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1 The max-flow min-cut theorem A directed graph or digraph is a pair ,, where is a finite set and ⊆. A network is a quadruple ,,,, where is a digraph, ,∈ are distinct, and :→0,∞. s … source t … sink c … capacity function Example

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The max-flow min-cut theorem

A directed graph or digraph is a pair � � ��, ��, where � is a

finite set and � ⊆ � �.

A network is a quadruple � � ��, �, , ��, where � is a digraph,

�, ∈ �� are distinct, and �: �� → �0,∞�. s … source t … sink c … capacity function

Example

Page 2: The max-flow min-cut theorem - Georgia Institute of Technologypeople.math.gatech.edu › ~thomas › TEACH › 6014 › lecture4.pdf · (Max-flow min-cut theorem, Ford & Fulkerson)

Notation. For � ⊆ �� ≔ �� ∈ ��: �has tail in�,head in�� ≔ �� ∈ ��: �has head in�,tail in��

�� , �� . If !: �� → ", then

!�� ≔ # !��,$∈%&�'�

!�� ≔ # !��$∈%(�'�

Example

Page 3: The max-flow min-cut theorem - Georgia Institute of Technologypeople.math.gatech.edu › ~thomas › TEACH › 6014 › lecture4.pdf · (Max-flow min-cut theorem, Ford & Fulkerson)

A flow in � is a mapping !: �� → R such that

(i) 0 * !�� * �� ∀� ∈ �� (capacity constraints)

(ii) !�� !�� ∀ ∈ �� , � (conservation conditions)

Example

Page 4: The max-flow min-cut theorem - Georgia Institute of Technologypeople.math.gatech.edu › ~thomas › TEACH › 6014 › lecture4.pdf · (Max-flow min-cut theorem, Ford & Fulkerson)

Lemma. If ! is a flow in a network � � ��, �, , �� and � ⊆ �� with � ∈ �, ∉ �, then

!�� !�� !�� !��

Definition. !�� !�� is the value of !, denoted by val�!�. Proof. !�� !�� ∀ ∈ �� , �. Sum over all ∈ � � ��

!�� !��, �0� + !��, �� + # !��$2345672

$894:8'��4�

�

� !�� !��0 , �� + !��, �� + # !��$2345672

$894:8'��4�

!��, �0� + !��, �� !��, !��0 , �� + !��, �� !��

Page 5: The max-flow min-cut theorem - Georgia Institute of Technologypeople.math.gatech.edu › ~thomas › TEACH › 6014 › lecture4.pdf · (Max-flow min-cut theorem, Ford & Fulkerson)

Corollary. If ! is a flow in a network � � ��, �, , ��, then

!�� !�� !�� !�� Proof. Apply previous lemma to � � �� . □

A cut in a network � is a set of edges of the form �� for some

set � ⊆ �� with � ∈ �, ∉ �. The capacity of a cut < is

cap�<� ≔ #��$∈?

If < � ��, then cap�<� � ��.

Corollary. val�!� * cap�<� for every flow ! and every cut < in

�.

Proof. Let < � ��. val�!� � !�� !�� !�� !��

� # !��$∈%&�'�

� # !��$∈%(�'�

* # ��$∈%&�'�

� cap�<�

□

Page 6: The max-flow min-cut theorem - Georgia Institute of Technologypeople.math.gatech.edu › ~thomas › TEACH › 6014 › lecture4.pdf · (Max-flow min-cut theorem, Ford & Fulkerson)

Lemma. In any network, there exists a flow of maximum value.

Proof. Let � � ��, �, , �� be a network, and let

@ ≔ sup�val�!� ∶ !isaflowin�}.

There exists a sequence of flows !I, !J, … such that

LM�!8� ≥ @ � I8.

Let �I, �J, … , �O be the edges of �. The sequence �!8��I��8PIQ has

a convergent subsequence �!8��I��8∈RS. Let !��I� be the limit.

The sequence �!8��J��8∈RS has a convergent subsequence

�!8��J��8∈RT. Let !��J� be the limit. After U steps we construct a

flow !. Then ! is a flow of value @. □

Page 7: The max-flow min-cut theorem - Georgia Institute of Technologypeople.math.gatech.edu › ~thomas › TEACH › 6014 › lecture4.pdf · (Max-flow min-cut theorem, Ford & Fulkerson)

Theorem. (Max-flow min-cut theorem, Ford & Fulkerson) In any

network � there exists a flow !∗ and cut <∗ such that

val�!∗� � cap�<∗�. If the capacity function is integral (takes on integer values only),

then !∗ can be chosen integral.

Proof. Let � � ��, �, , ��, and let ! be a flow of maximum value.

By an augmenting path we mean a path W in the underlying

undirected multigraph such that

(i) � is an end of W

(ii) !�� X �� for every “forward” edge � ∈ ��W� (iii) !�� Y 0 for every “backward” edge � ∈ ��W�

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