<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
optimization	O
theory	O
,	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
states	O
that	O
in	O
a	O
flow	B-Algorithm
network	I-Algorithm
,	O
the	O
maximum	O
amount	O
of	O
flow	O
passing	O
from	O
the	O
source	O
to	O
the	O
sink	O
is	O
equal	O
to	O
the	O
total	O
weight	O
of	O
the	O
edges	O
in	O
a	O
minimum	O
cut	O
,	O
i.e.	O
,	O
the	O
smallest	O
total	O
weight	O
of	O
the	O
edges	O
which	O
if	O
removed	O
would	O
disconnect	O
the	O
source	O
from	O
the	O
sink	O
.	O
</s>
<s>
This	O
is	O
a	O
special	O
case	O
of	O
the	O
duality	B-Algorithm
theorem	O
for	O
linear	B-Algorithm
programs	I-Algorithm
and	O
can	O
be	O
used	O
to	O
derive	O
Menger	O
's	O
theorem	O
and	O
the	O
Kőnig	O
–	O
Egerváry	O
theorem	O
.	O
</s>
<s>
The	O
theorem	O
equates	O
two	O
quantities	O
:	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
through	O
a	O
network	O
,	O
and	O
the	O
minimum	O
capacity	O
of	O
a	O
cut	O
of	O
the	O
network	O
.	O
</s>
<s>
The	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
asks	O
for	O
the	O
largest	O
flow	O
on	O
a	O
given	O
network	O
.	O
</s>
<s>
Maximum	B-Algorithm
Flow	I-Algorithm
Problem	I-Algorithm
.	I-Algorithm
</s>
<s>
The	O
other	O
half	O
of	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
refers	O
to	O
a	O
different	O
aspect	O
of	O
a	O
network	O
:	O
the	O
collection	O
of	O
cuts	O
.	O
</s>
<s>
The	O
main	O
theorem	O
links	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
value	O
with	O
the	O
minimum	O
cut	O
capacity	O
of	O
the	O
network	O
.	O
</s>
<s>
Max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
The	O
value	O
of	O
the	O
flow	O
is	O
equal	O
to	O
the	O
capacity	O
of	O
the	O
cut	O
,	O
showing	O
that	O
the	O
flow	O
is	O
a	O
maximal	B-Algorithm
flow	I-Algorithm
and	O
the	O
cut	O
is	O
a	O
minimal	B-Algorithm
cut	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
the	O
flow	O
through	O
each	O
of	O
the	O
two	O
arrows	O
that	O
connect	O
S	O
to	O
T	O
is	O
at	O
full	O
capacity	O
;	O
this	O
is	O
always	O
the	O
case	O
:	O
a	O
minimal	B-Algorithm
cut	I-Algorithm
represents	O
a	O
'	O
bottleneck	O
 '	O
of	O
the	O
system	O
.	O
</s>
<s>
The	O
max-flow	B-Algorithm
problem	I-Algorithm
and	O
min-cut	O
problem	O
can	O
be	O
formulated	O
as	O
two	O
primal-dual	B-Algorithm
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
The	O
max-flow	B-Algorithm
LP	O
is	O
straightforward	O
.	O
</s>
<s>
The	O
dual	O
LP	O
is	O
obtained	O
using	O
the	O
algorithm	O
described	O
in	O
dual	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
:	O
the	O
variables	O
and	O
sign	O
constraints	O
of	O
the	O
dual	O
correspond	O
to	O
the	O
constraints	O
of	O
the	O
primal	O
,	O
and	O
the	O
constraints	O
of	O
the	O
dual	O
correspond	O
to	O
the	O
variables	O
and	O
sign	O
constraints	O
of	O
the	O
primal	O
.	O
</s>
<s>
The	O
equality	O
in	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
follows	O
from	O
the	O
strong	B-Algorithm
duality	I-Algorithm
theorem	I-Algorithm
in	I-Algorithm
linear	I-Algorithm
programming	I-Algorithm
,	O
which	O
states	O
that	O
if	O
the	O
primal	O
program	O
has	O
an	O
optimal	O
solution	O
,	O
x*	O
,	O
then	O
the	O
dual	O
program	O
also	O
has	O
an	O
optimal	O
solution	O
,	O
y*	O
,	O
such	O
that	O
the	O
optimal	O
values	O
formed	O
by	O
the	O
two	O
solutions	O
are	O
equal	O
.	O
</s>
<s>
The	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
can	O
be	O
formulated	O
as	O
the	O
maximization	O
of	O
the	O
electrical	O
current	O
through	O
a	O
network	O
composed	O
of	O
nonlinear	O
resistive	O
elements	O
.	O
</s>
<s>
In	O
this	O
new	O
definition	O
,	O
the	O
generalized	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
states	O
that	O
the	O
maximum	O
value	O
of	O
an	O
s-t	O
flow	O
is	O
equal	O
to	O
the	O
minimum	O
capacity	O
of	O
an	O
s-t	O
cut	O
in	O
the	O
new	O
sense	O
.	O
</s>
<s>
By	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
,	O
one	O
can	O
solve	O
the	O
problem	O
as	O
a	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
.	I-Algorithm
</s>
<s>
It	O
was	O
not	O
long	O
after	O
this	O
until	O
the	O
main	O
result	O
,	O
Theorem	O
5.1	O
,	O
which	O
we	O
call	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
,	O
was	O
conjectured	O
and	O
established	O
.	O
</s>
<s>
Consider	O
the	O
flow	O
computed	O
for	O
by	O
Ford	B-Algorithm
–	I-Algorithm
Fulkerson	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
the	O
residual	O
graph	O
obtained	O
for	O
(	O
after	O
the	O
final	O
flow	O
assignment	O
by	O
Ford	B-Algorithm
–	I-Algorithm
Fulkerson	I-Algorithm
algorithm	I-Algorithm
)	O
,	O
define	O
two	O
subsets	O
of	O
vertices	O
as	O
follows	O
:	O
</s>
<s>
Also	O
,	O
the	O
flow	O
was	O
obtained	O
by	O
Ford-Fulkerson	B-Algorithm
algorithm	I-Algorithm
,	O
so	O
it	O
is	O
the	O
max-flow	B-Algorithm
of	O
the	O
network	O
as	O
well	O
.	O
</s>
<s>
Also	O
,	O
since	O
any	O
flow	O
in	O
the	O
network	O
is	O
always	O
less	O
than	O
or	O
equal	O
to	O
capacity	O
of	O
every	O
cut	O
possible	O
in	O
a	O
network	O
,	O
the	O
above	O
described	O
cut	O
is	O
also	O
the	O
min-cut	O
which	O
obtains	O
the	O
max-flow	B-Algorithm
.	O
</s>
<s>
A	O
corollary	O
from	O
this	O
proof	O
is	O
that	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
through	O
any	O
set	O
of	O
edges	O
in	O
a	O
cut	O
of	O
a	O
graph	O
is	O
equal	O
to	O
the	O
minimum	O
capacity	O
of	O
all	O
previous	O
cuts	O
.	O
</s>
