On minimum flow and transitive reduction

Abstract

In general, a flow problem G=(V,E,b,c) is given by a directed graph G=(V, E) and the two functions b and c on the set of edges, where b(e) means a lower and c(e) an upper bound. The aim is: Find a maximum (minimum) flow function f subject to the condition that

$$0 \leqslant b(e) \leqslant f(e) \leqslant c(e) \leqslant \infty \forall e \in E.$$

Here we search a minimum flow for the special case c(e)=∞. We show for this special case: There is a subgraph G'=(V, E') of G=(V, E) and a function b' on E' such that the flow problem G'=(V, E', b') is equivalent to G=(V,E,b). The subgraph G'=(V, E') is well-known as the transitive reduction of G. Since G' and b' are computable efficiently and in general |E'| is much smaller than |E|, we find a minimum flow f for an acyclic digraph G in average time O(n ²·log² n). If G is strongly connected we need only worst case time O(n ²).