Abstract
An incremental algorithm may yield an enormous computational time saving to solve a network flow problem. It updates the solution to an instance of a problem for a unit change in the input. In this paper we have proposed an efficient incremental implementation of maximum flow problem after inserting an edge in the network G. The algorithm has the time complexity of O((Δn)2 m), where Δn is the number of affected vertices and m is the number of edges in the network. We have also discussed the incremental algorithm for deletion of an edge in the network G.
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Ahuja, R. K., Magnati, T. and Orlin, J. B.: Network Flows, Prentice-Hall, Englewood Cliffs, 1993.
Cheriyan, J. and Maheshawari, S. N.: Analysis of preflow push algorithms for maximum network flow, SIAM J. Comput. 18 (1989), 1057–1086.
Cherkasky, R. V.: An algorithm for constructing maximum flows in networks with complexity of O(V2 \(\sqrt {\left( E \right)}\)), Math. Methods Solution Econ. Probl. 7 (1977), 112–125.
Dinic, E. A.: Algorithm for solution of a problem of a maximum flow in networks with power estimation, Soviet Math. Dokl. 11 (1970), 1277–1280.
Edmonds, J. and Karp, R. M.: Theoretical improvements in algorithmic efficiency for network flow problems, J. Assoc. Comput. Mach. 19 (1972), 248–264.
Ford, L. R. and Fulkerson, D. R.:Maximal flows through a network, IRE Trans. Inform. Theory 2 (1956), 117–119.
Gabow, H. N.: Scaling algorithms for network problems, J. Comput. System Sci. 31 (1985), 148–168.
Galil, Z.: An O(V5/3 E 2/3) algorithm for the maximum flow problem, Acta Inform. 14 (1980), 221–242.
Galil, Z. and Naamad, A.: An O(EV (log V ) 2 ) algorithm for the maximum flow problem, J. Comput. System Sci. 21 (1980), 203–217.
Goldberg, A. V.: A new max-flow algorithm, Tech. Rep. MIT/LCS/TM-291, Laboratory for Comp. Sci., MIT, Cambridge, Mass., 1985.
Goldberg, A. V. and Tarjan R. E.: A new approach to the maximum flow problem, J. Assoc. Comput. Mach. 35 (1988), 921–940.
Goldberg, A. V.: Processor efficient implementation of a maximum flow algorithm, Inform. Process. Lett. 38 (1991), 179–185.
Karzanov, A. V.: Determining the maximal flow in a network by the method of preflows, Soviet Math. Dokl. 15 (1974), 434–437.
Malhotra, V. M., Kumar, P. and Maheshwari, M. N.: An O(|V|3) algorithm for finding maximum flows in networks, Inform. Process. Lett. 57 (1978), 1251–1254.
Shiloach, Y. and Vishkin, U.: An O(n 2 log n) parallel max-flow algorithm, J. Algorithms 3 (1982), 128–146.
Tarjan, R. E.: A simple version of Karzanov's blocking flow algorithm, Oper. Res. Lett. 2 (1984) 265–268.
Goldberg, A. V. and Tarjan, R. E.: A new approach to the maximum flow problem, Proc. 18th ACM Sympos. Theory of Computing, ACM, New York, 1986, pp. 136–146.
Yang, H. H. and Wang, D. F.: Processor efficient implementation of a maximum flow algorithm, IEEE Trans. CAD 15 (1996), 1533–1540.
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Kumar, S., Gupta, P. An Incremental Algorithm for the Maximum Flow Problem. Journal of Mathematical Modelling and Algorithms 2, 1–16 (2003). https://doi.org/10.1023/A:1023607406540
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DOI: https://doi.org/10.1023/A:1023607406540