<s>
Graph	B-Algorithm
cut	I-Algorithm
optimization	I-Algorithm
is	O
a	O
combinatorial	O
optimization	O
method	O
applicable	O
to	O
a	O
family	O
of	O
functions	O
of	O
discrete	O
variables	O
,	O
named	O
after	O
the	O
concept	O
of	O
cut	B-Algorithm
in	O
the	O
theory	O
of	O
flow	B-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
Thanks	O
to	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
,	O
determining	O
the	O
minimum	O
cut	B-Algorithm
over	O
a	O
graph	O
representing	O
a	O
flow	B-Algorithm
network	I-Algorithm
is	O
equivalent	O
to	O
computing	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
over	O
the	O
network	O
.	O
</s>
<s>
then	O
it	O
is	O
possible	O
to	O
find	O
the	O
global	O
optimum	O
of	O
in	O
polynomial	O
time	O
by	O
computing	O
a	O
minimum	O
cut	B-Algorithm
of	O
the	O
graph	O
.	O
</s>
<s>
The	O
mapping	O
between	O
cuts	O
and	O
variable	O
assignments	O
is	O
done	O
by	O
representing	O
each	O
variable	O
with	O
one	O
node	O
in	O
the	O
graph	O
and	O
,	O
given	O
a	O
cut	B-Algorithm
,	O
each	O
variable	O
will	O
have	O
a	O
value	O
of	O
0	O
if	O
the	O
corresponding	O
node	O
belongs	O
to	O
the	O
component	O
connected	O
to	O
the	O
source	O
,	O
or	O
1	O
if	O
it	O
belong	O
to	O
the	O
component	O
connected	O
to	O
the	O
sink	O
.	O
</s>
<s>
Not	O
all	O
pseudo-Boolean	O
functions	O
can	O
be	O
represented	O
by	O
a	O
flow	B-Algorithm
network	I-Algorithm
,	O
and	O
in	O
the	O
general	O
case	O
the	O
global	O
optimization	O
problem	O
is	O
NP-hard	O
.	O
</s>
<s>
Graph	B-Algorithm
cut	I-Algorithm
optimization	I-Algorithm
can	O
be	O
extended	O
to	O
functions	O
of	O
discrete	O
variables	O
with	O
a	O
finite	O
number	O
of	O
values	O
,	O
that	O
can	O
be	O
approached	O
with	O
iterative	O
algorithms	O
with	O
strong	O
optimality	O
properties	O
,	O
computing	O
one	O
graph	O
cut	B-Algorithm
at	O
each	O
iteration	O
.	O
</s>
<s>
Graph	B-Algorithm
cut	I-Algorithm
optimization	I-Algorithm
is	O
an	O
important	O
tool	O
for	O
inference	O
over	O
graphical	B-Algorithm
models	I-Algorithm
such	O
as	O
Markov	O
random	O
fields	O
or	O
conditional	B-General_Concept
random	I-General_Concept
fields	I-General_Concept
,	O
and	O
it	O
has	O
applications	B-Algorithm
in	I-Algorithm
computer	I-Algorithm
vision	I-Algorithm
problems	I-Algorithm
such	O
as	O
image	B-Algorithm
segmentation	I-Algorithm
,	O
denoising	O
,	O
registration	B-Algorithm
and	O
stereo	B-Algorithm
matching	I-Algorithm
.	O
</s>
<s>
A	O
pseudo-Boolean	O
function	O
is	O
said	O
to	O
be	O
representable	O
if	O
there	O
exists	O
a	O
graph	O
with	O
non-negative	O
weights	O
and	O
with	O
source	O
and	O
sink	O
nodes	O
and	O
respectively	O
,	O
and	O
there	O
exists	O
a	O
set	O
of	O
nodes	O
such	O
that	O
,	O
for	O
each	O
tuple	O
of	O
values	O
assigned	O
to	O
the	O
variables	O
,	O
equals	O
(	O
up	O
to	O
a	O
constant	O
)	O
the	O
value	O
of	O
the	O
flow	O
determined	O
by	O
a	O
minimum	O
cut	B-Algorithm
of	O
the	O
graph	O
such	O
that	O
if	O
and	O
if	O
.	O
</s>
<s>
After	O
building	O
a	O
graph	O
representing	O
a	O
pseudo-Boolean	O
function	O
,	O
it	O
is	O
possible	O
to	O
compute	O
a	O
minimum	O
cut	B-Algorithm
using	O
one	O
among	O
the	O
various	O
algorithms	O
developed	O
for	O
flow	B-Algorithm
networks	I-Algorithm
,	O
such	O
as	O
Ford	B-Algorithm
–	I-Algorithm
Fulkerson	I-Algorithm
,	O
Edmonds	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
,	O
and	O
Boykov	O
–	O
Kolmogorov	O
algorithm	O
.	O
</s>
<s>
Max-flow	B-Algorithm
algorithms	O
such	O
as	O
BoykovKolmogorov	O
's	O
are	O
very	O
efficient	O
in	O
practice	O
for	O
sequential	O
computation	O
,	O
but	O
they	O
are	O
difficult	O
to	O
parallelise	O
,	O
making	O
them	O
not	O
suitable	O
for	O
distributed	B-Architecture
computing	I-Architecture
applications	O
and	O
preventing	O
them	O
from	O
exploiting	O
the	O
potential	O
of	O
modern	O
CPUs	O
.	O
</s>
<s>
Parallel	O
max-flow	B-Algorithm
algorithms	O
were	O
developed	O
,	O
such	O
as	O
push-relabel	B-Algorithm
and	O
jump-flood	O
,	O
that	O
can	O
also	O
take	O
advantage	O
of	O
hardware	O
acceleration	O
in	O
GPGPU	B-Architecture
implementations	O
.	O
</s>
<s>
In	O
the	O
general	O
case	O
,	O
optimization	O
of	O
such	O
functions	O
is	O
a	O
NP-hard	O
problem	O
,	O
and	O
stochastic	B-Algorithm
optimization	I-Algorithm
methods	O
such	O
as	O
simulated	B-Algorithm
annealing	I-Algorithm
are	O
sensitive	O
to	O
local	O
minima	O
and	O
in	O
practice	O
they	O
can	O
generate	O
arbitrarily	O
sub-optimal	O
results	O
.	O
</s>
<s>
In	O
both	O
cases	O
,	O
the	O
optimization	O
problem	O
in	O
the	O
innermost	O
loop	O
can	O
be	O
solved	O
exactly	O
and	O
efficiently	O
with	O
a	O
graph	O
cut	B-Algorithm
.	O
</s>
<s>
Generally	O
speaking	O
,	O
the	O
problem	O
of	O
optimizing	O
a	O
non-submodular	O
pseudo-Boolean	O
function	O
is	O
NP-hard	O
and	O
cannot	O
be	O
solved	O
in	O
polynomial	O
time	O
with	O
a	O
simple	O
graph	O
cut	B-Algorithm
.	O
</s>
<s>
In	O
case	O
of	O
quadratic	O
non-submodular	O
functions	O
,	O
it	O
is	O
possible	O
to	O
compute	O
in	O
polynomial	O
time	O
a	O
partial	O
solution	O
using	O
algorithms	O
such	O
as	O
QPBO	B-Algorithm
.	O
</s>
<s>
Higher-order	O
functions	O
can	O
be	O
reduced	O
in	O
polynomial	O
time	O
to	O
a	O
quadratic	O
form	O
that	O
can	O
be	O
optimised	O
with	O
QPBO	B-Algorithm
.	O
</s>
<s>
For	O
instance	O
in	O
computer	B-Application
vision	I-Application
applications	O
,	O
where	O
each	O
variable	O
represents	O
a	O
pixel	B-Algorithm
or	O
voxel	B-Algorithm
of	O
the	O
image	O
,	O
higher-order	O
interactions	O
can	O
be	O
used	O
to	O
model	O
texture	O
information	O
,	O
that	O
would	O
be	O
difficult	O
to	O
capture	O
using	O
only	O
quadratic	O
functions	O
.	O
</s>
