<s>
In	O
computer	B-General_Concept
science	I-General_Concept
(	O
particularly	O
algorithmics	B-Data_Structure
)	O
,	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
(	O
PTAS	O
)	O
is	O
a	O
type	O
of	O
approximation	B-Algorithm
algorithm	I-Algorithm
for	O
optimization	O
problems	O
(	O
most	O
often	O
,	O
NP-hard	O
optimization	O
problems	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
the	O
Euclidean	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
,	O
a	O
PTAS	O
would	O
produce	O
a	O
tour	O
with	O
length	O
at	O
most	O
,	O
with	O
being	O
the	O
length	O
of	O
the	O
shortest	O
tour	O
.	O
</s>
<s>
One	O
way	O
of	O
addressing	O
this	O
is	O
to	O
define	O
the	O
efficient	B-Algorithm
polynomial-time	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
or	O
EPTAS	B-Algorithm
,	O
in	O
which	O
the	O
running	O
time	O
is	O
required	O
to	O
be	O
for	O
a	O
constant	O
independent	O
of	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
an	O
EPTAS	B-Algorithm
runs	O
in	O
FPT	B-General_Concept
time	O
where	O
the	O
parameter	O
is	O
ε	O
.	O
</s>
<s>
Even	O
more	O
restrictive	O
,	O
and	O
useful	O
in	O
practice	O
,	O
is	O
the	O
fully	B-Algorithm
polynomial-time	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
or	O
FPTAS	B-Algorithm
,	O
which	O
requires	O
the	O
algorithm	O
to	O
be	O
polynomial	O
in	O
both	O
the	O
problem	O
size	O
and	O
.	O
</s>
<s>
Consequently	O
,	O
under	O
this	O
assumption	O
,	O
APX-hard	B-Algorithm
problems	O
do	O
not	O
have	O
PTASs	O
.	O
</s>
<s>
Furthermore	O
,	O
a	O
PTAS	O
can	O
run	O
in	O
FPT	B-General_Concept
time	O
for	O
some	O
parameterization	O
of	O
the	O
problem	O
,	O
which	O
leads	O
to	O
a	O
parameterized	B-General_Concept
approximation	O
scheme	O
.	O
</s>
<s>
Some	O
problems	O
which	O
do	O
not	O
have	O
a	O
PTAS	O
may	O
admit	O
a	O
randomized	B-General_Concept
algorithm	I-General_Concept
with	O
similar	O
properties	O
,	O
a	O
polynomial-time	B-Algorithm
randomized	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
or	O
PRAS	O
.	O
</s>
<s>
Like	O
a	O
PTAS	O
,	O
a	O
PRAS	O
must	O
have	O
running	O
time	O
polynomial	O
in	O
,	O
but	O
not	O
necessarily	O
in	O
;	O
with	O
further	O
restrictions	O
on	O
the	O
running	O
time	O
in	O
,	O
one	O
can	O
define	O
an	O
efficient	B-Algorithm
polynomial-time	I-Algorithm
randomized	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
or	O
EPRAS	B-Algorithm
similar	O
to	O
the	O
EPTAS	B-Algorithm
,	O
and	O
a	O
fully	B-Algorithm
polynomial-time	I-Algorithm
randomized	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
or	O
FPRAS	B-Algorithm
similar	O
to	O
the	O
FPTAS	B-Algorithm
.	O
</s>
<s>
PTAS	O
is	O
a	O
subset	O
of	O
APX	B-Algorithm
,	O
and	O
unless	O
P	O
=	O
NP	O
,	O
it	O
is	O
a	O
strict	O
subset	O
.	O
</s>
<s>
Membership	O
in	O
PTAS	O
can	O
be	O
shown	O
using	O
a	O
PTAS	B-Algorithm
reduction	I-Algorithm
,	O
L-reduction	B-Algorithm
,	O
or	O
P-reduction	O
,	O
all	O
of	O
which	O
preserve	O
PTAS	O
membership	O
,	O
and	O
these	O
may	O
also	O
be	O
used	O
to	O
demonstrate	O
PTAS-completeness	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
showing	O
non-membership	O
in	O
PTAS	O
(	O
namely	O
,	O
the	O
nonexistence	O
of	O
a	O
PTAS	O
)	O
,	O
may	O
be	O
done	O
by	O
showing	O
that	O
the	O
problem	O
is	O
APX-hard	B-Algorithm
,	O
after	O
which	O
the	O
existence	O
of	O
a	O
PTAS	O
would	O
show	O
P	O
=	O
NP	O
.	O
</s>
<s>
APX-hardness	O
is	O
commonly	O
shown	O
via	O
PTAS	B-Algorithm
reduction	I-Algorithm
or	O
AP-reduction	O
.	O
</s>
