<s>
In	O
graph	O
theory	O
,	O
a	O
cut	O
is	O
a	O
partition	O
of	O
the	O
vertices	O
of	O
a	O
graph	O
into	O
two	O
disjoint	B-Algorithm
subsets	I-Algorithm
.	O
</s>
<s>
In	O
a	O
flow	B-Algorithm
network	I-Algorithm
,	O
an	O
s	O
–	O
t	O
cut	O
is	O
a	O
cut	O
that	O
requires	O
the	O
source	O
and	O
the	O
sink	O
to	O
be	O
in	O
different	O
subsets	O
,	O
and	O
its	O
cut-set	O
only	O
consists	O
of	O
edges	O
going	O
from	O
the	O
source	O
's	O
side	O
to	O
the	O
sink	O
's	O
side	O
.	O
</s>
<s>
The	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
proves	O
that	O
the	O
maximum	O
network	B-Algorithm
flow	I-Algorithm
and	O
the	O
sum	O
of	O
the	O
cut-edge	O
weights	O
of	O
any	O
minimum	O
cut	O
that	O
separates	O
the	O
source	O
and	O
the	O
sink	O
are	O
equal	O
.	O
</s>
<s>
There	O
are	O
polynomial-time	O
methods	O
to	O
solve	O
the	O
min-cut	O
problem	O
,	O
notably	O
the	O
Edmonds	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
A	O
cut	O
is	O
maximum	O
if	O
the	O
size	B-Algorithm
of	I-Algorithm
the	I-Algorithm
cut	I-Algorithm
is	O
not	O
smaller	O
than	O
the	O
size	O
of	O
any	O
other	O
cut	O
.	O
</s>
<s>
The	O
illustration	O
on	O
the	O
right	O
shows	O
a	O
maximum	O
cut	O
:	O
the	O
size	B-Algorithm
of	I-Algorithm
the	I-Algorithm
cut	I-Algorithm
is	O
equal	O
to	O
5	O
,	O
and	O
there	O
is	O
no	O
cut	O
of	O
size	O
6	O
,	O
or	O
|E|	O
(	O
the	O
number	O
of	O
edges	O
)	O
,	O
because	O
the	O
graph	O
is	O
not	O
bipartite	O
(	O
there	O
is	O
an	O
odd	O
cycle	O
)	O
.	O
</s>
<s>
The	O
max-cut	O
problem	O
is	O
also	O
APX-hard	B-Algorithm
,	O
meaning	O
that	O
there	O
is	O
no	O
polynomial-time	O
approximation	O
scheme	O
for	O
it	O
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
However	O
,	O
it	O
can	O
be	O
approximated	O
to	O
within	O
a	O
constant	O
approximation	B-Algorithm
ratio	I-Algorithm
using	O
semidefinite	O
programming	O
.	O
</s>
<s>
Note	O
that	O
min-cut	O
and	O
max-cut	O
are	O
not	O
dual	O
problems	O
in	O
the	O
linear	B-Algorithm
programming	I-Algorithm
sense	O
,	O
even	O
though	O
one	O
gets	O
from	O
one	O
problem	O
to	O
other	O
by	O
changing	O
min	O
to	O
max	O
in	O
the	O
objective	O
function	O
.	O
</s>
<s>
The	O
sparsest	B-Algorithm
cut	I-Algorithm
problem	I-Algorithm
is	O
to	O
bipartition	O
the	O
vertices	O
so	O
as	O
to	O
minimize	O
the	O
ratio	O
of	O
the	O
number	O
of	O
edges	O
across	O
the	O
cut	O
divided	O
by	O
the	O
number	O
of	O
vertices	O
in	O
the	O
smaller	O
half	O
of	O
the	O
partition	O
.	O
</s>
<s>
The	O
problem	O
is	O
known	O
to	O
be	O
NP-hard	O
,	O
and	O
the	O
best	O
known	O
approximation	B-Algorithm
algorithm	I-Algorithm
is	O
an	O
approximation	O
due	O
to	O
.	O
</s>
<s>
The	O
family	O
of	O
all	O
cut	B-Algorithm
sets	I-Algorithm
of	O
an	O
undirected	O
graph	O
is	O
known	O
as	O
the	O
cut	O
space	O
of	O
the	O
graph	O
.	O
</s>
<s>
It	O
forms	O
a	O
vector	O
space	O
over	O
the	O
two-element	O
finite	O
field	O
of	O
arithmetic	O
modulo	O
two	O
,	O
with	O
the	O
symmetric	O
difference	O
of	O
two	O
cut	B-Algorithm
sets	I-Algorithm
as	O
the	O
vector	O
addition	O
operation	O
,	O
and	O
is	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
the	O
cycle	O
space	O
.	O
</s>
<s>
If	O
the	O
edges	O
of	O
the	O
graph	O
are	O
given	O
positive	O
weights	O
,	O
the	O
minimum	O
weight	O
basis	O
of	O
the	O
cut	O
space	O
can	O
be	O
described	O
by	O
a	O
tree	O
on	O
the	O
same	O
vertex	O
set	O
as	O
the	O
graph	O
,	O
called	O
the	O
Gomory	B-Algorithm
–	I-Algorithm
Hu	I-Algorithm
tree	I-Algorithm
.	O
</s>
