<s>
In	O
computability	O
theory	O
and	O
computational	O
complexity	O
theory	O
,	O
especially	O
the	O
study	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
an	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
is	O
an	O
algorithm	O
for	O
transforming	O
one	O
optimization	O
problem	O
into	O
another	O
problem	O
,	O
such	O
that	O
the	O
distance	O
of	O
solutions	O
from	O
optimal	O
is	O
preserved	O
to	O
some	O
degree	O
.	O
</s>
<s>
Approximation-preserving	B-Algorithm
reductions	I-Algorithm
are	O
a	O
subset	O
of	O
more	O
general	O
reductions	O
in	O
complexity	O
theory	O
;	O
the	O
difference	O
is	O
that	O
approximation-preserving	B-Algorithm
reductions	I-Algorithm
usually	O
make	O
statements	O
on	O
approximation	O
problems	O
or	O
optimization	O
problems	O
,	O
as	O
opposed	O
to	O
decision	O
problems	O
.	O
</s>
<s>
Intuitively	O
,	O
problem	O
A	O
is	O
reducible	B-Algorithm
to	O
problem	O
B	O
via	O
an	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
if	O
,	O
given	O
an	O
instance	O
of	O
problem	O
A	O
and	O
a	O
(	O
possibly	O
approximate	O
)	O
solver	O
for	O
problem	O
B	O
,	O
one	O
can	O
convert	O
the	O
instance	O
of	O
problem	O
A	O
into	O
an	O
instance	O
of	O
problem	O
B	O
,	O
apply	O
the	O
solver	O
for	O
problem	O
B	O
,	O
and	O
recover	O
a	O
solution	O
for	O
problem	O
A	O
that	O
also	O
has	O
some	O
guarantee	O
of	O
approximation	O
.	O
</s>
<s>
Unlike	O
reductions	O
on	O
decision	O
problems	O
,	O
an	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
must	O
preserve	O
more	O
than	O
the	O
truth	O
of	O
the	O
problem	O
instances	O
when	O
reducing	O
from	O
one	O
problem	O
to	O
another	O
.	O
</s>
<s>
An	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
is	O
a	O
pair	O
of	O
functions	O
(	O
which	O
often	O
must	O
be	O
computable	O
in	O
polynomial	O
time	O
)	O
,	O
such	O
that	O
:	O
</s>
<s>
preserves	O
some	O
guarantee	O
of	O
the	O
solution	O
's	O
performance	O
,	O
or	O
approximation	B-Algorithm
ratio	I-Algorithm
,	O
defined	O
as	O
.	O
</s>
<s>
There	O
are	O
many	O
different	O
types	O
of	O
approximation-preserving	B-Algorithm
reductions	I-Algorithm
,	O
all	O
of	O
which	O
have	O
a	O
different	O
guarantee	O
(	O
the	O
third	O
point	O
in	O
the	O
definition	O
above	O
)	O
.	O
</s>
<s>
However	O
,	O
unlike	O
with	O
other	O
reductions	O
,	O
approximation-preserving	B-Algorithm
reductions	I-Algorithm
often	O
overlap	O
in	O
what	O
properties	O
they	O
demonstrate	O
on	O
optimization	O
problems	O
(	O
e.g.	O
</s>
<s>
Not	O
all	O
types	O
of	O
approximation-preserving	B-Algorithm
reductions	I-Algorithm
can	O
be	O
used	O
to	O
show	O
membership	O
in	O
all	O
approximability	B-Algorithm
complexity	O
classes	O
,	O
the	O
most	O
notable	O
of	O
which	O
are	O
PTAS	B-Algorithm
and	O
APX	B-Algorithm
.	O
</s>
<s>
Some	O
reductions	O
shown	O
below	O
only	O
preserve	O
membership	O
in	O
APX	B-Algorithm
or	O
PTAS	B-Algorithm
,	O
but	O
not	O
the	O
other	O
.	O
</s>
<s>
Because	O
of	O
this	O
,	O
careful	O
choice	O
must	O
be	O
made	O
when	O
selecting	O
an	O
approximation-preserving	B-Algorithm
reductions	I-Algorithm
,	O
especially	O
for	O
the	O
purpose	O
of	O
proving	O
completeness	O
of	O
a	O
problem	O
within	O
a	O
complexity	O
class	O
.	O
</s>
<s>
Crescenzi	O
suggests	O
that	O
the	O
three	O
most	O
ideal	O
styles	O
of	O
reduction	O
,	O
for	O
both	O
ease	O
of	O
use	O
and	O
proving	O
power	O
,	O
are	O
PTAS	B-Algorithm
reduction	I-Algorithm
,	O
AP	O
reduction	O
,	O
and	O
L-reduction	B-Algorithm
.	O
</s>
<s>
The	O
reduction	O
descriptions	O
that	O
follow	O
are	O
from	O
Crescenzi	O
's	O
survey	O
of	O
approximation-preserving	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
Strict	O
reduction	O
is	O
the	O
simplest	O
type	O
of	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
In	O
a	O
strict	O
reduction	O
,	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
a	O
solution	O
y	O
 '	O
to	O
an	O
instance	O
x	O
 '	O
of	O
a	O
problem	O
B	O
must	O
be	O
at	O
most	O
as	O
good	O
as	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
the	O
corresponding	O
solution	O
y	O
to	O
instance	O
x	O
of	O
problem	O
A	O
.	O
</s>
<s>
Strict	O
reduction	O
preserves	O
membership	O
in	O
both	O
PTAS	B-Algorithm
and	O
APX	B-Algorithm
.	O
</s>
<s>
L-reductions	B-Algorithm
preserve	O
membership	O
in	O
PTAS	B-Algorithm
as	O
well	O
as	O
APX	B-Algorithm
(	O
but	O
only	O
for	O
minimization	O
problems	O
in	O
the	O
case	O
of	O
the	O
latter	O
)	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
they	O
cannot	O
be	O
used	O
in	O
general	O
to	O
prove	O
completeness	O
results	O
about	O
APX	B-Algorithm
,	O
Log-APX	B-Algorithm
,	O
or	O
Poly-APX	B-Algorithm
,	O
but	O
nevertheless	O
they	O
are	O
valued	O
for	O
their	O
natural	O
formulation	O
and	O
ease	O
of	O
use	O
in	O
proofs	O
.	O
</s>
<s>
PTAS-reduction	O
is	O
another	O
commonly	O
used	O
reduction	O
scheme	O
.	O
</s>
<s>
Though	O
it	O
preserves	O
membership	O
in	O
PTAS	B-Algorithm
,	O
it	O
does	O
not	O
do	O
so	O
for	O
APX	B-Algorithm
.	O
</s>
<s>
Nevertheless	O
,	O
APX-completeness	O
is	O
defined	O
in	O
terms	O
of	O
PTAS	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
PTAS-reductions	O
are	O
a	O
generalization	O
of	O
P-reductions	O
,	O
shown	O
below	O
,	O
with	O
the	O
only	O
difference	O
being	O
that	O
the	O
function	O
is	O
allowed	O
to	O
depend	O
on	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
.	O
</s>
<s>
A-reduction	O
and	O
P-reduction	O
are	O
similar	O
reduction	O
schemes	O
that	O
can	O
be	O
used	O
to	O
show	O
membership	O
in	O
APX	B-Algorithm
and	O
PTAS	B-Algorithm
respectively	O
.	O
</s>
<s>
The	O
existence	O
of	O
a	O
P-reduction	O
implies	O
the	O
existence	O
of	O
a	O
PTAS-reduction	O
.	O
</s>
<s>
E-reduction	O
,	O
which	O
is	O
a	O
generalization	O
of	O
strict	O
reduction	O
but	O
implies	O
both	O
A-reduction	O
and	O
P-reduction	O
,	O
is	O
an	O
example	O
of	O
a	O
less	O
restrictive	O
reduction	O
style	O
that	O
preserves	O
membership	O
not	O
only	O
in	O
PTAS	B-Algorithm
and	O
APX	B-Algorithm
,	O
but	O
also	O
the	O
larger	O
classes	O
Log-APX	B-Algorithm
and	O
Poly-APX	B-Algorithm
.	O
</s>
<s>
AP-reductions	O
are	O
used	O
to	O
define	O
completeness	O
in	O
the	O
classes	O
Log-APX	B-Algorithm
and	O
Poly-APX	B-Algorithm
.	O
</s>
<s>
They	O
are	O
a	O
special	O
case	O
of	O
PTAS	B-Algorithm
reduction	I-Algorithm
,	O
meeting	O
the	O
following	O
restrictions	O
.	O
</s>
<s>
with	O
the	O
additional	O
generalization	O
that	O
the	O
function	O
is	O
allowed	O
to	O
depend	O
on	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
,	O
as	O
in	O
PTAS-reduction	O
.	O
</s>
<s>
An	O
additional	O
restriction	O
actually	O
needs	O
to	O
be	O
imposed	O
for	O
AP-reduction	O
to	O
preserve	O
Log-APX	B-Algorithm
and	O
Poly-APX	B-Algorithm
membership	O
,	O
as	O
E-reduction	O
does	O
:	O
for	O
fixed	O
problem	O
size	O
,	O
the	O
computation	O
time	O
of	O
must	O
be	O
non-increasing	O
as	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
increases	O
.	O
</s>
