Network Optimization

Flow Decomposition

2

The initial flow

1

2 4

53

6

2

3

47

3

4 48

59

A deficit node (more flow leaving than entering).An excess node (more flow entering than leaving)

A balanced node (flow in = flow out)

2 -2

3

Find a Path or Cycle W

1

2 4

53

6

2

3

47

3

4 48

59

2 -21

Carry out a depth first search. Stop when a node with excess is reached or when there is a cycle.

Select a node with deficit if there is one.

Determine the capacity of the walk W.

The capacity of 1-2-4-5-3-1 is 2.

2 4

53

1

4

Updates

1

2 4

53

6

2

3

47

3

4 48

59

2 -21

Subtract the flow in W from the current flow.

Add the flow in W to the decomposition.

0

25

6

7

cycle flows path flows

2 units around 1-2-4-5-3-1

5

Find a path or cycle W

1

2 4

53

63

25

3

4 46

57

2 -21Carry out a dfs.

Select a node with deficit if there is one.

Determine the capacity of W.

2 4

53

The capacity of 2-4-5-3-2 is 3.

6

Updates

Add the cycle flow to the decomposition

update the current flow

cycle flows path flows

2 units around 1-2-4-5-3-13 units around 2-4-5-3-2

1 63

25

3

4 46

57

-212 0

2

3

4

2 4

53

7

Find a path or cycle using dfs

1 6

22

3

4 43

54

-212

2 4

53

Carry out a dfs.

Select a node with deficit if there is one.

Determine the capacity of W.

6

22

3

-212

2 4

capacity of a path = min {arc capacity, excess, deficit} = 2

8

Updates

22

3

4 43

54

-22

53

Add the path flow to the decomposition

update the current flow

cycle flows path flows

2 units around 1-2-4-5-3-13 units around 2-4-5-3-2

2 units in 1-2-4-6

00

1

00 1 6

2 4

1 6

9

Find a path or cycle using dfs

1

4 43

5453

1 6

2 4

1 6

Carry out a dfs.

Select a node with deficit if there is one. Otherwise, select any node with flow leaving.

Determine the capacity of W.

3

5

4

6

3

The capacity is 1

10

Updates

1

4 43

5453

1 6

2 4

1 6

Add the cycle flow to the decomposition

update the current flow

2 units around 1-2-4-5-3-13 units around 2-4-5-3-2

2 units in 1-2-4-6

1 unit around 3-4-6-5-3

0

3

43

cycle flows path flows

11

Find a path or cycle using dfs

3 43

4353

1 6

2 4

1 6

Carry out a dfs.

Select a node with deficit if there is one. Otherwise, select any node with flow leaving.

Determine the capacity of W.

3

The capacity of 3-4-5-3 is 3

12

Updates

3 43

4353

1 6

2 4

1 6

Add the cycle flow to the decomposition

update the current flow

2 units around 1-2-4-5-3-13 units around 2-4-5-3-2

2 units in 1-2-4-61 unit around 3-4-6-5-33 units around 3-4-5-3

cycle flows path flows

13

Find a path or cycle using dfs

4

453

1 6

2 4

1 6

Carry out a dfs.

Select a node with deficit if there is one. Otherwise, select any node with flow leaving.

Determine the capacity of W.

5

14

Updates and the final flow decomposition

53

1 6

2 4

1 64

4

2 units around 1-2-4-5-3-13 units around 2-4-5-3-2 2 units in 1-2-4-61 unit around 3-4-6-5-33 units around 3-4-5-3

Add the cycle flow to the decomposition

update the current flow

4 units around 5-6-5

cycle flows path flows

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15.082J / 6.855J / ESD.78J Network OptimizationFall 2010

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