Ford-Fulkerson Algorithm
A digraph G = (V,E), with an integer-valued function c (capacity
function) define on its edges is called a capacitated network. Two
distinguished vertices exist. The first, vertex s has in-degree 0 is
called the source and the second, vertex t has out-degree 0 is called
the sink. The capacity of edge (i,j) is c(i,j)0.
A flow in the network is an integer-valued function f defined on the
edges of G satisfying 0 f(i,j) c(i,j) for every edge (i,j) in E.
Conservation Condition
For every vertex j in V, where j is not the source s or the sink t, the
sum of the flow into j equals the sum of the flow out of j.
A flow that satisfies the conservation condition is called a feasible
flow.
Let f be a feasible flow in a network G. The flow of the network,
denoted by f(G) is the sum of flows coming out of the source s.
The Ford-Fulkerson algorithm determines the maximum flow of the network.
Let f be a feasible flow in G. Edge (i,j) is said to be
a) saturated if f(i,j) = c(i,j)
b) free if f(i,j) = 0
c) positive if 0 < f(i,j) < c(i,j).
An augmenting path is an alternating sequence of vertices and edges of
the form s, e1, v1, e2, v2, …, ek, t in which no vertex is repeated and no
forward edge is saturated and no backward edge is free.
s=
1
3
2
5
4
t=6
27
12 15
24
6
12
12
8
6
