3
The maximum fow problem
Although it is a special case of the minimumcost network Fow problem, the
maximum fow problem
is of great importance in its own right. This problem
can be motivated by the following setting. Imagine that you have a network
of pipes that are used to ship, for example, oil from its source, to where is it
is re±ned. Each pipe in the network can maintain a certain capacity of Fow
(per second), which depends on its crosssection, and other less signi±cant
factors. At what rate can we deliver oil to the re±nery?
The network of pipes corresponds to a directed graph
G
= (
N, E
). Each
directed edge (
i, j
)
∈
E
has a speci±ed capacity
u
ij
.
There is a speci±ed
source node
s
∈
N
and a speci±ed sink node
t
∈
N
. This is the entire input
to the maximum Fow problem.
To specify a solution for this input, we must give a Fow value
x
ij
for each
directed edge (
i.j
)
∈
E
. Such a solution
x
is feasible if
1. 0
≤
x
ij
≤
u
ij
for each (
i.j
)
∈
E
; and
2.
X
j
:(
j,i
)
∈
E
x
ji
=
X
j
:(
i,j
)
∈
E
x
ij
for each node
i
∈
N
 {
s, t
}
(i.e., each node that is neither the source nor the sink).
The ±rst type of constraints, the inequalities, are called
capacity constraints
and the second type, the equations, are called
fow conservation constraints
.
(These are exactly the constraints of the LP formulation that we gave in the
previous section.)
The value of a Fow
x
is the total net Fow into
t
, which is equal to
X
j
:(
j,t
)
∈
E
x
jt

X
k
:(
t,k
)
∈
E
x
tk
.
This is the objective function for the maximum Fow problem; in other words,
we wish to ±nd a feasible Fow of maximum value.
How big can the optimal Fow value be?
Partition the vertices
N
into a
set
S
containing the source
s
and a set
T
containing the sink
t
.
(The sets
S
and
T
form a partition of
N
if each node in
N
is in
exactly
one of
S
and
T
.) We shall call such a partition (
S, T
) a
cut
; note that the de±nition of a
cut requires that
s
∈
S
and
t
∈
T
. Observe that every unit of Fow that goes
from node
s
to node
t
must at some point pass along an edge (
i, j
)
∈
E
where
19
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∈
S
and
j
∈
T
.
However, there are only so many directed edges of this
form, and each such edge (
i, j
) has an upper bound
u
ij
on the total Fow that
can use it. If we let
D
(
S, T
) denote the set of directed edges (
i, j
) for which
i
∈
S
and
j
∈
T
, then the
capacity
of the cut (
S, T
) is equal to
∑
(
i,j
)
∈
D
(
S,T
)
u
ij
.
Since each unit of Fow from
s
to
t
“uses up” one unit of the capacity of the
cut (
S, T
), the value of the maximum Fow is at most the capacity of the
cut.
This claim is true for
any
cut (
S, T
).
±or the same input, some cuts
may have large capacity, and some cuts may have small capacity. However,
for each cut, its capacity places a limit on the value of the maximum Fow.
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 Spring '07
 SHMOYS/LEWIS
 Flow network, Maximum flow problem, Maxflow mincut theorem, uij, feasible flow

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