<s>
The	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
is	O
a	O
classical	O
question	O
in	O
combinatorics	O
,	O
computer	B-General_Concept
science	I-General_Concept
,	O
operations	O
research	O
,	O
and	O
complexity	O
theory	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
elements	O
(	O
called	O
the	O
universe	O
)	O
and	O
a	O
collection	O
of	O
sets	O
whose	O
union	O
equals	O
the	O
universe	O
,	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
is	O
to	O
identify	O
the	O
smallest	O
sub-collection	O
of	O
whose	O
union	O
equals	O
the	O
universe	O
.	O
</s>
<s>
In	O
the	O
set	B-Algorithm
covering	I-Algorithm
decision	O
problem	O
,	O
the	O
input	O
is	O
a	O
pair	O
and	O
an	O
integer	O
;	O
the	O
question	O
is	O
whether	O
there	O
is	O
a	O
set	B-Algorithm
covering	I-Algorithm
of	O
size	O
or	O
less	O
.	O
</s>
<s>
In	O
the	O
set	B-Algorithm
covering	I-Algorithm
optimization	O
problem	O
,	O
the	O
input	O
is	O
a	O
pair	O
,	O
and	O
the	O
task	O
is	O
to	O
find	O
a	O
set	B-Algorithm
covering	I-Algorithm
that	O
uses	O
the	O
fewest	O
sets	O
.	O
</s>
<s>
The	O
decision	O
version	O
of	O
set	B-Algorithm
covering	I-Algorithm
is	O
NP-complete	O
,	O
and	O
the	O
optimization/search	O
version	O
of	O
set	B-Algorithm
cover	I-Algorithm
is	O
NP-hard	O
.	O
</s>
<s>
It	O
is	O
a	O
problem	O
"	O
whose	O
study	O
has	O
led	O
to	O
the	O
development	O
of	O
fundamental	O
techniques	O
for	O
the	O
entire	O
field	O
"	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
If	O
each	O
set	O
is	O
assigned	O
a	O
cost	B-General_Concept
,	O
it	O
becomes	O
a	O
weighted	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
minimum	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
can	O
be	O
formulated	O
as	O
the	O
following	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
(	O
ILP	O
)	O
.	O
</s>
<s>
This	O
ILP	O
belongs	O
to	O
the	O
more	O
general	O
class	O
of	O
ILPs	O
for	O
covering	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
The	O
integrality	O
gap	O
of	O
this	O
ILP	O
is	O
at	O
most	O
,	O
so	O
its	O
relaxation	B-Algorithm
gives	O
a	O
factor	O
-	O
approximation	B-Algorithm
algorithm	I-Algorithm
for	O
the	O
minimum	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
(	O
where	O
is	O
the	O
size	O
of	O
the	O
universe	O
)	O
.	O
</s>
<s>
In	O
weighted	O
set	B-Algorithm
cover	I-Algorithm
,	O
the	O
sets	O
are	O
assigned	O
weights	O
.	O
</s>
<s>
Then	O
the	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
describing	O
weighted	O
set	B-Algorithm
cover	I-Algorithm
is	O
identical	O
to	O
the	O
one	O
given	O
above	O
,	O
except	O
that	O
the	O
objective	O
function	O
to	O
minimize	O
is	O
.	O
</s>
<s>
Set	B-Algorithm
covering	I-Algorithm
is	O
equivalent	O
to	O
the	O
hitting	O
set	O
problem	O
.	O
</s>
<s>
There	O
is	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
for	O
polynomial	O
time	O
approximation	O
of	O
set	B-Algorithm
covering	I-Algorithm
that	O
chooses	O
sets	O
according	O
to	O
one	O
rule	O
:	O
at	O
each	O
stage	O
,	O
choose	O
the	O
set	O
that	O
contains	O
the	O
largest	O
number	O
of	O
uncovered	O
elements	O
.	O
</s>
<s>
This	O
method	O
can	O
be	O
implemented	O
in	O
time	O
linear	O
in	O
the	O
sum	O
of	O
sizes	O
of	O
the	O
input	O
sets	O
,	O
using	O
a	O
bucket	B-Application
queue	I-Application
to	O
prioritize	O
the	O
sets	O
.	O
</s>
<s>
It	O
achieves	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
,	O
where	O
is	O
the	O
size	O
of	O
the	O
set	O
to	O
be	O
covered	O
.	O
</s>
<s>
This	O
greedy	B-Algorithm
algorithm	I-Algorithm
actually	O
achieves	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
where	O
is	O
the	O
maximum	O
cardinality	O
set	O
of	O
.	O
</s>
<s>
There	O
is	O
a	O
standard	O
example	O
on	O
which	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
achieves	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
.	O
</s>
<s>
A	O
tighter	O
analysis	O
for	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
shows	O
that	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
exactly	O
.	O
</s>
<s>
If	O
each	O
element	O
occurs	O
in	O
at	O
most	O
sets	O
,	O
then	O
a	O
solution	O
can	O
be	O
found	O
in	O
polynomial	O
time	O
that	O
approximates	O
the	O
optimum	O
to	O
within	O
a	O
factor	O
of	O
using	O
LP	B-Algorithm
relaxation	I-Algorithm
.	O
</s>
<s>
If	O
the	O
constraint	O
is	O
replaced	O
by	O
for	O
all	O
in	O
in	O
the	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
shown	O
above	O
,	O
then	O
it	O
becomes	O
a	O
(	O
non-integer	O
)	O
linear	O
program	O
.	O
</s>
<s>
When	O
refers	O
to	O
the	O
size	O
of	O
the	O
universe	O
,	O
showed	O
that	O
set	B-Algorithm
covering	I-Algorithm
cannot	O
be	O
approximated	O
in	O
polynomial	O
time	O
to	O
within	O
a	O
factor	O
of	O
,	O
unless	O
NP	O
has	O
quasi-polynomial	O
time	O
algorithms	O
.	O
</s>
<s>
Feige	O
(	O
1998	O
)	O
improved	O
this	O
lower	O
bound	O
to	O
under	O
the	O
same	O
assumptions	O
,	O
which	O
essentially	O
matches	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
achieved	O
by	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Relaxing	B-Algorithm
the	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
for	O
weighted	O
set	B-Algorithm
cover	I-Algorithm
stated	O
above	O
,	O
one	O
may	O
use	O
randomized	B-Algorithm
rounding	I-Algorithm
to	O
get	O
an	O
-factor	O
approximation	O
.	O
</s>
<s>
Non	O
weighted	O
set	B-Algorithm
cover	I-Algorithm
can	O
be	O
adapted	O
to	O
the	O
weighted	O
case	O
.	O
</s>
<s>
Hitting	O
set	O
is	O
an	O
equivalent	O
reformulation	O
of	O
Set	B-Algorithm
Cover	I-Algorithm
.	O
</s>
<s>
Edge	O
cover	O
is	O
a	O
special	O
case	O
of	O
Set	B-Algorithm
Cover	I-Algorithm
.	O
</s>
<s>
Geometric	O
set	B-Algorithm
cover	I-Algorithm
is	O
a	O
special	O
case	O
of	O
Set	B-Algorithm
Cover	I-Algorithm
when	O
the	O
universe	O
is	O
a	O
set	O
of	O
points	O
in	O
and	O
the	O
sets	O
are	O
induced	O
by	O
the	O
intersection	O
of	O
the	O
universe	O
and	O
geometric	O
shapes	O
(	O
e.g.	O
,	O
disks	O
,	O
rectangles	O
)	O
.	O
</s>
<s>
The	O
Dominating	O
set	O
problem	O
was	O
shown	O
to	O
be	O
NP	O
complete	O
through	O
a	O
reduction	O
from	O
Set	B-Algorithm
cover	I-Algorithm
.	O
</s>
<s>
Exact	O
cover	O
problem	O
is	O
to	O
choose	O
a	O
set	B-Algorithm
cover	I-Algorithm
with	O
no	O
element	O
included	O
in	O
more	O
than	O
one	O
covering	O
set	O
.	O
</s>
<s>
Red	O
Blue	O
Set	B-Algorithm
Cover	I-Algorithm
.	O
</s>
