<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
kernelization	B-General_Concept
is	O
a	O
technique	O
for	O
designing	O
efficient	O
algorithms	O
that	O
achieve	O
their	O
efficiency	O
by	O
a	O
preprocessing	O
stage	O
in	O
which	O
inputs	O
to	O
the	O
algorithm	O
are	O
replaced	O
by	O
a	O
smaller	O
input	O
,	O
called	O
a	O
"	O
kernel	O
"	O
.	O
</s>
<s>
Kernelization	B-General_Concept
is	O
often	O
achieved	O
by	O
applying	O
a	O
set	O
of	O
reduction	O
rules	O
that	O
cut	O
away	O
parts	O
of	O
the	O
instance	O
that	O
are	O
easy	O
to	O
handle	O
.	O
</s>
<s>
In	O
parameterized	B-General_Concept
complexity	I-General_Concept
theory	I-General_Concept
,	O
it	O
is	O
often	O
possible	O
to	O
prove	O
that	O
a	O
kernel	O
with	O
guaranteed	O
bounds	O
on	O
the	O
size	O
of	O
a	O
kernel	O
(	O
as	O
a	O
function	O
of	O
some	O
parameter	O
associated	O
to	O
the	O
problem	O
)	O
can	O
be	O
found	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
When	O
this	O
is	O
possible	O
,	O
it	O
results	O
in	O
a	O
fixed-parameter	O
tractable	O
algorithm	O
whose	O
running	O
time	O
is	O
the	O
sum	O
of	O
the	O
(	O
polynomial	O
time	O
)	O
kernelization	B-General_Concept
step	O
and	O
the	O
(	O
non-polynomial	O
but	O
bounded	O
by	O
the	O
parameter	O
)	O
time	O
to	O
solve	O
the	O
kernel	O
.	O
</s>
<s>
Indeed	O
,	O
every	O
problem	O
that	O
can	O
be	O
solved	O
by	O
a	O
fixed-parameter	O
tractable	O
algorithm	O
can	O
be	O
solved	O
by	O
a	O
kernelization	B-General_Concept
algorithm	O
of	O
this	O
type	O
.	O
</s>
<s>
This	O
is	O
also	O
true	O
for	O
approximate	O
kernelization	B-General_Concept
.	O
</s>
<s>
A	O
standard	O
example	O
for	O
a	O
kernelization	B-General_Concept
algorithm	O
is	O
the	O
kernelization	B-General_Concept
of	O
the	O
vertex	O
cover	O
problem	O
by	O
S	O
.	O
Buss	O
.	O
</s>
<s>
This	O
kernelization	B-General_Concept
may	O
be	O
implemented	O
in	O
linear	O
time	O
.	O
</s>
<s>
Once	O
the	O
kernel	O
has	O
been	O
constructed	O
,	O
the	O
vertex	O
cover	O
problem	O
may	O
be	O
solved	O
by	O
a	O
brute	B-Algorithm
force	I-Algorithm
search	I-Algorithm
algorithm	O
that	O
tests	O
whether	O
each	O
subset	O
of	O
the	O
kernel	O
is	O
a	O
cover	O
of	O
the	O
kernel	O
.	O
</s>
<s>
More	O
complex	O
kernelization	B-General_Concept
procedures	O
can	O
improve	O
this	O
bound	O
,	O
by	O
finding	O
smaller	O
kernels	O
,	O
at	O
the	O
expense	O
of	O
greater	O
running	O
time	O
in	O
the	O
kernelization	B-General_Concept
step	O
.	O
</s>
<s>
In	O
the	O
vertex	O
cover	O
example	O
,	O
kernelization	B-General_Concept
algorithms	O
are	O
known	O
that	O
produce	O
kernels	O
with	O
at	O
most	O
vertices	O
.	O
</s>
<s>
Another	O
kernelization	B-General_Concept
algorithm	O
achieving	O
that	O
bound	O
is	O
based	O
on	O
what	O
is	O
known	O
as	O
the	O
crown	O
reduction	O
rule	O
and	O
uses	O
alternating	O
path	O
arguments	O
.	O
</s>
<s>
The	O
currently	O
best	O
known	O
kernelization	B-General_Concept
algorithm	O
in	O
terms	O
of	O
the	O
number	O
of	O
vertices	O
is	O
due	O
to	O
and	O
achieves	O
vertices	O
for	O
any	O
fixed	O
constant	O
.	O
</s>
<s>
In	O
the	O
literature	O
,	O
there	O
is	O
no	O
clear	O
consensus	O
on	O
how	O
kernelization	B-General_Concept
should	O
be	O
formally	O
defined	O
and	O
there	O
are	O
subtle	O
differences	O
in	O
the	O
uses	O
of	O
that	O
expression	O
.	O
</s>
<s>
In	O
the	O
notation	O
of	O
,	O
a	O
parameterized	B-General_Concept
problem	O
is	O
a	O
subset	O
describing	O
a	O
decision	O
problem	O
.	O
</s>
<s>
The	O
output	O
of	O
kernelization	B-General_Concept
is	O
called	O
a	O
kernel	O
.	O
</s>
<s>
In	O
the	O
notation	O
of	O
,	O
a	O
parameterized	B-General_Concept
problem	O
consists	O
of	O
a	O
decision	O
problem	O
and	O
a	O
function	O
,	O
the	O
parameterization	O
.	O
</s>
<s>
First	O
the	O
kernelization	B-General_Concept
algorithm	O
,	O
which	O
runs	O
in	O
time	O
for	O
some	O
c	O
,	O
is	O
invoked	O
to	O
generate	O
a	O
kernel	O
of	O
size	O
.	O
</s>
<s>
Assume	O
that	O
the	O
question	O
is	O
non-trivial	O
,	O
meaning	O
that	O
there	O
is	O
at	O
least	O
one	O
instance	O
that	O
is	O
in	O
the	O
language	O
,	O
called	O
,	O
and	O
at	O
least	O
one	O
instance	O
that	O
is	O
not	O
in	O
the	O
language	O
,	O
called	O
;	O
otherwise	O
,	O
replacing	O
any	O
instance	O
by	O
the	O
empty	O
string	O
is	O
a	O
valid	O
kernelization	B-General_Concept
.	O
</s>
<s>
Bidimensional	B-General_Concept
problems	O
:	O
Many	O
parameterized	B-General_Concept
versions	O
of	O
bidimensional	B-General_Concept
problems	O
have	O
linear	O
kernels	O
on	O
planar	O
graphs	O
,	O
and	O
more	O
generally	O
,	O
on	O
graphs	O
excluding	O
some	O
fixed	O
graph	O
as	O
a	O
minor	O
.	O
</s>
<s>
The	O
vertex	O
cover	O
problem	O
parameterized	B-General_Concept
by	O
the	O
feedback	O
vertex	O
number	O
of	O
the	O
input	O
graph	O
has	O
a	O
polynomial	O
kernelization	B-General_Concept
:	O
There	O
is	O
a	O
polynomial-time	O
algorithm	O
that	O
,	O
given	O
a	O
graph	O
whose	O
feedback	O
vertex	O
number	O
is	O
,	O
outputs	O
a	O
graph	O
on	O
vertices	O
such	O
that	O
a	O
minimum	O
vertex	O
cover	O
in	O
can	O
be	O
transformed	O
into	O
a	O
minimum	O
vertex	O
cover	O
for	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
The	O
kernelization	B-General_Concept
algorithm	O
therefore	O
guarantees	O
that	O
instances	O
with	O
a	O
small	O
feedback	O
vertex	O
number	O
are	O
reduced	O
to	O
small	O
instances	O
.	O
</s>
