<s>
L-reduction	B-Algorithm
(	O
"	O
linear	O
reduction	O
"	O
)	O
is	O
a	O
transformation	O
of	O
optimization	O
problems	O
which	O
linearly	O
preserves	O
approximability	B-Algorithm
features	O
;	O
it	O
is	O
one	O
type	O
of	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
L-reductions	B-Algorithm
in	O
studies	O
of	O
approximability	B-Algorithm
of	O
optimization	O
problems	O
play	O
a	O
similar	O
role	O
to	O
that	O
of	O
polynomial	B-Algorithm
reductions	I-Algorithm
in	O
the	O
studies	O
of	O
computational	O
complexity	O
of	O
decision	O
problems	O
.	O
</s>
<s>
The	O
term	O
L	O
reduction	O
is	O
sometimes	O
used	O
to	O
refer	O
to	O
log-space	B-Algorithm
reductions	I-Algorithm
,	O
by	O
analogy	O
with	O
the	O
complexity	O
class	O
L	O
,	O
but	O
this	O
is	O
a	O
different	O
concept	O
.	O
</s>
<s>
A	O
pair	O
of	O
functions	O
f	O
and	O
g	O
is	O
an	O
L-reduction	B-Algorithm
if	O
all	O
of	O
the	O
following	O
conditions	O
are	O
met	O
:	O
</s>
<s>
An	O
L-reduction	B-Algorithm
from	O
problem	O
A	O
to	O
problem	O
B	O
implies	O
an	O
AP-reduction	O
when	O
A	O
and	O
B	O
are	O
minimization	O
problems	O
and	O
a	O
PTAS	B-Algorithm
reduction	I-Algorithm
when	O
A	O
and	O
B	O
are	O
maximization	O
problems	O
.	O
</s>
<s>
In	O
both	O
cases	O
,	O
when	O
B	O
has	O
a	O
PTAS	O
and	O
there	O
is	O
an	O
L-reduction	B-Algorithm
from	O
A	O
to	O
B	O
,	O
then	O
A	O
also	O
has	O
a	O
PTAS	O
.	O
</s>
<s>
This	O
enables	O
the	O
use	O
of	O
L-reduction	B-Algorithm
as	O
a	O
replacement	O
for	O
showing	O
the	O
existence	O
of	O
a	O
PTAS-reduction	O
;	O
Crescenzi	O
has	O
suggested	O
that	O
the	O
more	O
natural	O
formulation	O
of	O
L-reduction	B-Algorithm
is	O
actually	O
more	O
useful	O
in	O
many	O
cases	O
due	O
to	O
ease	O
of	O
usage	O
.	O
</s>
<s>
Let	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
B	O
be	O
.	O
</s>
<s>
Begin	O
with	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
A	O
,	O
.	O
</s>
<s>
We	O
can	O
remove	O
absolute	O
values	O
around	O
the	O
third	O
condition	O
of	O
the	O
L-reduction	B-Algorithm
definition	O
since	O
we	O
know	O
A	O
and	O
B	O
are	O
minimization	O
problems	O
.	O
</s>
<s>
Thus	O
,	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
A	O
is	O
.	O
</s>
<s>
Let	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
B	O
be	O
.	O
</s>
<s>
Begin	O
with	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
A	O
,	O
.	O
</s>
<s>
We	O
can	O
remove	O
absolute	O
values	O
around	O
the	O
third	O
condition	O
of	O
the	O
L-reduction	B-Algorithm
definition	O
since	O
we	O
know	O
A	O
and	O
B	O
are	O
maximization	O
problems	O
.	O
</s>
<s>
Thus	O
,	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
A	O
is	O
.	O
</s>
<s>
If	O
,	O
then	O
,	O
which	O
meets	O
the	O
requirements	O
for	O
PTAS	B-Algorithm
reduction	I-Algorithm
but	O
not	O
AP-reduction	O
.	O
</s>
<s>
L-reductions	B-Algorithm
also	O
imply	O
P-reduction	O
.	O
</s>
<s>
One	O
may	O
deduce	O
that	O
L-reductions	B-Algorithm
imply	O
PTAS	B-Algorithm
reductions	I-Algorithm
from	O
this	O
fact	O
and	O
the	O
fact	O
that	O
P-reductions	O
imply	O
PTAS	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
L-reductions	B-Algorithm
preserve	O
membership	O
in	O
APX	O
for	O
the	O
minimizing	O
case	O
only	O
,	O
as	O
a	O
result	O
of	O
implying	O
AP-reductions	O
.	O
</s>
