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

Chapter 3: Introduction to Graphs and Network Coding 21
Definition 3.8 Max-flow Problem:
maximize x T S
subject to
∑
x ij −
∑
x ji = 0, ∀i ∈ V (G) with i ≠ S and i ≠ T, (3.2)
{j|(i,j)∈E(G)}
{j|(j,i)∈E(G)}
∑
x Sj =
∑
x iT = x T S , (3.3)
{j|(S,j)∈E(G)}
{i|(i,T )∈E(G)}
b ij ≤ x ij ≤ c ij , ∀(i, j) ∈ E(G) with (i, j) ≠ (T, S). (3.4)
The Ford-Fulkerson algorithm can be used to solve the max-flow problem. The basic
idea is to recursively increase the flow by finding a path from S to T that is unblocked with
respect to the flow vector until we can verify that the maximum flow is achieved. Such
verification can be done by checking whether there exists a “saturated cut” separating S
from T [17]. The following four definitions explain the meaning of the term “saturated
cut.”
Definition 3.9 A cut Q in a graph G is a partition of the node set V (G) into two
nonempty subsets, a set C and its complement V (G) − C. We use the notation
Q = [C, V (G) − C]. (3.5)
Note that the partition is ordered such that the cut [C, V (G) − C] is distinct from [V (G) −
C, C]. For a cut [C, V (G) − C], using the notation
Q + = {(i, j) ∈ E(G)|i ∈ C, j ∉ C}, (3.6)
Q − = {(i, j) ∈ E(G)|i ∉ C, j ∈ C}, (3.7)
we call Q + and Q − the sets of forward and backward arcs of the cut, respectively.
Definition 3.10 The flux F (Q) across a cut Q = [C, V (G) − C] is the total net flow