6.6: Maximum flowFrank Stajano Thomas Sauerwald

Lent 2016

Illustration of the Ford-Fulkerson Method

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Graph G = (V , E , c):

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Residual Graph Gf = (V , Ef , cf ):

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|f | = 0|f | = 8|f | = 10|f | = 16

|f | = 18

|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Outline

Introduction

Ford-Fulkerson

A Glimpse at the Max-Flow Min-Cut Theorem

Analysis of Ford-Fulkerson

6.6: Maximum flow T.S. 2

History of the Maximum Flow Problem [Harris, Ross (1955)]

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Maximum Flow is 163,000 tons per day!

6.6: Maximum flow T.S. 3

History of the Maximum Flow Problem [Harris, Ross (1955)]

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Maximum Flow is 163,000 tons per day!

6.6: Maximum flow T.S. 3

Flow Network

Abstraction for material (one commodity!) flowing through the edges

G = (V ,E) directed graph without parallel edges

distinguished nodes: source s and sink t

every edge e has a capacity c(e)

Flow Network

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|f | = 5 + 10 + 10 = 25|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

Abstraction for material (one commodity!) flowing through the edges

G = (V ,E) directed graph without parallel edges

distinguished nodes: source s and sink t

every edge e has a capacity c(e)

Flow Network

Capacity function c : V × V → R+

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|f | = 5 + 10 + 10 = 25|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

Abstraction for material (one commodity!) flowing through the edges

G = (V ,E) directed graph without parallel edges

distinguished nodes: source s and sink t

every edge e has a capacity c(e)

Flow Network

Capacity function c : V × V → R+ c(u, v) = 0 ⇔ (u, v) 6∈ E

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|f | = 5 + 10 + 10 = 25|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:

For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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|f | = 5 + 10 + 10 = 25|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)

For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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|f | = 5 + 10 + 10 = 25|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)

For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)

For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

Flow Conservation

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

Flow Conservation

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

Flow Conservation

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

Flow Conservation

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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|f | = 5 + 10 + 10 = 25|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)∑v∈V f (s, v) =

∑v∈V f (v , t)

Flow

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How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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|f | = 5 + 10 + 10 = 25|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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|f | = 5 + 10 + 10 = 25

|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

Flow Network

A flow is a function f : V × V → R that satisfies:For every u, v ∈ V , f (u, v) ≤ c(u, v)For every u, v ∈ V , f (u, v) = −f (v , u)For every u ∈ V \ {s, t}, ∑v∈V f (u, v) = 0

The value of a flow is defined as |f | = ∑v∈V f (s, v)

Flow

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13/15

0/4

8/9

0/4

2/15

8/8

3/6

13/30

8/100/15

0/15 10/1

0

10/10

|f | = 5 + 10 + 10 = 25|f | = 8 + 10 + 10 = 28

How to find a Maximum Flow?

6.6: Maximum flow T.S. 4

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0

|f | = 8|f | = 10|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0

|f | = 8|f | = 10|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0

|f | = 8|f | = 10|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0

|f | = 8

|f | = 10|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0

|f | = 8

|f | = 10|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10

��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0

|f | = 8

|f | = 10|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10

��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0|f | = 8

|f | = 10

|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10

��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0|f | = 8

|f | = 10

|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10

��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0|f | = 8

|f | = 10

|f | = 16|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6

��0/10

10/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0|f | = 8|f | = 10

|f | = 16

|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6

��0/10

10/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0|f | = 8|f | = 10

|f | = 16

|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?

Greedy did not succeed!

6.6: Maximum flow T.S. 5

A First Attempt

Start with f (u, v) = 0 everywhereRepeat as long as possible:

Find a (s, t)-path p where each edge e = (u, v) has f (u, v) < c(u, v)Augment flow along p

Greedy Algorithm

s

2

3

4

5 t

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6

��0/1010/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

|f | = 0|f | = 8|f | = 10|f | = 16

|f | = 19

10/10

9/10

0/2 5/6

4/4

6/8

9/9

9/10

10/10

Is this optimal?

Greedy did not succeed!

6.6: Maximum flow T.S. 5

Outline

Introduction

Ford-Fulkerson

A Glimpse at the Max-Flow Min-Cut Theorem

Analysis of Ford-Fulkerson

6.6: Maximum flow T.S. 6

Residual Graph

Edge e = (u, v) ∈ Eflow f (u, v) and capacity c(u, v)

Original Edge

cf (u, v) =

c(u, v)− f (u, v) if (u, v) ∈ E ,f (v , u) if (v , u) ∈ E ,0 otherwise.

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v6/17

Residual Gf :

u v

11

6

6.6: Maximum flow T.S. 7

Residual Graph

Edge e = (u, v) ∈ Eflow f (u, v) and capacity c(u, v)

Original Edge

cf (u, v) =

c(u, v)− f (u, v) if (u, v) ∈ E ,f (v , u) if (v , u) ∈ E ,0 otherwise.

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v6/17

Residual Gf :

u v

11

6

6.6: Maximum flow T.S. 7

Residual Graph

Edge e = (u, v) ∈ Eflow f (u, v) and capacity c(u, v)

Original Edge

cf (u, v) =

c(u, v)− f (u, v) if (u, v) ∈ E ,f (v , u) if (v , u) ∈ E ,0 otherwise.

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v6/17

Residual Gf :

u v

11

6

6.6: Maximum flow T.S. 7

Residual Graph

Edge e = (u, v) ∈ Eflow f (u, v) and capacity c(u, v)

Original Edge

cf (u, v) =

c(u, v)− f (u, v) if (u, v) ∈ E ,f (v , u) if (v , u) ∈ E ,0 otherwise.

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v6/17

Residual Gf :

u v

11

6

6.6: Maximum flow T.S. 7

Residual Graph

Edge e = (u, v) ∈ Eflow f (u, v) and capacity c(u, v)

Original Edge

cf (u, v) =

c(u, v)− f (u, v) if (u, v) ∈ E ,f (v , u) if (v , u) ∈ E ,0 otherwise.

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v6/17

Residual Gf :

u v

11

6

6.6: Maximum flow T.S. 7

Residual Graph with anti-parallel edges

Edge e = (u, v) ∈ E (& possibly e′ = (v , u) ∈ E)flow f (u, v) and capacity c(u, v)

Original Edge

For every pair (u, v) ∈ V × V ,

cf (u, v) = c(u, v)− f (u, v).

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v

6/17

2/4

Residual Gf :

u v

??

??

17-(6-2)

4-(2-6)

13

8

6.6: Maximum flow T.S. 7

Residual Graph with anti-parallel edges

Edge e = (u, v) ∈ E (& possibly e′ = (v , u) ∈ E)flow f (u, v) and capacity c(u, v)

Original Edge

For every pair (u, v) ∈ V × V ,

cf (u, v) = c(u, v)− f (u, v).

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v

6/17

2/4

Residual Gf :

u v

??

??

17-(6-2)

4-(2-6)

13

8

6.6: Maximum flow T.S. 7

Residual Graph with anti-parallel edges

Edge e = (u, v) ∈ E (& possibly e′ = (v , u) ∈ E)flow f (u, v) and capacity c(u, v)

Original Edge

For every pair (u, v) ∈ V × V ,

cf (u, v) = c(u, v)− f (u, v).

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v

6/17

2/4

Residual Gf :

u v

??

??

17-(6-2)

4-(2-6)

13

8

6.6: Maximum flow T.S. 7

Residual Graph with anti-parallel edges

Edge e = (u, v) ∈ E (& possibly e′ = (v , u) ∈ E)flow f (u, v) and capacity c(u, v)

Original Edge

For every pair (u, v) ∈ V × V ,

cf (u, v) = c(u, v)− f (u, v).

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v

6/17

2/4

Residual Gf :

u v

??

??

17-(6-2)

4-(2-6)

13

8

6.6: Maximum flow T.S. 7

Residual Graph with anti-parallel edges

Edge e = (u, v) ∈ E (& possibly e′ = (v , u) ∈ E)flow f (u, v) and capacity c(u, v)

Original Edge

For every pair (u, v) ∈ V × V ,

cf (u, v) = c(u, v)− f (u, v).

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v

6/17

2/4

Residual Gf :

u v

??

??

17-(6-2)

4-(2-6)

13

8

6.6: Maximum flow T.S. 7

Residual Graph with anti-parallel edges

Edge e = (u, v) ∈ E (& possibly e′ = (v , u) ∈ E)flow f (u, v) and capacity c(u, v)

Original Edge

For every pair (u, v) ∈ V × V ,

cf (u, v) = c(u, v)− f (u, v).

Residual Capacity

Gf = (V ,Ef , cf ), Ef := {(u, v) : cf (u, v) > 0}Residual Graph

Graph G:

u v

6/17

2/4

Residual Gf :

u v

??

??

17-(6-2)

4-(2-6)

13

8

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/101/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/10

1/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/6

1/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/10

1/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/6

1/8

1/121/14

1/6

0/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/10

1/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/6

0/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/10

1/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

Example of a Residual Graph (Handout)

Flow network Gs

2

3

4

5

t

1/14

1/6

2/3

4/105/12

5/8

4/7

3/4

1/3

3/7

3/4

1/3

3/7

2/4

0/3

2/7

2/4

0/3

2/7

4/105/12

0/101/12

0/10

1/12

5/8

4/7

1/8

0/7

1/8

0/7

1/8

1/121/14

1/61/8

1/121/14

1/60/8

0/120/14

0/6

0/8

0/120/14

0/6

Residual Graph Gfs

2

3

4

5

t

1111

131

51

78

13

21

43

1 2

By successively eliminating cycles we can sim-plify and reduce the “transportation” cost of a flow.

6.6: Maximum flow T.S. 7

The Ford-Fulkerson Method (“Enhanced Greedy”)

0: def fordFulkerson(G)1: initialize flow to 0 on all edges2: while an augmenting path in Gf can be found:3: push as much extra flow as possible through it

Augmenting path: Pathfrom source to sink in Gf

If f ′ is a flow in Gf and f a flowin G, then f + f ′ is a flow in G

Questions:How to find an augmenting path?Does this method terminate?If it terminates, how good is the solution?

Using BFS or DFS, we can find anaugmenting path in O(V + E) time.

6.6: Maximum flow T.S. 8

The Ford-Fulkerson Method (“Enhanced Greedy”)

0: def fordFulkerson(G)1: initialize flow to 0 on all edges2: while an augmenting path in Gf can be found:3: push as much extra flow as possible through it

Augmenting path: Pathfrom source to sink in Gf

If f ′ is a flow in Gf and f a flowin G, then f + f ′ is a flow in G

Questions:How to find an augmenting path?Does this method terminate?If it terminates, how good is the solution?

Using BFS or DFS, we can find anaugmenting path in O(V + E) time.

6.6: Maximum flow T.S. 8

The Ford-Fulkerson Method (“Enhanced Greedy”)

0: def fordFulkerson(G)1: initialize flow to 0 on all edges2: while an augmenting path in Gf can be found:3: push as much extra flow as possible through it

Augmenting path: Pathfrom source to sink in Gf

If f ′ is a flow in Gf and f a flowin G, then f + f ′ is a flow in G

Questions:How to find an augmenting path?Does this method terminate?If it terminates, how good is the solution?

Using BFS or DFS, we can find anaugmenting path in O(V + E) time.

6.6: Maximum flow T.S. 8

The Ford-Fulkerson Method (“Enhanced Greedy”)

0: def fordFulkerson(G)1: initialize flow to 0 on all edges2: while an augmenting path in Gf can be found:3: push as much extra flow as possible through it

Augmenting path: Pathfrom source to sink in Gf

If f ′ is a flow in Gf and f a flowin G, then f + f ′ is a flow in G

Questions:How to find an augmenting path?Does this method terminate?If it terminates, how good is the solution?

Using BFS or DFS, we can find anaugmenting path in O(V + E) time.

6.6: Maximum flow T.S. 8

The Ford-Fulkerson Method (“Enhanced Greedy”)

0: def fordFulkerson(G)1: initialize flow to 0 on all edges2: while an augmenting path in Gf can be found:3: push as much extra flow as possible through it

Augmenting path: Pathfrom source to sink in Gf

If f ′ is a flow in Gf and f a flowin G, then f + f ′ is a flow in G

Questions:How to find an augmenting path?Does this method terminate?If it terminates, how good is the solution?

Using BFS or DFS, we can find anaugmenting path in O(V + E) time.

6.6: Maximum flow T.S. 8

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0

|f | = 8|f | = 10|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0

|f | = 8|f | = 10|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0

|f | = 8|f | = 10|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0

|f | = 8

|f | = 10|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/10

8/10

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0

|f | = 8

|f | = 10|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10

��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0

|f | = 8

|f | = 10|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10

��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8

|f | = 10

|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10

��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8

|f | = 10

|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10

��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8

|f | = 10

|f | = 16|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/10

10/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10

|f | = 16

|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/10

10/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10

|f | = 16

|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10

��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10

��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10

|f | = 16

|f | = 18|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/10

10/10

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10

��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10|f | = 16

|f | = 18

|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/10

10/10

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10

��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10|f | = 16

|f | = 18

|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10

��8/10 ��8/9

��8/8

��2/4��8/1010

/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10|f | = 16

|f | = 18

|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/10

10/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10|f | = 16|f | = 18

|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/10

10/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10|f | = 16|f | = 18

|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/10

10/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10|f | = 16|f | = 18

|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Illustration of the Ford-Fulkerson Method

s

2

3

4

5 t

Graph G = (V ,E , c):

0/10

0/10

0/2 0/6

0/4

0/8

0/9

0/10

0/10

��

0/10 ��0/8

��0/108/1

0

0/10

0/2 0/6

0/4

8/8

0/9

0/10

8/10

��

8/10 ��0/2

��0/9��8/10

10/10

0/10

2/2 0/6

0/4

8/8

2/9

0/10

10/10��0/10��2/9

��0/6��0/1010

/10

6/10

2/2 6/6

0/4

8/8

8/9

6/10

10/10��6/10

��2/2

��0/4

��6/1010/1

0

8/10

0/2 6/6

2/4

8/8

8/9

8/10

10/10��8/10 ��8/9

��8/8

��2/4��8/10

10/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

s

2

3

4

5 t

Residual Graph Gf = (V ,Ef , cf ):

10

10

2 6

4

8

9

10

10

2

8

10

2 6

4

8

9

10

2

8

10

10

2 6

4

8

7

2

10

10

10

4

6

2 6

4

8

1

8

4

6

10

10

2

8

2 6

2

2

8

1

8

2

8

10

10

1

9

2 6

1

37

1

9

1

9

10

|f | = 0|f | = 8|f | = 10|f | = 16|f | = 18

|f | = 19

Is this a max-flow?

6.6: Maximum flow T.S. 9

Outline

Introduction

Ford-Fulkerson

A Glimpse at the Max-Flow Min-Cut Theorem

Analysis of Ford-Fulkerson

6.6: Maximum flow T.S. 10

From Flows to Cuts

A cut (S,T ) is a partition of V into S and T = V \ S such that s ∈ Sand t ∈ T .

The capacity of a cut (S,T ) is the sum of capacities of the edgesfrom S to T :

c(S,T ) =∑

u∈S,v∈T

c(u, v) =∑

(u,v)∈E(S,T )

c(u, v)

A minimum cut of a network is a cut whose capacity is minimumover all cuts of the network.

Cut

s

2

3

4

5 t

Graph G = (V ,E , c):

10

10

2 6

4

8

9

10

10

c({s, 3}, {2, 4, 5, t}) =

10 + 9 = 19

|f | = 19

10/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

10− 0 + 9 = 199 + 7− 6 + 9 = 19

6.6: Maximum flow T.S. 11

From Flows to Cuts

A cut (S,T ) is a partition of V into S and T = V \ S such that s ∈ Sand t ∈ T .

The capacity of a cut (S,T ) is the sum of capacities of the edgesfrom S to T :

c(S,T ) =∑

u∈S,v∈T

c(u, v) =∑

(u,v)∈E(S,T )

c(u, v)

A minimum cut of a network is a cut whose capacity is minimumover all cuts of the network.

Cut

s

2

3

4

5 t

Graph G = (V ,E , c):

10

10

2 6

4

8

9

10

10

c({s, 3}, {2, 4, 5, t}) =

10 + 9 = 19

|f | = 19

10/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

10− 0 + 9 = 199 + 7− 6 + 9 = 19

6.6: Maximum flow T.S. 11

From Flows to Cuts

A cut (S,T ) is a partition of V into S and T = V \ S such that s ∈ Sand t ∈ T .The capacity of a cut (S,T ) is the sum of capacities of the edgesfrom S to T :

c(S,T ) =∑

u∈S,v∈T

c(u, v) =∑

(u,v)∈E(S,T )

c(u, v)

A minimum cut of a network is a cut whose capacity is minimumover all cuts of the network.

Cut

s

2

3

4

5 t

Graph G = (V ,E , c):

10

10

2 6

4

8

9

10

10

c({s, 3}, {2, 4, 5, t}) =

10 + 9 = 19

|f | = 19

10/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

10− 0 + 9 = 199 + 7− 6 + 9 = 19

6.6: Maximum flow T.S. 11

From Flows to Cuts

A cut (S,T ) is a partition of V into S and T = V \ S such that s ∈ Sand t ∈ T .The capacity of a cut (S,T ) is the sum of capacities of the edgesfrom S to T :

c(S,T ) =∑

u∈S,v∈T

c(u, v) =∑

(u,v)∈E(S,T )

c(u, v)

A minimum cut of a network is a cut whose capacity is minimumover all cuts of the network.

Cut

s

2

3

4

5 t

Graph G = (V ,E , c):

10

10

2 6

4

8

9

10

10

c({s, 3}, {2, 4, 5, t}) = 10 + 9 = 19

|f | = 19

10/10

9/10

0/2 6/6

3/4

7/8

9/9

9/10

10/10

10− 0 + 9 = 199 + 7− 6 + 9 = 19

6.6: Maximum flow T.S. 11

From Flows to Cuts

A cut (S,T ) is a partition of V into S and T = V \ S such that s ∈ Sand t ∈ T .The capacity of a cut (S,T ) is the sum of capacities of the edgesfrom S to T :

c(S,T ) =∑

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