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**A simple version of Karzanov’s blocking flow algorithm.**
*(English)*
Zbl 0542.05057

A network is a directed graph G with two distinguished vertices, a source s and a sink t, and a positive capacity c(v,w) on every edge [v,w]. A flow is blocking if there is a saturated edge on every path from s to t. E. A. Dinits [Sov. Math., Dokl. 11, 1277-1280 (1970); translation from Dokl. Akad. Nauk SSSR 194, 754-757 (1970; Zbl 0219.90046)] has shown that the classic maximum flow problem on a graph of n vertices and m edges can be reduced to a sequence of, at most n-1 so- called ”blocking flow” problems on acyclic graphs. For dense graphs, the best time bound known for the blocking flow problem is \(O(n^ 2)\) [A. V. Karzanov, Sov. Math. Dokl. 15(1974), 434-437 (1975; Zbl 0303.90014); V. M. Malhotra, M. P. Kumar and S. N. Maheshwari, Inf. Process. Lett. 7, 277-278 (1978; Zbl 0391.90041)]. In this paper the author presents a version of Karzanov’s algorithm that is easy to implement.

Reviewer: L.Caccetta

### MSC:

05C99 | Graph theory |

68W99 | Algorithms in computer science |

90B10 | Deterministic network models in operations research |

Full Text:
DOI

### References:

[1] | Cherkasky, R.V., Algorithm of construction of maximal flow in networks with complexity of O(V2√E) operations, Mathematical methods of solution of economical problems, 7, 112-125, (1977), (in Russian) |

[2] | Dinic, E.A., Algorithm for solution of a problem of maximum flow in a network with power estimation, Soviet math. dokl., 11, 1277-1280, (1970) · Zbl 0219.90046 |

[3] | Even, S., Graph algorithms, (1979), Computer Science Press Potomac, MD · Zbl 0441.68072 |

[4] | Gail, Z., An \( O(V\^{}\{53\}E\^{}\{23\})\) algorithm for the maximal flow problem, Acta informatica, 14, 221-242, (1980) |

[5] | Galil, Z.; Naamad, A., An O(EV log2V) algorithm for the maximal flow problem, J. computer and system sciences, 21, 203-217, (1980) · Zbl 0449.90094 |

[6] | Karzanov, A.V., Determining the maximal flow in a network by the method of preflows, Soviet math. dokl., 15, 434-437, (1974) · Zbl 0303.90014 |

[7] | Knuth, D.E., () |

[8] | Lawler, E.L., Combinatorial optimization: networks and matroids, (1976), Holt, Rinehart and Winston New York · Zbl 0358.68059 |

[9] | Malhotra, V.M.; Kumar, M.P.; Maheshwari, S.N., An O(|V|3) algorithm for finding maximum flows in networks, (), 277-278 · Zbl 0391.90041 |

[10] | Shiloach, Y.; Vishkin, U., An O(n2log n) parallel MAX-flow algorithm, J. algorithms, 3, 128-146, (1982) · Zbl 0483.90044 |

[11] | Sleator, D.D., An O(nm log n) algorithm for maximum network flow, () |

[12] | D.D. Sleator and R.E. Tarjan, “A data structure for dynamic trees”, J. Computer and System Sciences, to appear; |

[13] | Tarjan, R.E., Finding dominators in directed graphs, SIAM J. comput., 3, 62-89, (1974) · Zbl 0296.68030 |

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