Network Coding and Wireless Physical-layer ... - Jacobs University

20 Chapter 3: Introduction to Graphs and Network Coding
The length of the cycle C is the number of edges used for connecting the nodes, which
is k + 1 [79].
Definition 3.7 An acyclic graph is a directed graph without any cycle.
3.2 The Max-Flow/Min-Cut Theorem
Now, we would like to introduce the max-flow problem in a digital communication network.
In this problem, we have a graph with two special nodes, the source S and the sink T .
This is an optimization problem whose objective is to maximize the amount of information
flowing from S to T . Speaking in terms of flows and divergences, the objective is to find
a flow vector that makes the divergences of all nodes other than S and T equal 0 while
maximizing the divergence of S. In [17], the max-flow problem is formulated as a special
case of the minimum cost flow problem.
By adding an artificial arc from T to S, as
shown by the dashed line in Fig. 3.3, the max-flow problem becomes a minimum cost flow
problem when the cost coefficient at every edge is zero, except the cost of the feedback
edge T S, which is -1. In this case, the problem can be formulated in Definition 3.8 [17].
S
T
Artificial feedback arc with cost coefficient of -1
Figure 3.3: Illustration of the max-flow problem as a special case of the minimum cost
flow problem [17]