<s>
Approximate	B-Algorithm
max-flow	I-Algorithm
min-cut	I-Algorithm
theorems	I-Algorithm
are	O
mathematical	O
propositions	O
in	O
network	B-Algorithm
flow	I-Algorithm
theory	O
.	O
</s>
<s>
They	O
deal	O
with	O
the	O
relationship	O
between	O
maximum	B-Algorithm
flow	I-Algorithm
rate	O
(	O
"	O
max-flow	B-Algorithm
"	O
)	O
and	O
minimum	O
cut	O
(	O
"	O
min-cut	O
"	O
)	O
in	O
a	O
multi-commodity	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
theorems	O
have	O
enabled	O
the	O
development	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
for	O
use	O
in	O
graph	O
partition	O
and	O
related	O
problems	O
.	O
</s>
<s>
A	O
"	O
commodity	O
"	O
in	O
a	O
network	B-Algorithm
flow	I-Algorithm
problem	O
is	O
a	O
pair	O
of	O
source	O
and	O
sink	O
nodes	O
.	O
</s>
<s>
In	O
a	O
multi-commodity	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
,	O
there	O
are	O
commodities	O
,	O
each	O
with	O
its	O
own	O
source	O
,	O
sink	O
,	O
and	O
demand	O
.	O
</s>
<s>
Specially	O
,	O
a	O
1-commodity	O
(	O
or	O
single	O
commodity	O
)	O
flow	B-Algorithm
problem	I-Algorithm
is	O
also	O
known	O
as	O
a	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
According	O
to	O
the	O
Ford	B-Algorithm
–	I-Algorithm
Fulkerson	I-Algorithm
algorithm	I-Algorithm
,	O
the	O
max-flow	B-Algorithm
and	O
min-cut	O
are	O
always	O
equal	O
in	O
a	O
1-commodity	O
flow	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
In	O
a	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
,	O
max-flow	B-Algorithm
is	O
the	O
maximum	O
value	O
of	O
,	O
where	O
is	O
the	O
common	O
fraction	O
of	O
each	O
commodity	O
that	O
is	O
routed	O
,	O
such	O
that	O
units	O
of	O
commodity	O
can	O
be	O
simultaneously	O
routed	O
for	O
each	O
without	O
violating	O
any	O
capacity	O
constraints	O
.	O
</s>
<s>
Max-flow	B-Algorithm
is	O
always	O
upper	O
bounded	O
by	O
the	O
min-cut	O
for	O
a	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
In	O
a	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
,	O
there	O
is	O
a	O
commodity	O
for	O
every	O
pair	O
of	O
nodes	O
and	O
the	O
demand	O
for	O
every	O
commodity	O
is	O
the	O
same	O
.	O
</s>
<s>
In	O
a	O
product	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
,	O
there	O
is	O
a	O
nonnegative	O
weight	O
for	O
each	O
node	O
in	O
graph	O
.	O
</s>
<s>
The	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
is	O
a	O
special	O
case	O
of	O
the	O
product	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
for	O
which	O
the	O
weight	O
is	O
set	O
to	O
1	O
for	O
all	O
nodes	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
dual	O
of	O
a	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
for	O
a	O
graph	O
is	O
the	O
problem	O
of	O
apportioning	O
a	O
fixed	O
amount	O
of	O
weight	O
(	O
where	O
weights	O
can	O
be	O
considered	O
as	O
distances	O
)	O
to	O
the	O
edges	O
of	O
such	O
that	O
to	O
maximize	O
the	O
cumulative	O
distance	O
between	O
the	O
source	O
and	O
sink	O
pairs	O
.	O
</s>
<s>
The	O
research	O
on	O
the	O
relationship	O
between	O
the	O
max-flow	B-Algorithm
and	O
min-cut	O
of	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
has	O
obtained	O
great	O
interest	O
since	O
Ford	O
and	O
Fulkerson	O
's	O
result	O
for	O
1-commodity	O
flow	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
showed	O
that	O
the	O
max-flow	B-Algorithm
and	O
min-cut	O
are	O
always	O
equal	O
for	O
two	O
commodities	O
.	O
</s>
<s>
Okamura	O
and	O
Seymour	O
illustrated	O
a	O
4-commodity	O
flow	B-Algorithm
problem	I-Algorithm
with	O
max-flow	B-Algorithm
equals	O
to	O
3/4	O
and	O
min-cut	O
equals	O
1	O
.	O
</s>
<s>
Shahrokhi	O
and	O
Matula	O
also	O
proved	O
that	O
the	O
max-flow	B-Algorithm
and	O
min-cut	O
are	O
equal	O
provided	O
the	O
dual	O
of	O
the	O
flow	B-Algorithm
problem	I-Algorithm
satisfies	O
a	O
certain	O
cut	O
condition	O
in	O
a	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
For	O
a	O
general	O
network	B-Algorithm
flow	I-Algorithm
problem	O
,	O
the	O
max-flow	B-Algorithm
is	O
within	O
a	O
factor	O
of	O
of	O
the	O
min-cut	O
since	O
each	O
commodity	O
can	O
be	O
optimized	O
separately	O
using	O
of	O
the	O
capacity	O
of	O
each	O
edge	O
.	O
</s>
<s>
For	O
any	O
,	O
there	O
is	O
an	O
-node	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
with	O
max-flow	B-Algorithm
and	O
min-cut	O
for	O
which	O
.	O
</s>
<s>
For	O
any	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
,	O
,	O
where	O
is	O
the	O
max-flow	B-Algorithm
and	O
is	O
the	O
min-cut	O
of	O
the	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
To	O
prove	O
Theorem	O
2	O
,	O
both	O
the	O
max-flow	B-Algorithm
and	O
the	O
min-cut	O
should	O
be	O
discussed	O
.	O
</s>
<s>
For	O
the	O
max-flow	B-Algorithm
,	O
the	O
techniques	O
from	O
duality	O
theory	O
of	O
linear	O
programming	O
have	O
to	O
be	O
employed	O
.	O
</s>
<s>
According	O
to	O
the	O
duality	O
theory	O
of	O
linear	O
programming	O
,	O
an	O
optimal	O
distance	O
function	O
results	O
in	O
a	O
total	O
weight	O
that	O
is	O
equal	O
to	O
the	O
max-flow	B-Algorithm
of	O
the	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Stage	O
1	O
:	O
Consider	O
the	O
dual	O
of	O
uniform	O
commodity	O
flow	B-Algorithm
problem	I-Algorithm
and	O
use	O
the	O
optimal	O
solution	O
to	O
define	O
a	O
graph	O
with	O
distance	O
labels	O
on	O
the	O
edges	O
.	O
</s>
<s>
For	O
any	O
product	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
with	O
commodities	O
,	O
,	O
where	O
is	O
the	O
max-flow	B-Algorithm
and	O
is	O
the	O
min-cut	O
of	O
the	O
product	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
In	O
a	O
directed	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
,	O
each	O
edge	O
has	O
a	O
direction	O
,	O
and	O
the	O
flow	O
is	O
restricted	O
to	O
move	O
in	O
the	O
specified	O
direction	O
.	O
</s>
<s>
In	O
a	O
directed	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
,	O
the	O
demand	O
is	O
set	O
to	O
1	O
for	O
every	O
directed	O
edge	O
.	O
</s>
<s>
For	O
any	O
directed	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
with	O
nodes	O
,	O
,	O
where	O
is	O
the	O
max-flow	B-Algorithm
and	O
is	O
the	O
min-cut	O
of	O
the	O
uniform	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Similarly	O
,	O
for	O
product	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
,	O
we	O
have	O
the	O
following	O
extended	O
theorem	O
:	O
</s>
<s>
For	O
any	O
directed	O
product	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
with	O
commodities	O
,	O
,	O
where	O
is	O
the	O
max-flow	B-Algorithm
and	O
is	O
the	O
directed	O
min-cut	O
of	O
the	O
product	O
multicommodity	O
flow	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
above	O
theorems	O
are	O
very	O
useful	O
to	O
design	O
approximation	B-Algorithm
algorithms	I-Algorithm
for	O
NP-hard	O
problems	O
,	O
such	O
as	O
the	O
graph	O
partition	O
problem	O
and	O
its	O
variations	O
.	O
</s>
<s>
An	O
approximation	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
designed	O
for	O
this	O
problem	O
,	O
and	O
the	O
core	O
idea	O
is	O
that	O
has	O
a	O
-balanced	O
cut	O
of	O
size	O
,	O
then	O
we	O
find	O
a	O
-balanced	O
cut	O
of	O
size	O
for	O
any	O
where	O
and	O
.	O
</s>
<s>
An	O
approximation	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
introduced	O
and	O
the	O
result	O
is	O
times	O
optimal	O
.	O
</s>
<s>
It	O
remains	O
an	O
open	O
question	O
if	O
there	O
is	O
a	O
polylog	O
times	O
optimal	O
approximation	B-Algorithm
algorithm	I-Algorithm
for	O
.	O
</s>
