Lecture Notes Discrete Optimization - Applied Mathematics

6. Minimum Cost Flows
6.1 Introduction
We consider the minimum cost flow problem.
Minimum Cost Flow Problem:
Given: A directed graph G=(V,E) with capacities w : E → R + and costs c :
E →R + and a balance function b : V →R.
Goal: Compute a feasible flow f such that the overall cost ∑ e∈E c(e) f(e) is
minimized.
Here, a flow f : E →R + is said to be feasible if it respects the capacity constraints and the
total flow at every node u∈ V is equal to the balance b(u). More formally, f is feasible if
the following two conditions are satisfied:
1. Capacity constraint: for every(u,v)∈E, f(u,v)≤w(u,v).
2. Flow balance constraints: for every u∈ V,
∑ f(u,v)− ∑ f(v,u)=b(u).
(u,v)∈E (v,u)∈E
Intuitively, a positive balance indicates that node u has a supply of b(u) units of flow,
while a negative balance indicates that node u has a demand of −b(u) units of flow. A
feasible flow f that satisfies the flow balance constraints with b(u)=0 for every u∈ V is
called a circulation.
The minimum cost flow problem can naturally be formulated as a linear program:
minimize
subject to
∑ c(e) f(e)
e∈E
∑
(u,v)∈E
f(u,v)− ∑ f(v,u) = b(u) ∀u∈ V
(v,u)∈E
f(e) ≤ w(e) ∀e∈E
f(e) ≥ 0 ∀e∈E
(6)
We use c( f)=∑ e∈E c(e) f(e) to refer to the total cost of a feasible flow f . We make a few
assumptions throughout this section:
Assumption 6.1. Capacities, costs and balances are integral.
Note that we can enforce this assumption if all input numbers are rational numbers by
multiplying by a suitably large constant.
Assumption 6.2. The balance function satisfies ∑ u∈V b(u) = 0 and there is a feasible
flow satisfying these balances.
Note that we can test whether a feasible flow exists by a single max-flow computation
as follows: Augment the network by adding a super-source s and a super-target t. Add
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