<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
parameterized	B-General_Concept
complexity	I-General_Concept
is	O
a	O
branch	O
of	O
computational	O
complexity	O
theory	O
that	O
focuses	O
on	O
classifying	O
computational	O
problems	O
according	O
to	O
their	O
inherent	O
difficulty	O
with	O
respect	O
to	O
multiple	O
parameters	O
of	O
the	O
input	O
or	O
output	O
.	O
</s>
<s>
The	O
first	O
systematic	O
work	O
on	O
parameterized	B-General_Concept
complexity	I-General_Concept
was	O
done	O
by	O
.	O
</s>
<s>
Problems	O
in	O
which	O
some	O
parameter	O
is	O
fixed	O
are	O
called	O
parameterized	B-General_Concept
problems	O
.	O
</s>
<s>
A	O
parameterized	B-General_Concept
problem	O
that	O
allows	O
for	O
such	O
an	O
fpt-algorithm	O
is	O
said	O
to	O
be	O
a	O
fixed-parameter	O
tractable	O
problem	O
and	O
belongs	O
to	O
the	O
class	O
,	O
and	O
the	O
early	O
name	O
of	O
the	O
theory	O
of	O
parameterized	B-General_Concept
complexity	I-General_Concept
was	O
fixed-parameter	O
tractability	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
parameterized	B-General_Concept
complexity	I-General_Concept
can	O
be	O
seen	O
as	O
two-dimensional	O
complexity	O
theory	O
.	O
</s>
<s>
A	O
parameterized	B-General_Concept
problem	O
is	O
a	O
language	O
,	O
where	O
is	O
a	O
finite	O
alphabet	O
.	O
</s>
<s>
A	O
parameterized	B-General_Concept
problem	O
is	O
fixed-parameter	O
tractable	O
if	O
the	O
question	O
"	O
?	O
"	O
</s>
<s>
An	O
example	O
is	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
,	O
parameterised	O
by	O
the	O
number	O
of	O
variables	O
.	O
</s>
<s>
Kernelization	B-General_Concept
is	O
a	O
preprocessing	O
technique	O
that	O
reduces	O
the	O
original	O
instance	O
to	O
its	O
"	O
hard	O
kernel	O
"	O
,	O
a	O
possibly	O
much	O
smaller	O
instance	O
that	O
is	O
equivalent	O
to	O
the	O
original	O
instance	O
but	O
has	O
a	O
size	O
that	O
is	O
bounded	O
by	O
a	O
function	O
in	O
the	O
parameter	O
.	O
</s>
<s>
FPT	O
is	O
closed	O
under	O
a	O
parameterised	O
notion	O
of	O
reductions	B-Algorithm
called	O
fpt-reductions	O
.	O
</s>
<s>
Such	O
reductions	B-Algorithm
transform	O
an	O
instance	O
of	O
some	O
problem	O
into	O
an	O
equivalent	O
instance	O
of	O
another	O
problem	O
(	O
with	O
)	O
and	O
can	O
be	O
computed	O
in	O
time	O
where	O
is	O
a	O
polynomial	O
.	O
</s>
<s>
Moreover	O
,	O
it	O
contains	O
all	O
optimisation	O
problems	O
in	O
NP	O
that	O
allow	O
an	O
efficient	B-Algorithm
polynomial-time	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
(	O
EPTAS	B-Algorithm
)	O
.	O
</s>
<s>
A	O
parameterized	B-General_Concept
problem	O
is	O
in	O
the	O
class	O
W[i],	O
if	O
every	O
instance	O
can	O
be	O
transformed	O
(	O
in	O
fpt-time	O
)	O
to	O
a	O
combinatorial	O
circuit	O
that	O
has	O
weft	O
at	O
most	O
i	O
,	O
such	O
that	O
if	O
and	O
only	O
if	O
there	O
is	O
a	O
satisfying	O
assignment	O
to	O
the	O
inputs	O
that	O
assigns	O
1	O
to	O
exactly	O
k	O
inputs	O
.	O
</s>
<s>
can	O
be	O
defined	O
using	O
the	O
family	O
of	O
Weighted	O
Weft--Depth	O
-	O
SAT	B-Algorithm
problems	I-Algorithm
for	O
:	O
</s>
<s>
is	O
the	O
class	O
of	O
parameterized	B-General_Concept
problems	O
that	O
fpt-reduce	O
to	O
this	O
problem	O
,	O
and	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
for	O
the	O
problem	O
Weighted	O
-Normalize	O
SAT	O
is	O
complete	O
for	O
under	O
fpt-reductions	O
.	O
</s>
<s>
XP	O
is	O
the	O
class	O
of	O
parameterized	B-General_Concept
problems	O
that	O
can	O
be	O
solved	O
in	O
time	O
for	O
some	O
computable	O
function	O
.	O
</s>
<s>
para-NP	O
is	O
the	O
class	O
of	O
parameterized	B-General_Concept
problems	O
that	O
can	O
be	O
solved	O
by	O
a	O
nondeterministic	O
algorithm	O
in	O
time	O
for	O
some	O
computable	O
function	O
.	O
</s>
<s>
A	O
parameterized	B-General_Concept
problem	O
that	O
is	O
-hard	O
cannot	O
belong	O
to	O
the	O
class	O
,	O
unless	O
.	O
</s>
<s>
A	O
classic	O
example	O
of	O
a	O
-hard	O
parameterized	B-General_Concept
problem	O
is	O
graph	O
coloring	O
,	O
parameterized	B-General_Concept
by	O
the	O
number	O
of	O
colors	O
,	O
which	O
is	O
already	O
-hard	O
for	O
(	O
see	O
Graph	O
coloring	O
#Computational	O
complexity	O
)	O
.	O
</s>
