Flows
More Flow
An important value we will be worried about is the
value of flow
f
=

f

=
∑
v
∈
V
f
(
s, v
)
: The total flow out of the source.
The Maximum Flow Problem
Given
G
= (
V, E
)
. source node
s
∈
V
, sink node
t
∈
V
, edge capacities
c
. Find a flow whose value is maximum.
Jeff Linderoth
(Lehigh University)
IE170:Lecture 24
Lecture Notes
5 / 24
Flows
Lemma, Lemma, Lemma
Recall Shorthand
f
(
X, Y
) =
x
∈
X y
∈
Y
f
(
x, y
)
.
1
f
(
X, X
) = 0
∀
X
⊆
V
2
f
(
X, Y
) =

f
(
Y, X
)
∀
X, Y
⊆
V
3
Let
X, Y, Z
⊂
V
be such that
X
∩
Y
=
∅
, then
f
(
X
∪
Y, Z
)
=
f
(
X, Z
) +
f
(
Y, Z
)
f
(
Z, X
∪
Y
)
=
f
(
Z, X
) +
f
(
Z, Y
)
4

f

=
f
(
V, t
)
Jeff Linderoth
(Lehigh University)
IE170:Lecture 24
Lecture Notes
6 / 24
Flows
Cuts
A
cut
of a (flow) network
G
= (
V, E
)
is a partition of
V
into
S
and
T
=
V
\
S
such that
s
∈
S
and
t
∈
T
For flow
f
, net flow across a cut is
f
(
S, T
)
and the cuts capacity is
c
(
S, T
) =
∑
u
∈
S
∑
v
∈
T
c
(
u, v
)
A
minimum cut
of
G
is a cut whose capacity is minimum
Jeff Linderoth
(Lehigh University)
IE170:Lecture 24
Lecture Notes
7 / 24
Flows
A Simple Upper Bound
Flow Across Cuts Lemma
For any cut
(
S, T
)
, f
(
S, T
) =

f

Coronary :)
The value of any flow is no more than the capacity of any cut

f

=
f
(
S, T
) =
u
∈
S v
∈
T
f
(
u, v
)
≤
u
∈