On flows in simple bidirected and skew-symmetric networks

On flows in simple bidirected and skew-symmetric networks We consider a simple directed network. Results of Karzanov, Even, and Tarjan show that the blocking flow method constructs a maximum integer flow in this network in O(m min (m 1/2, n 2/3)) time (hereinafter, n denotes the number of nodes, and m the number of arcs or edges). For the bidirected case, Gabow proposed a reduction to solve the maximum integer flow problem in O(m 3/2) time. We show that, with a variant of the blocking flow method, this problem can also be solved in O(mn 2/3) time. Hence, the gap between the complexities of directed and bidirected cases is eliminated. Our results are described in the equivalent terms of skew-symmetric networks. To obtain the time bound of O(mn 2/3), we prove that the value of an integer s-s′ flow in a skew-symmetric network without parallel arcs does not exceed O(Un 2/d 2), where d is the length of the shortest regular s-s′ path in the split network and U is the maximum arc capacity. We also show that any acyclic integer flow of value v in a skew-symmetric network without parallel arcs can be positive on at most O(n√v) arcs. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

On flows in simple bidirected and skew-symmetric networks

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Publisher
Nauka/Interperiodica
Copyright
Copyright © 2006 by Pleiades Publishing, Inc.
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946006040089
Publisher site
See Article on Publisher Site

Abstract

We consider a simple directed network. Results of Karzanov, Even, and Tarjan show that the blocking flow method constructs a maximum integer flow in this network in O(m min (m 1/2, n 2/3)) time (hereinafter, n denotes the number of nodes, and m the number of arcs or edges). For the bidirected case, Gabow proposed a reduction to solve the maximum integer flow problem in O(m 3/2) time. We show that, with a variant of the blocking flow method, this problem can also be solved in O(mn 2/3) time. Hence, the gap between the complexities of directed and bidirected cases is eliminated. Our results are described in the equivalent terms of skew-symmetric networks. To obtain the time bound of O(mn 2/3), we prove that the value of an integer s-s′ flow in a skew-symmetric network without parallel arcs does not exceed O(Un 2/d 2), where d is the length of the shortest regular s-s′ path in the split network and U is the maximum arc capacity. We also show that any acyclic integer flow of value v in a skew-symmetric network without parallel arcs can be positive on at most O(n√v) arcs.

Journal

Problems of Information TransmissionSpringer Journals

Published: Jan 24, 2006

References

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