Ford Fulkerson Algorithm

FORD FULKERSON
ALGORITHM
Adarsh V R
ME Scholar, UVCE
K R Circle, Bangalore

 Flow network is a directed graph G=(V,E) such that each
edge has a non-negative capacity c(u,v)≥0.
 Two distinguished vertices exist in G namely :
• Source (denoted by s) : In-degree of this vertex is 0.
• Sink (denoted by t) : Out-degree of this vertex is 0.
 Flow in a network is an integer-valued function f defined on
the edges of G satisfying 0 ≤ f(u,v) ≤ c(u,v), for every
Edge(u,v) in E.
 Augmented Path is a path from source s to sink t in a
residual graph.
 Residual Graph is graph after sending the flow through the
network with edges having remaining capacities (residual
capacity).
2

• FORD-FULKERSON(G,s,t)
• for each edge (u,v)  E[G]
• do f[u,v] 0
• f[v,u] 0
• while there exists a path p from s to t in the residual
network Gf
• do cf(p) min{cf(u,v): (u,v) is in p}
• for each edge (u,v) in p
• do f[u,v] f[u,v]+cf(p) 3
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Ford Fulkerson Algorithm

 After every step in the algorithm the following is
maintained:
• Capacity Constraints : ∀ 𝑢, 𝑣 𝜖 𝐸 𝑓 𝑢, 𝑣 ≤ 𝑐(𝑢, 𝑣)
 The flow along an edge can not exceed its capacity.
• Skew Symmetry : ∀ 𝑢, 𝑣 𝜖 𝐸 𝑓 𝑢, 𝑣 = −𝑓(𝑣, 𝑢)
 The net flow from u to v must be the opposite of the net flow from v to u
• Flow Conservation :
 Unless u is s or t. The net flow to a node is zero, except for the source, which
"produces" flow, and the sink, which "consumes" flow.
4

When the algorithm terminates?
All paths from s to t are blocked by either a
• Full forward edge
• Empty backward edge
5

EXAMPLE:
s
2
3
4
5 t10
10
9
8
4
10
1062
0
0
0
0 0 0
0
0
G:
Flow value = 0
0
flow
capacity
6

s
2
3
4
5 t10
10
9
8
4
10
1062
0
0
0
0 0 0
0
0
G:
s
2
3
4
5 t10 9
4
1062
Gf:
10 8
10
8 8
8
X X
X
0
Flow value = 0
capacity
residual capacity
flow
7

s
2
3
4
5 t10
10
9
8
4
10
1062
8
0
0
0 0 8
8
0 0
G:
s
2
3
4
5 t10
4
106
Gf:
8
8
8
9
22
2
10
2
10
X
X
X2X
Flow value = 8
8

0
s
2
3
4
5 t10
10
9
8
4
10
1062
10
0
0
0 2 10
8
2
G:
s
2
3
4
5 t
4
2
Gf:
10
810
2
10 7
106
X
6
6
6
X
X
8X
Flow value = 10
9

s
2
3
4
5 t10
10
9
8
4
10
1062
10
0
6
6 8 10
8
2
G:
s
2
3
4
5 t1
6
Gf:
10
810
8
6
6
6
4
4
4
2
X
8
2
8
X
X
0
X
Flow value = 16
10

s
2
3
4
5 t10
10
9
8
4
10
1062
10
2
8
8 8 10
8
0
G:
s
2
3
4
5 t
62
Gf:
10
10
8
6
8
8
2
2 1
2
8 2
X
9
7 9
X
X
9X
X 3
Flow value = 18
11

s
2
3
4
5 t10
10
9
8
4
10
1062
10
3
9
9 9 10
7
0
G:
s
2
3
4
5 t1 9
1
162
Gf:
10
710
6
9
9
3
1
Flow value = 19
12

s
2
3
4
5 t10
10
9
8
4
10
1062
10
3
9
9 9 10
7
0
G:
s
2
3
4
5 t1 9
1
162
Gf:
10
710
6
9
9
3
1
Flow value = 19
13

Ford Fulkerson Algorithm

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