningham. Optimal attack and reinforcement of a network.
*J. Assoc. Comput. Mach.*32 (1985) 549–561Google Scholar - 3.E.A. Dinic. Algorithm for a solution of a problem of maximum flow in a network with power estimation.
*Soviet Math. Dokl.*2 (1970) 1277–1280.Google Scholar - 4.J. Edmonds and R. M. Karp. Theoretical improvements in algorithmic efficiency for network flow problems.
*J. of ACM*19 (1972) 248–264.Google Scholar - 5.P. Elias, A. Feinstein, and C.E. Shannon. Note on maximum flow through a network.
*IRE Transactions on Information Theory*IT-2 (1956) 117–119.Google Scholar - 6.L.R. Ford and D.R. Fulkerson. Maximal flow through a network.
*Canadian J. of Math.*, 8 (1956) 399–404.Google Scholar - 7.D.R. Fulkerson. Networks, frames, blocking systems, in:
*Mathematics of the Decision Sciences, Part 1*(G.B. Dantzig and A.F. Veinott, JR., eds.), Lectures in Applied Mathematics Vol. 11, American Mathematical Society, Providence, R.I. (1968) 303–334.Google Scholar - 8.D.R Fulkerson and G.B. Dantzig. Computation of maximum flow in networks. Naval Reasearch Logistics Quarterly 2 (1955) 277–283.Google Scholar
- 9.G. Gallo, M.D. Grigoriadis, R. Tarjan. A fast parametric maximum flow algorithm and applications.
*SIAM J. Computing*, Vol. 18, No. 1 (1989) 30–55.Google Scholar - 10.A.V. Goldberg and R.E. Tarjan. A new approach to the maximum flow problem.
*J. Assoc. Comput. Mach*. 35 (1988).Google Scholar - 11.D. Gusfield. On scheduling transmissions in a network. Tech. Report YALEU DCS TR 481, Department of Computer Science, Yale University, New Haven, CT, 1986.Google Scholar
- 12.A. Itai and M. Rodeh. Scheduling transmissions in a network.
*J. of Algorithms*6 (1985) 409–429.Google Scholar - 13.S.T. McCormick and T.R. Ervolina. Computing maximum mean cuts. UBC Faculty of Commerce Working Paper 90-MCS-011, 1990.Google Scholar
- 14.N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms.
*J.Assoc.Comput.Mach.*, 30 (1983) 852–865.Google Scholar - 15.C.H. Norton, S.A. Plotkin, and Eva Tardos. Using separation algorithms in fixed dimension.
*Journal of Algorithms*13 (1992) 79–98.16.Google Scholar - 16.T. Radzik. Parametric flows, weighted means of cuts, and fractional combinatorial optimization. Working paper, School of Operations Research, Cornell University.Google Scholar
- 17.A. Schrijver.
*Theory of Linear and Integer Programming*. John Wiley & Sons, New York, 1986.Google Scholar - 18.D.D. Sleator and E.T. Tarjan. A data structure for dynamic trees. 13th ACM Symops. Theory of Computing. Milwaukee, Wisconsin (1981) 114–122.Google Scholar
- 19.V. Srinivasin and G.L. Thompson. An operator theory of parametric programming for the transportation problem.
*Naval Research Logistics Quarterly*19 (1972) 205–252.Google Scholar - 20.E. Tardos. A strongly polynomial algorithm to solve combinatorial linear programs.
*Operations Research*34 (1986) 250–256.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1995