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

22 Chapter 3: Introduction to Graphs and Network Coding
coming out of C such that
F (Q) =
∑
x ij −
∑
x ij . (3.8)
(i,j)∈Q + (i,j)∈Q −
Definition 3.11 Given lower and upper flow bounds b ij and c ij for each edge (i, j), the
capacity C(Q) of a cut Q is
∑
C(Q) = c ij −
∑
b ij . (3.9)
{j|(i,j)∈Q + } {j|(i,j)∈Q − }
Definition 3.12 Q is said to be a saturated cut with respect to a set of flow if and only
if F (Q) = C(Q).
Using the Ford-Fulkerson algorithm, the first saturated cut that we find is called
the minimum cut or min-cut representing the network bottleneck. This saturated cut is
“minimum” in the sense that its flux is minimum among all possible saturated cuts. As
suggested by intuition as well as the following max-flow/min-cut theorem, the capacity
of the minimum cut equals the maximum flow of the network. The theorem is formally
proved in [17].
Theorem 3.1 Max-Flow/Min-Cut Theorem:
(a) If x ∗ is an optimal solution of the max-flow problem, then the divergence out of S
corresponding to x ∗ is equal to the minimum cut capacity over all cuts separating S from
T .
(b) If all lower arc flow bounds are zero, the max-flow problem has an optimal solution,
and the maximal divergence out of S is equal to the minimum cut capacity over all cuts
separating S from T .
Figure 3.4 illustrates the max-flow/min-cut theorem. We can see from (b) that the
capacity of the minimum cut is 5, equaling the maximum flow to T in (c).