Lecture Notes Discrete Optimization - Applied Mathematics

Using this and the definition of h, we obtain
0= ∑
(u,v)∈E + g(u,v)− ∑
(u,v)∈E − g(v,u)− ∑
= ∑
(u,v)∈E f
g(u,v)− ∑
(v,u)∈E f
g(v,u)− ∑
= 2(
∑
(u,v)∈E f
g(u,v)− ∑
(v,u)∈E f
g(v,u)
(v,u)∈E + g(v,u)+ ∑
g(v,u)+ ∑
(v,u)∈E f
)
.
(v,u)∈E − g(u,v)
(u,v)∈E f
g(u,v)
Here the second equality follows from the observations above: for every (u,v)∈E + we
have (u,v) ∈ E f , for every (u,v) ∈ E − we have (v,u)∈E f , and g(u,v) is non-zero only
on the edges in E + and E − . We conclude that g is a circulation of G f .
The proof now follows from Lemma 6.3. (Note that there are at most m edges with
positive flow in g. Thus, g can be decomposed into at most m cycles flows.)
6.3 Cycle Canceling Algorithm
Lemma 6.2 shows that if we are able to find a cycle C in G f of negative cost c(C) < 0,
then we can augment r f (C) units of flow along this cycle and obtain a flow f ′ of cost
strictly smaller than c( f). This observation gives rise to our first optimality condition:
Theorem 6.1 (Negative cycle optimality condition). A feasible flow f of G is a minimum
cost flow if and only if G f does not contain a negative cost cycle.
Proof. Suppose f is a minimum cost flow and G f contains a negative cost cycle C. By
Lemma 6.2, we can augment r f (C) units of flow along C and obtain a feasible flow f ′
with
c( f ′ )=c( f)+r f (C)·c(C)