On flows in simple bidirected and skew-symmetric networks

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.