ICALP 1988: Automata, Languages and Programming pp 535-546

# On minimum flow and transitive reduction

• Klaus Simon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 317)

## 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(n2·log2n). If G is strongly connected we need only worst case time O(n2).

## Key words

Minimum flow transitive closure transitive reduction random digraph

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