Lecture Notes Discrete Optimization - Applied Mathematics

mizes the ratio c(C)/|C|. A minimum mean cost cycle can be computed in time O(nm).
Using this idea, one can show that the resulting algorithm has an overall running time of
O(n 2 m 3 logn).
6.4 Successive Shortest Path Algorithm
We derive an alternative optimality condition. Suppose we associate a potential π(u) with
every node u∈ V. Define the reduced cost c π (u,v) of an edge(u,v) as
c π (u,v)=c(u,v)−π(u)+π(v).
Note that this definition is applicable to both the original network and the residual graph.
Theorem 6.4 (Reduced cost optimality conditions). A feasible flow f of G is a minimum
cost flow if and only if there exist some node potentials π : V →R such that c π (u,v)≥0
for every edge(u,v)∈E f of G f .
Proof. Suppose that there exist node potentials such that c π (u,v) ≥ 0 for every edge
(u,v) ∈ E f of the residual graph G f . Let C be an arbitrary simple directed cycle in G f .
Then
∑
e∈C
c(e)= ∑ c π (e)≥0.
e∈C
We conclude that G f does not contain any negative cost cycle. By Theorem 6.1, f is a
minimum cost flow.
Let f be a minimum cost flow. By Theorem 6.1, G f contains no negative cycle. Let
δ : V → R be the shortest path distances from an arbitrarily chosen source node s ∈ V
to every other node u ∈ V (with respect to c). Note that δ is well-defined because G f
contains no negative cycles. The distance function δ must satisfy the triangle inequality
(see Lemma 4.2), i.e., for every edge (u,v) ∈ E f , δ(v) ≤ δ(u)+c(u,v). Define π(u)=
−δ(u) for every node u∈ V. With this definition, we have for every(u,v)∈E f :
which concludes the proof.
c π (u,v)=c(u,v)−π(u)+π(v)= c(u,v)+δ(u)−δ(v)≥ 0,
We next introduce the notion of a pseudoflow. A pseudoflow x of G is a function x : E →
R + that satisfies the nonnegativity and capacity constraints; it need not satisfy the flow
balance constraints. Given a pseudoflow x, define the excess of a node u∈ V as
exs(u)=b(u)+ ∑ x(v,u)− ∑ x(u,v).
(v,u)∈E (u,v)∈E
Intuitively, exs(u)>0means that node u has an excess of exs(u) units of flow; exs(u)