<s>
In	O
combinatorics	O
and	O
computer	B-General_Concept
science	I-General_Concept
,	O
covering	O
problems	O
are	O
computational	O
problems	O
that	O
ask	O
whether	O
a	O
certain	O
combinatorial	O
structure	O
'	O
covers	O
 '	O
another	O
,	O
or	O
how	O
large	O
the	O
structure	O
has	O
to	O
be	O
to	O
do	O
that	O
.	O
</s>
<s>
Covering	O
problems	O
are	O
minimization	O
problems	O
and	O
usually	O
integer	B-Algorithm
linear	I-Algorithm
programs	I-Algorithm
,	O
whose	O
dual	O
problems	O
are	O
called	O
packing	O
problems	O
.	O
</s>
<s>
The	O
most	O
prominent	O
examples	O
of	O
covering	O
problems	O
are	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
,	O
which	O
is	O
equivalent	O
to	O
the	O
hitting	O
set	O
problem	O
,	O
and	O
its	O
special	O
cases	O
,	O
the	O
vertex	O
cover	O
problem	O
and	O
the	O
edge	O
cover	O
problem	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
linear	B-Algorithm
programming	I-Algorithm
,	O
one	O
can	O
think	O
of	O
any	O
linear	B-Algorithm
program	I-Algorithm
as	O
a	O
covering	O
problem	O
if	O
the	O
coefficients	O
in	O
the	O
constraint	O
matrix	B-Architecture
,	O
the	O
objective	O
function	O
,	O
and	O
right-hand	O
side	O
are	O
nonnegative	O
.	O
</s>
<s>
More	O
precisely	O
,	O
consider	O
the	O
following	O
general	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
:	O
</s>
<s>
Such	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
is	O
called	O
a	O
covering	O
problem	O
if	O
for	O
all	O
and	O
.	O
</s>
<s>
Finally	O
,	O
an	O
optimal	O
solution	O
to	O
the	O
above	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
is	O
a	O
covering	O
of	O
minimal	O
cost	O
.	O
</s>
<s>
For	O
Petri	B-Operating_System
nets	I-Operating_System
,	O
for	O
example	O
,	O
the	O
covering	O
problem	O
is	O
defined	O
as	O
the	O
question	O
if	O
for	O
a	O
given	O
marking	O
,	O
there	O
exists	O
a	O
run	O
of	O
the	O
net	O
,	O
such	O
that	O
some	O
larger	O
(	O
or	O
equal	O
)	O
marking	O
can	O
be	O
reached	O
.	O
</s>
<s>
In	O
particular	O
,	O
in	O
the	O
rainbow	B-Algorithm
covering	I-Algorithm
problem	O
,	O
each	O
of	O
the	O
original	O
objects	O
has	O
a	O
"	O
color	O
"	O
,	O
and	O
it	O
is	O
required	O
that	O
the	O
covering	O
contains	O
exactly	O
one	O
(	O
or	O
at	O
most	O
one	O
)	O
object	O
of	O
each	O
color	O
.	O
</s>
<s>
Rainbow	B-Algorithm
covering	I-Algorithm
was	O
studied	O
e.g.	O
</s>
<s>
The	O
Rainbow	B-Algorithm
covering	I-Algorithm
problem	O
is	O
the	O
problem	O
of	O
finding	O
a	O
rainbow	O
set	O
Q	O
that	O
is	O
a	O
covering	O
of	O
P	O
.	O
</s>
<s>
The	O
problem	O
is	O
NP-hard	O
(	O
by	O
reduction	O
from	O
linear	B-Algorithm
SAT	I-Algorithm
)	O
.	O
</s>
<s>
Conflict-free	O
set	B-Algorithm
cover	I-Algorithm
is	O
the	O
problem	O
of	O
finding	O
a	O
conflict-free	O
subset	O
of	O
O	O
that	O
is	O
a	O
covering	O
of	O
P	O
.	O
Banik	O
,	O
Panolan	O
,	O
Raman	O
,	O
Sahlot	O
and	O
Saurabh	O
prove	O
the	O
following	O
for	O
the	O
special	O
case	O
in	O
which	O
the	O
conflict-graph	O
has	O
bounded	O
arboricity	O
:	O
</s>
<s>
If	O
the	O
geometric	O
cover	O
problem	O
is	O
fixed-parameter	B-General_Concept
tractable	O
(	O
FPT	O
)	O
,	O
then	O
the	O
conflict-free	O
geometric	O
cover	O
problem	O
is	O
FPT	O
.	O
</s>
