Algorithm Design

382
Chapter 7 Network Flow
out of s*, and every edge into t*, is completely saturated with flow: Thus, if
we delete these edges, we obtain a circulation [ in G with [in(v) - [°ut(v) = dv
for each node v. Further, if there is a flow of value D in G’, then there is such
a flow that takes integer values.
In summary, we have proved the following.
(7.50) There is a [easible circulation with demands {dr} in G i[ and only i[ the
maximum s*-t* [low in G ~ has value D. I[ all capacities and demands in G are
integers, and there is a [easible circulation, then there is a [easible circulation
that is integer-valued.
At the end of Section 7.5, we used the Max-Flow Min-Cut Theorem to
derive the characterization (7.40) of bipartite graphs that do not have perfect
matchings. We can give an analogous characterization for graphs that do not
have a feasible circulation. The characterization uses the notion of a cut,
adapted to the present setting. In the context of circulation problems with
demands, a cut (A, B) is any partition of the node set V into two sets, with no
restriction on which side of the partition the sources and sinks fall. We include
the characterization here without a proof.
(7.51) The graph G has a [easible circulation with demands {dr} i[ and only
i[ [or all cuts (A, B),
E d~