<s>
In	O
computational	O
complexity	O
theory	O
,	O
a	O
PTAS	B-Algorithm
reduction	I-Algorithm
is	O
an	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
that	O
is	O
often	O
used	O
to	O
perform	O
reductions	B-Algorithm
between	O
solutions	O
to	O
optimization	O
problems	O
.	O
</s>
<s>
It	O
preserves	O
the	O
property	O
that	O
a	O
problem	O
has	O
a	O
polynomial	B-Algorithm
time	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
(	O
PTAS	O
)	O
and	O
is	O
used	O
to	O
define	O
completeness	O
for	O
certain	O
classes	O
of	O
optimization	O
problems	O
such	O
as	O
APX	B-Algorithm
.	O
</s>
<s>
Notationally	O
,	O
if	O
there	O
is	O
a	O
PTAS	B-Algorithm
reduction	I-Algorithm
from	O
a	O
problem	O
A	O
to	O
a	O
problem	O
B	O
,	O
we	O
write	O
.	O
</s>
<s>
With	O
ordinary	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reductions	I-Algorithm
,	O
if	O
we	O
can	O
describe	O
a	O
reduction	B-Algorithm
from	O
a	O
problem	O
A	O
to	O
a	O
problem	O
B	O
,	O
then	O
any	O
polynomial-time	O
solution	O
for	O
B	O
can	O
be	O
composed	O
with	O
that	O
reduction	B-Algorithm
to	O
obtain	O
a	O
polynomial-time	O
solution	O
for	O
the	O
problem	O
A	O
.	O
</s>
<s>
Similarly	O
,	O
our	O
goal	O
in	O
defining	O
PTAS	B-Algorithm
reductions	I-Algorithm
is	O
so	O
that	O
given	O
a	O
PTAS	B-Algorithm
reduction	I-Algorithm
from	O
an	O
optimization	O
problem	O
A	O
to	O
a	O
problem	O
B	O
,	O
a	O
PTAS	O
for	O
B	O
can	O
be	O
composed	O
with	O
the	O
reduction	B-Algorithm
to	O
obtain	O
a	O
PTAS	O
for	O
the	O
problem	O
A	O
.	O
</s>
<s>
Formally	O
,	O
we	O
define	O
a	O
PTAS	B-Algorithm
reduction	I-Algorithm
from	O
A	O
to	O
B	O
using	O
three	O
polynomial-time	O
computable	O
functions	O
,	O
f	O
,	O
g	O
,	O
and	O
α	O
,	O
with	O
the	O
following	O
properties	O
:	O
</s>
<s>
L-reductions	B-Algorithm
imply	O
PTAS	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
As	O
a	O
result	O
,	O
one	O
may	O
show	O
the	O
existence	O
of	O
a	O
PTAS	B-Algorithm
reduction	I-Algorithm
via	O
a	O
L-reduction	B-Algorithm
instead	O
.	O
</s>
<s>
PTAS	B-Algorithm
reductions	I-Algorithm
are	O
used	O
to	O
define	O
completeness	O
in	O
APX	B-Algorithm
,	O
the	O
class	O
of	O
optimization	O
problems	O
with	O
constant-factor	B-Algorithm
approximation	I-Algorithm
algorithms	I-Algorithm
.	O
</s>
