- The classic algorithm for
finding the
maximum flow is called:
- The Ford-Fulkerson Algorithm
Wikipedia page: click here
- The Ford-Fulkerson Algorithm
- The Ford-Fulkerson Algorithm:
for ( each edge e ) { f(e) = 0; // Start with 0 flow on each edge } while ( there is a "flow augmenting path" ) { increase the flow along this "flow augmenting path"; }To understand the algorithm, we need to learn about:
- What is a flow augmenting path ???
- Flow augmenting path:
- Flow augmenting path = a path from source S to sink T where you can increase the amount of flow of the commodity
- I
distinguish 2 types of
flow augmenting paths:
- I call the first type of
flow augmenting paths:
- Purely unsaturated path
- I call the second type of
flow augmenting paths:
- Hybrid paths
- I call the first type of
flow augmenting paths:
- Flow augmenting path - type #1:
- A path from source S → sink T consisting entirely if unsaturated edges is an flow augmenting path
Example:
- The following path is an
flow augmenting path of
type#1:
Source S ---> A ---> D ---> Sink T
in the following basic network:
- Every edge on the path from source S → sink T is unsaturated (i.e.: f(e) < c(e))
- Definition:
slack
Slack( path ) = maximum amount of flow that can be increased along the given path
- Slack on a
flow augmenting path of
type #1:
Slack ( flow augmenting path type 1 ) = min e ∈ path ( c(e) - f(e) )Example:
- Consider the flow augmenting path
S
→
A
→
D
→
T
in the following
basic network:
The slack of this path is:
Slack ( S --> A --> D --> T ) = min ( (3-2), (5-2), 4-2) ) = min ( 1, 3, 2 ) = 1
- Consider the flow augmenting path
S
→
A
→
D
→
T
in the following
basic network:
- Fact:
- The flow along a flow augmenting path can be increased by an amount equal to the slack without violating the feasibility condition f(e) ≤ c(e)
Example: increase flow along a "flow augmenting path"
- A flow augmenting path:
- The slack on this
flow augmenting path =
1
(The bottleneck edge is (S,A))
- After increasing the flow by
1
along the
flow augmenting path:
- Consider the flowing network flow:
- Observation:
- The current flow amount
from S → T =
5
- There are no more paths
from S → T
that consists
entirely of
unsaturated edges:
- The only posible
increase is through
S → C:
- The only path
from
S → T
through
S → C is:
- The slack on this path
is:
Slack = 2 - 2 (on the edge E --> T) = 0
- The only posible
increase is through
S → C:
- The current flow amount
from S → T =
5
- $64,000 question:
- Is this solution the maximum possible flow through the network ???
Answer: no !!
There is a better solution:
(Total flow = 6 !!!)
- The flow augmenting path used
(reconstructed by
comparing the
before and after values
of the flows):
- State before the change:
- State after the change:
- Difference:
- State before the change:
- $64,000 question:
- How can such a path be an flow augmenting path ???
- Definition:
forward and
backward edges
- Let P be
a path from
source S to
sink T
- Let e be
an edge
on path P
- The edge e
is an forward edge if:
- The path P uses edge e in the forward direction
- The edge e
is an backward edge if:
- The path P uses edge e in the backward direction
Example:
- Consider the following path
from
source S to
sink T:
Forward edges:
S --> C, C --> E, B --> F, F --> TBackward edge:
B --> E
- Let P be
a path from
source S to
sink T
- Fact:
- The flow
on a (any) path is
increased by:
- Increasing the flow on the forward edges of the path and (simultaneously)
- Decreasing the flow on the back edges of the path !!!!
Example:
To increase flow, we must:
- Increase flow on the green edges and
- Decrease flow on the red edge !!!!
- Intuitive explanation:
- When you cut back on the amount you push backward, you will increase the amount that you push forward !!!
- The flow
on a (any) path is
increased by:
- Flow augmenting path:
- Flow augmenting path =
a path from
source S
→ sink T
that consists of:
- forward edges that are
unsaturated
(i.e.,
f(e) < c(e))
and/or
- backward edges with non-zero flow (i.e., f(e) > 0)
- forward edges that are
unsaturated
(i.e.,
f(e) < c(e))
and/or
- The slack of
a flow augmenting path P:
{ min forward e ( c(e) - f(e) ) slack( P ) = min of : { { min <sub>backward e ( f(e) )
- Flow augmenting path =
a path from
source S
→ sink T
that consists of: