<s>
If	O
is	O
submodular	O
then	O
QPBO	B-Algorithm
produces	O
a	O
global	O
optimum	O
equivalently	O
to	O
graph	B-Algorithm
cut	I-Algorithm
optimization	I-Algorithm
,	O
while	O
if	O
contains	O
non-submodular	O
terms	O
then	O
the	O
algorithm	O
produces	O
a	O
partial	O
solution	O
with	O
specific	O
optimality	O
properties	O
,	O
in	O
both	O
cases	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
QPBO	B-Algorithm
is	O
a	O
useful	O
tool	O
for	O
inference	O
on	O
Markov	O
random	O
fields	O
and	O
conditional	B-General_Concept
random	I-General_Concept
fields	I-General_Concept
,	O
and	O
has	O
applications	O
in	O
computer	B-Application
vision	I-Application
problems	O
such	O
as	O
image	B-Algorithm
segmentation	I-Algorithm
and	O
stereo	B-Algorithm
matching	I-Algorithm
.	O
</s>
<s>
then	O
the	O
function	O
can	O
be	O
efficiently	O
optimised	O
with	O
graph	B-Algorithm
cut	I-Algorithm
optimization	I-Algorithm
.	O
</s>
<s>
It	O
is	O
indeed	O
possible	O
to	O
represent	O
it	O
with	O
a	O
non-negative	O
weighted	O
graph	O
,	O
and	O
the	O
global	O
minimum	O
can	O
be	O
found	O
in	O
polynomial	O
time	O
by	O
computing	O
a	O
minimum	O
cut	O
of	O
the	O
graph	O
,	O
which	O
can	O
be	O
computed	O
with	O
algorithms	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
Kolmogorov	O
's	O
.	O
</s>
<s>
QPBO	B-Algorithm
builds	O
an	O
extended	O
graph	O
,	O
introducing	O
a	O
set	O
of	O
auxiliary	O
variables	O
ideally	O
equivalent	O
to	O
the	O
negation	O
of	O
the	O
variables	O
in	O
the	O
problem	O
.	O
</s>
<s>
QPBO	B-Algorithm
produces	O
a	O
solution	O
where	O
each	O
variable	O
assumes	O
one	O
of	O
three	O
possible	O
values	O
:	O
true	O
,	O
false	O
,	O
and	O
undefined	O
,	O
noted	O
in	O
the	O
following	O
as	O
1	O
,	O
0	O
,	O
and	O
respectively	O
.	O
</s>
<s>
Partial	O
optimality	O
:	O
if	O
is	O
submodular	O
,	O
then	O
QPBO	B-Algorithm
produces	O
a	O
global	O
minimum	O
exactly	O
,	O
equivalent	O
to	O
graph	B-Algorithm
cut	I-Algorithm
,	O
and	O
all	O
variables	O
have	O
a	O
non-undefined	O
value	O
;	O
if	O
submodularity	O
is	O
not	O
satisfied	O
,	O
the	O
result	O
will	O
be	O
a	O
partial	O
solution	O
where	O
a	O
subset	O
of	O
the	O
variables	O
have	O
a	O
non-undefined	O
value	O
.	O
</s>
<s>
Persistence	O
:	O
given	O
a	O
solution	O
generated	O
by	O
QPBO	B-Algorithm
and	O
an	O
arbitrary	O
assignment	O
of	O
values	O
to	O
the	O
variables	O
,	O
if	O
a	O
new	O
solution	O
is	O
constructed	O
by	O
replacing	O
with	O
for	O
each	O
,	O
then	O
.	O
</s>
<s>
The	O
minimum	O
cut	O
of	O
the	O
graph	O
can	O
be	O
computed	O
with	O
a	O
max-flow	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
It	O
is	O
always	O
possible	O
to	O
reduce	O
a	O
higher-order	O
function	O
to	O
a	O
quadratic	O
function	O
which	O
is	O
equivalent	O
with	O
respect	O
to	O
the	O
optimisation	O
,	O
problem	O
known	O
as	O
"	O
higher-order	O
clique	O
reduction	O
"	O
(	O
HOCR	O
)	O
,	O
and	O
the	O
result	O
of	O
such	O
reduction	O
can	O
be	O
optimized	O
with	O
QPBO	B-Algorithm
.	O
</s>
