Maximum Flow

Haidong Xue (Haydon)10/28/2013

What is a flow network?

• A no-selfloop, weak connected directed graph (V, E)

• Among vertices in V– There is one and only one source vertex – There is one and only one sink vertex – For each vertex v, there is a path

• No antiparallel edges in E– if

• A Capacity is defined on each edge:

V

E

What is a flow network?

• A flow on a flow network

– i.e. does not exceed (capasity constraint)

– (flow conservation)

Note: In some definition of flow networks, negative flows are allowed;in these slides, we keep consistent with the textbook

What is a flow network?• Are they flow networks with valid flows?

11/16

10/13

1/4

12/12

11/14

4/9 7/7

15/20

4/4

Red: flowGreen: capacity

11/16

8/13

1/4

12/12

4/9

15/2011/16

13/13

6/4

17/12

11/14

4/9 7/7

20/20

4/4

11/16

8/13

1/4

12/12

11/14

4/9 7/7

15/20

4/4

Not connected

Multiple Sinks

Violate the capacity constraint

Violate the flow conservation rule

What is a flow network?• Are they flow networks with valid flows?

Red: flowGreen: capacity

have a self-loop have an antiparallel edge

11/16

8/13

1/4

12/12

11/14

4/9 7/7

15/20

4/4

0/7

11/16

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0/10 1/4

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4/9 7/7

15/20

4/4

s t

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1/4

12/12

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4/9 7/7

15/20

4/4

1 source; 1 sink;no self-loop; no antiparallel edge;no flow exceed capacity;for all non-source non-sink vertices, total flow is 0.

Page 710 Fig 26.1(b)

What if the real problem has antiparallel edges or multiple sinks or sources?

v1

v2

10 4

v1

v2

10

4V’

10Add an intermediate vertex to eliminate antiparallel edges

s1

s2

s3

s4

s5

t3

t2

t1

ts

Add a super-source and a super sink

10

12

8

147

11

2

5

3

156

20

13

18

Why do we disallow antiparallel edges?

What is the max-flow problem?

s t

11/16

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1/4

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4/9 7/7

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4/4

The flow of a flow network, : the net output of the sourcei.e., -

In the above example, = 11+8=19

What is the max-flow problem?

s t

9/16

8/13

1/4

10/12

9/14

2/9 7/7

15/20

2/4

Given a flow network, there could be more than one valid flows:

= 17

s t

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1/4

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4/9 7/7

15/20

4/4

= 19

s t

12/16

11/13

4

12/12

11/14

9 7/7

19/20

4/4

= 23 (it is a max-flow)

The maximum-flow problem:Given a flow network, find a valid flow that has the maximum

Find a maximum flow

• Input: a flow network

• Output: a flow net work with max-flow

s t

16

13

4

12

14

9 7

20

4

s t

12/16

11/13

4

12/12

11/14

9 7/7

19/20

4/4

Find a maximum flow- residual networks• What are the possible modifications can be made on

an edge that does not break the capacity constraint?

s t

4/16

4/13

4/4

8/12

4/14

4/9 7

4/20

4/4

Increase it at most by 10 = c-f=14-4;Decrease it at most by 4=f.

To represent a possible modifications, residual networks are introduced

Find a maximum flow - residual networks

s t

𝑣3

𝑣4𝑣2

𝑣111/16

8/13

1/4

12/12

11/14

4/9 7/7

15/20

4/4

A flow network with a valid flow

A residual network of where

s t

𝑣3

𝑣4𝑣2

𝑣15

11

5

83 1

12

5 4

3

11

75

15

4

Find a maximum flow – augmenting paths

s t

𝑣3

𝑣4𝑣2

𝑣111/16

8/13

1/4

12/12

11/14

4/9 7/7

15/20

4/4

s t

𝑣3

𝑣4𝑣2

𝑣15

11

5

83 1

12

5 4

3

11

75

15

4

𝐺 𝐺 𝑓 𝑜𝑓 𝐺An augmenting path is a simple path from s to t in the residual network

s t

𝑣3

𝑣4𝑣2

𝑣15

11

5

83 1

12

5 4

3

11

75

15

4

Find a maximum flow – augmenting paths

An augmenting path is a simple path from s to t in the residual network

s t

𝑣3

𝑣4𝑣2

𝑣15

11

5

83 1

12

5 4

3

11

75

15

4

The residual capacity of an augmenting path:

𝑐 𝑓 (𝑝 )=4

The flow defined by

4/

4/4/

Guaranteed by Corollary 26.3 , modifying according to this flow on does not break the flow conservation rule

Find a maximum flow – augmenting paths

s t

𝑣3

𝑣4𝑣2

𝑣111/16

8/13

1/4

12/12

11/14

4/9 7/7

15/20

4/4

𝐺 h𝑤𝑖𝑡 𝑓 𝐺 𝑓 𝑜𝑓 𝐺 h𝑤𝑖𝑡 h𝑝𝑎𝑡 𝑝𝑎𝑛𝑑 𝑓𝑙𝑜𝑤 𝑓 𝑝

s t

𝑣3

𝑣4𝑣2

𝑣15

11

5

83 1

12

5 4

3

11

75

15

44/

4/4/

Modifyaccording to

s t

𝑣3

𝑣4𝑣2

𝑣111/16

12/13

1/4

12/12

11/14

0/97/7

19/20

4/4

𝐺 h𝑤𝑖𝑡 ↑ { } rsub { } 𝑓 𝑓 𝑝

Find a maximum flow

• Corollary 26.3 guarantees that:– is a valid flow in

• Theorem 26.6 (Max-flow min-cut theorem) further tells that:– If there is no augmenting path in , in is a

maximum flow

Do you have any idea to develop an algorithm to find a max-flow?

Find a maximum flow – the basic Ford-Fulkerson algorithm

FORD-FULKERSON() for each ; while there exist pathfrom toin for each in p if + else

Initialize to 0

Repeatedly find a , and apply the

modification until there is no

augmenting path

Is it correct? What is the time complexity?

Find a maximum flow – Analysis of F-F

• Is it correct in general?– No!

– A counter example:http://en.wikipedia.org/wiki/Ford%E2%80%93Fulkerson_algorithm#Non-terminating_example

• Is it correct with integer capacities?– Yes. – To find a, if using DFS or BFS that is O(|V|+|E|)=O(|E|)– At most, need to it times– Time complexity:

Find a maximum flow – Edmonds-Karp algorithm

EDMONDS-KARP() for each ; while there exist a BFS pathfrom toin for each in p if + else

Find a maximum flow – Analysis of E-K

• Theorem 26.8– The total number of flow augmentations performed by the

algorithm is O(|V||E|)• Time complexity

Find a maximum flow – Example

s t

16

13

4

12

14

9 7

20

4

𝐺

s t

16

13

4

12

14

9 7

20

4

𝐺 𝑓

s t

16

13

4

12

14

9 7

20

4

12/

12/12/

s t

4

13

4

12

14

9 7

8

4

12 12

s t

16

13

4

12

14

9 7

20

4

12/

12/12/

4/4/

4/

s t

4

9

4

12

10

9 7

8

4

12 12

44

Find a maximum flow – Example𝐺 𝐺 𝑓

s t

16

13

4

12

14

9 7

20

4

12/

12/12/

4/4/

4/

s t

4

9

4

12

10

9 7

8

4

12 12

44

s t

16

13

4

12

14

9 7

20

4

12/

12/

4/11/11/

7/

19/

s t

4

2

4

12

3

9 7

1

4

12 19

1111

cannot be reached by A max-flow

Maximum bipartite matching

• A problem can be converted to the max-flow problem

• Bipartite Graph– A undirected graph (V, E)– , – All edges in E go betweenand– Every vertex in V has at least one incident edge

Maximum bipartite matchingRLRLRL

Maximum bipartite matching

RL

A matching ,

RL RL

|𝑀|=1 |𝑀|=2 |𝑀|=3

The maximum bipartite matching problem: find a M with the maximum

Maximum bipartite matchingConvert it to a max-flow problem

RL RL

s t

Set each edge with a capacity of 1

By Lemma 26.9,

Summary

• Max-flow problem– Flow network – The flow of – The residual network – A augmenting path of – , – Ford-Fulkerson, Edmonds-Karps

• Maximum bipartite Matching– Bipartite graph– Matching– Convert a bipartite graph to a flow net work

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