<s>
In	O
mathematics	O
,	O
a	O
submodular	B-Algorithm
set	I-Algorithm
function	I-Algorithm
(	O
also	O
known	O
as	O
a	O
submodular	B-Algorithm
function	I-Algorithm
)	O
is	O
a	O
set	O
function	O
whose	O
value	O
,	O
informally	O
,	O
has	O
the	O
property	O
that	O
the	O
difference	O
in	O
the	O
incremental	O
value	O
of	O
the	O
function	O
that	O
a	O
single	O
element	O
makes	O
when	O
added	O
to	O
an	O
input	O
set	O
decreases	O
as	O
the	O
size	O
of	O
the	O
input	O
set	O
increases	O
.	O
</s>
<s>
Submodular	B-Algorithm
functions	I-Algorithm
have	O
a	O
natural	O
diminishing	O
returns	O
property	O
which	O
makes	O
them	O
suitable	O
for	O
many	O
applications	O
,	O
including	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
game	O
theory	O
(	O
as	O
functions	O
modeling	O
user	O
preferences	O
)	O
and	O
electrical	O
networks	O
.	O
</s>
<s>
Recently	O
,	O
submodular	B-Algorithm
functions	I-Algorithm
have	O
also	O
found	O
immense	O
utility	O
in	O
several	O
real	O
world	O
problems	O
in	O
machine	O
learning	O
and	O
artificial	B-Application
intelligence	I-Application
,	O
including	O
automatic	B-Application
summarization	I-Application
,	O
multi-document	B-General_Concept
summarization	I-General_Concept
,	O
feature	B-General_Concept
selection	I-General_Concept
,	O
active	B-General_Concept
learning	I-General_Concept
,	O
sensor	O
placement	O
,	O
image	O
collection	O
summarization	O
and	O
many	O
other	O
domains	O
.	O
</s>
<s>
If	O
is	O
a	O
finite	O
set	O
,	O
a	O
submodular	B-Algorithm
function	I-Algorithm
is	O
a	O
set	O
function	O
,	O
where	O
denotes	O
the	O
power	O
set	O
of	O
,	O
which	O
satisfies	O
one	O
of	O
the	O
following	O
equivalent	O
conditions	O
.	O
</s>
<s>
A	O
nonnegative	O
submodular	B-Algorithm
function	I-Algorithm
is	O
also	O
a	O
subadditive	B-Algorithm
function	O
,	O
but	O
a	O
subadditive	B-Algorithm
function	O
need	O
not	O
be	O
submodular	B-Algorithm
.	O
</s>
<s>
Examples	O
of	O
monotone	O
submodular	B-Algorithm
functions	I-Algorithm
include	O
:	O
</s>
<s>
Then	O
for	O
any	O
we	O
have	O
that	O
is	O
a	O
submodular	B-Algorithm
function	I-Algorithm
,	O
where	O
is	O
the	O
entropy	O
of	O
the	O
set	O
of	O
random	O
variables	O
,	O
a	O
fact	O
known	O
as	O
Shannon	O
's	O
inequality	O
.	O
</s>
<s>
Then	O
the	O
rank	O
function	O
of	O
the	O
matroid	O
is	O
a	O
submodular	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
A	O
submodular	B-Algorithm
function	I-Algorithm
that	O
is	O
not	O
monotone	O
is	O
called	O
non-monotone	O
.	O
</s>
<s>
A	O
non-monotone	O
submodular	B-Algorithm
function	I-Algorithm
is	O
called	O
symmetric	O
if	O
for	O
every	O
we	O
have	O
that	O
.	O
</s>
<s>
Examples	O
of	O
symmetric	O
non-monotone	O
submodular	B-Algorithm
functions	I-Algorithm
include	O
:	O
</s>
<s>
Then	O
for	O
any	O
we	O
have	O
that	O
is	O
a	O
submodular	B-Algorithm
function	I-Algorithm
,	O
where	O
is	O
the	O
mutual	O
information	O
.	O
</s>
<s>
A	O
non-monotone	O
submodular	B-Algorithm
function	I-Algorithm
which	O
is	O
not	O
symmetric	O
is	O
called	O
asymmetric	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
submodular	B-Algorithm
functions	I-Algorithm
,	O
there	O
are	O
a	O
few	O
examples	O
of	O
continuous	O
extensions	O
that	O
are	O
commonly	O
used	O
,	O
which	O
are	O
described	O
as	O
follows	O
.	O
</s>
<s>
The	O
Lovász	O
extension	O
is	O
a	O
convex	O
function	O
if	O
and	O
only	O
if	O
is	O
a	O
submodular	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
For	O
the	O
extensions	O
discussed	O
above	O
,	O
it	O
can	O
be	O
shown	O
that	O
when	O
is	O
submodular	B-Algorithm
.	O
</s>
<s>
The	O
class	O
of	O
submodular	B-Algorithm
functions	I-Algorithm
is	O
closed	O
under	O
non-negative	O
linear	O
combinations	O
.	O
</s>
<s>
Consider	O
any	O
submodular	B-Algorithm
function	I-Algorithm
and	O
non-negative	O
numbers	O
.	O
</s>
<s>
Then	O
the	O
function	O
defined	O
by	O
is	O
submodular	B-Algorithm
.	O
</s>
<s>
For	O
any	O
submodular	B-Algorithm
function	I-Algorithm
,	O
the	O
function	O
defined	O
by	O
is	O
submodular	B-Algorithm
.	O
</s>
<s>
The	O
function	O
,	O
where	O
is	O
a	O
real	O
number	O
,	O
is	O
submodular	B-Algorithm
whenever	O
is	O
monotone	O
submodular	B-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
is	O
submodular	B-Algorithm
,	O
for	O
any	O
non	O
decreasing	O
concave	O
function	O
.	O
</s>
<s>
Submodular	B-Algorithm
functions	I-Algorithm
have	O
properties	O
which	O
are	O
very	O
similar	O
to	O
convex	O
and	O
concave	O
functions	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
an	O
optimization	O
problem	O
which	O
concerns	O
optimizing	O
a	O
convex	O
or	O
concave	O
function	O
can	O
also	O
be	O
described	O
as	O
the	O
problem	O
of	O
maximizing	O
or	O
minimizing	O
a	O
submodular	B-Algorithm
function	I-Algorithm
subject	O
to	O
some	O
constraints	O
.	O
</s>
<s>
The	O
hardness	O
of	O
minimizing	O
a	O
submodular	B-Algorithm
set	I-Algorithm
function	I-Algorithm
depends	O
on	O
constraints	O
imposed	O
on	O
the	O
problem	O
.	O
</s>
<s>
The	O
unconstrained	O
problem	O
of	O
minimizing	O
a	O
submodular	B-Algorithm
function	I-Algorithm
is	O
computable	O
in	O
polynomial	O
time	O
,	O
and	O
even	O
in	O
strongly-polynomial	O
time	O
.	O
</s>
<s>
The	O
problem	O
of	O
minimizing	O
a	O
submodular	B-Algorithm
function	I-Algorithm
with	O
a	O
cardinality	O
lower	O
bound	O
is	O
NP-hard	O
,	O
with	O
polynomial	O
factor	O
lower	O
bounds	O
on	O
the	O
approximation	O
factor	O
.	O
</s>
<s>
Unlike	O
the	O
case	O
of	O
minimization	O
,	O
maximizing	O
a	O
generic	O
submodular	B-Algorithm
function	I-Algorithm
is	O
NP-hard	O
even	O
in	O
the	O
unconstrained	O
setting	O
.	O
</s>
<s>
Thus	O
,	O
most	O
of	O
the	O
works	O
in	O
this	O
field	O
are	O
concerned	O
with	O
polynomial-time	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
including	O
greedy	B-Algorithm
algorithms	I-Algorithm
or	O
local	B-Algorithm
search	I-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
of	O
maximizing	O
a	O
non-negative	O
submodular	B-Algorithm
function	I-Algorithm
admits	O
a	O
1/2	O
approximation	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
of	O
maximizing	O
a	O
monotone	O
submodular	B-Algorithm
function	I-Algorithm
subject	O
to	O
a	O
cardinality	O
constraint	O
admits	O
a	O
approximation	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
of	O
maximizing	O
a	O
monotone	O
submodular	B-Algorithm
function	I-Algorithm
subject	O
to	O
a	O
matroid	O
constraint	O
(	O
which	O
subsumes	O
the	O
case	O
above	O
)	O
also	O
admits	O
a	O
approximation	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Apart	O
from	O
submodular	B-Algorithm
minimization	O
and	O
maximization	O
,	O
there	O
are	O
several	O
other	O
natural	O
optimization	O
problems	O
related	O
to	O
submodular	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Minimizing	O
the	O
difference	O
between	O
two	O
submodular	B-Algorithm
functions	I-Algorithm
is	O
not	O
only	O
NP	O
hard	O
,	O
but	O
also	O
inapproximable	O
.	O
</s>
<s>
Minimization/maximization	O
of	O
a	O
submodular	B-Algorithm
function	I-Algorithm
subject	O
to	O
a	O
submodular	B-Algorithm
level	O
set	O
constraint	O
(	O
also	O
known	O
as	O
submodular	B-Algorithm
optimization	O
subject	O
to	O
submodular	B-Algorithm
cover	O
or	O
submodular	B-Algorithm
knapsack	O
constraint	O
)	O
admits	O
bounded	O
approximation	O
guarantees	O
.	O
</s>
<s>
Partitioning	O
data	O
based	O
on	O
a	O
submodular	B-Algorithm
function	I-Algorithm
to	O
maximize	O
the	O
average	O
welfare	O
is	O
known	O
as	O
the	O
submodular	B-Algorithm
welfare	O
problem	O
,	O
which	O
also	O
admits	O
bounded	O
approximation	O
guarantees	O
(	O
see	O
welfare	O
maximization	O
)	O
.	O
</s>
<s>
Submodular	B-Algorithm
functions	I-Algorithm
naturally	O
occur	O
in	O
several	O
real	O
world	O
applications	O
,	O
in	O
economics	O
,	O
game	O
theory	O
,	O
machine	O
learning	O
and	O
computer	B-Application
vision	I-Application
.	O
</s>
<s>
Owing	O
to	O
the	O
diminishing	O
returns	O
property	O
,	O
submodular	B-Algorithm
functions	I-Algorithm
naturally	O
model	O
costs	O
of	O
items	O
,	O
since	O
there	O
is	O
often	O
a	O
larger	O
discount	O
,	O
with	O
an	O
increase	O
in	O
the	O
items	O
one	O
buys	O
.	O
</s>
<s>
Submodular	B-Algorithm
functions	I-Algorithm
model	O
notions	O
of	O
complexity	O
,	O
similarity	O
and	O
cooperation	O
when	O
they	O
appear	O
in	O
minimization	O
problems	O
.	O
</s>
