<s>
In	O
computational	O
complexity	O
theory	O
,	O
the	O
class	B-Algorithm
APX	I-Algorithm
(	O
an	O
abbreviation	O
of	O
"	O
approximable	O
"	O
)	O
is	O
the	O
set	O
of	O
NP	O
optimization	O
problems	O
that	O
allow	O
polynomial-time	O
approximation	B-Algorithm
algorithms	I-Algorithm
with	O
approximation	B-Algorithm
ratio	I-Algorithm
bounded	O
by	O
a	O
constant	O
(	O
or	O
constant-factor	B-Algorithm
approximation	I-Algorithm
algorithms	I-Algorithm
for	O
short	O
)	O
.	O
</s>
<s>
An	O
approximation	B-Algorithm
algorithm	I-Algorithm
is	O
called	O
an	O
-approximation	O
algorithm	O
for	O
input	O
size	O
if	O
it	O
can	O
be	O
proven	O
that	O
the	O
solution	O
that	O
the	O
algorithm	O
finds	O
is	O
at	O
most	O
a	O
multiplicative	O
factor	O
of	O
times	O
worse	O
than	O
the	O
optimal	O
solution	O
.	O
</s>
<s>
Here	O
,	O
is	O
called	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
.	O
</s>
<s>
Problems	O
in	O
APX	B-Algorithm
are	O
those	O
with	O
algorithms	O
for	O
which	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
a	O
constant	O
.	O
</s>
<s>
The	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
conventionally	O
stated	O
greater	O
than	O
1	O
.	O
</s>
<s>
A	O
problem	O
is	O
said	O
to	O
have	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
(	O
PTAS	O
)	O
if	O
for	O
every	O
multiplicative	O
factor	O
of	O
the	O
optimum	O
worse	O
than	O
1	O
there	O
is	O
a	O
polynomial-time	O
algorithm	O
to	O
solve	O
the	O
problem	O
to	O
within	O
that	O
factor	O
.	O
</s>
<s>
Unless	O
P	O
=	O
NP	O
there	O
exist	O
problems	O
that	O
are	O
in	O
APX	B-Algorithm
but	O
without	O
a	O
PTAS	O
,	O
so	O
the	O
class	O
of	O
problems	O
with	O
a	O
PTAS	O
is	O
strictly	O
contained	O
in	O
APX	B-Algorithm
.	O
</s>
<s>
A	O
problem	O
is	O
said	O
to	O
be	O
APX-hard	B-Algorithm
if	O
there	O
is	O
a	O
PTAS	B-Algorithm
reduction	I-Algorithm
from	O
every	O
problem	O
in	O
APX	B-Algorithm
to	O
that	O
problem	O
,	O
and	O
to	O
be	O
APX-complete	B-Algorithm
if	O
the	O
problem	O
is	O
APX-hard	B-Algorithm
and	O
also	O
in	O
APX	B-Algorithm
.	O
</s>
<s>
As	O
a	O
consequence	O
of	O
P	O
≠	O
NP	O
⇒	O
PTAS	O
≠	O
APX	B-Algorithm
,	O
if	O
P	O
≠	O
NP	O
is	O
assumed	O
,	O
no	O
APX-hard	B-Algorithm
problem	O
has	O
a	O
PTAS	O
.	O
</s>
<s>
In	O
practice	O
,	O
reducing	O
one	O
problem	O
to	O
another	O
to	O
demonstrate	O
APX-completeness	O
is	O
often	O
done	O
using	O
other	O
reduction	O
schemes	O
,	O
such	O
as	O
L-reductions	B-Algorithm
,	O
which	O
imply	O
PTAS	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
simplest	O
APX-complete	B-Algorithm
problems	O
is	O
MAX-3SAT-3	B-Application
,	O
a	O
variation	O
of	O
the	O
boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
In	O
this	O
problem	O
,	O
we	O
have	O
a	O
boolean	O
formula	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
where	O
each	O
variable	O
appears	O
at	O
most	O
3	O
times	O
,	O
and	O
we	O
wish	O
to	O
know	O
the	O
maximum	O
number	O
of	O
clauses	O
that	O
can	O
be	O
simultaneously	O
satisfied	O
by	O
a	O
single	O
assignment	O
of	O
true/false	O
values	O
to	O
the	O
variables	O
.	O
</s>
<s>
Other	O
APX-complete	B-Algorithm
problems	O
include	O
:	O
</s>
<s>
Max	O
independent	O
set	O
in	O
bounded-degree	O
graphs	O
(	O
here	O
,	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
depends	O
on	O
the	O
maximum	O
degree	O
of	O
the	O
graph	O
,	O
but	O
is	O
constant	O
if	O
the	O
max	O
degree	O
is	O
fixed	O
)	O
.	O
</s>
<s>
The	O
token	O
reconfiguration	O
problem	O
,	O
via	O
L-reduction	B-Algorithm
from	O
set	O
cover	O
.	O
</s>
<s>
PTAS	O
(	O
polynomial	B-Algorithm
time	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
)	O
consists	O
of	O
problems	O
that	O
can	O
be	O
approximated	O
to	O
within	O
any	O
constant	O
factor	O
besides	O
1	O
in	O
time	O
that	O
is	O
polynomial	O
to	O
the	O
input	O
size	O
,	O
but	O
the	O
polynomial	O
depends	O
on	O
such	O
factor	O
.	O
</s>
<s>
This	O
class	O
is	O
a	O
subset	O
of	O
APX	B-Algorithm
.	O
</s>
<s>
Unless	O
P	O
=	O
NP	O
,	O
there	O
exist	O
problems	O
in	O
APX	B-Algorithm
that	O
are	O
neither	O
in	O
PTAS	O
nor	O
APX-complete	B-Algorithm
.	O
</s>
<s>
Such	O
problems	O
can	O
be	O
thought	O
of	O
as	O
having	O
a	O
hardness	O
between	O
PTAS	O
problems	O
and	O
APX-complete	B-Algorithm
problems	O
,	O
and	O
may	O
be	O
called	O
APX-intermediate	O
.	O
</s>
<s>
The	O
bin	O
packing	O
problem	O
is	O
thought	O
to	O
be	O
APX-intermediate	O
.	O
</s>
<s>
Despite	O
not	O
having	O
a	O
known	O
PTAS	O
,	O
the	O
bin	O
packing	O
problem	O
has	O
several	O
"	O
asymptotic	O
PTAS	O
"	O
algorithms	O
,	O
which	O
behave	O
like	O
a	O
PTAS	O
when	O
the	O
optimum	O
solution	O
is	O
large	O
,	O
so	O
intuitively	O
it	O
may	O
be	O
easier	O
than	O
problems	O
that	O
are	O
APX-hard	B-Algorithm
.	O
</s>
<s>
One	O
other	O
example	O
of	O
a	O
potentially	O
APX-intermediate	O
problem	O
is	O
min	O
edge	O
coloring	O
.	O
</s>
<s>
One	O
can	O
also	O
define	O
a	O
family	O
of	O
complexity	O
classes	O
-APX	O
,	O
where	O
-APX	O
contains	O
problems	O
with	O
a	O
polynomial	O
time	O
approximation	B-Algorithm
algorithm	I-Algorithm
with	O
a	O
approximation	B-Algorithm
ratio	I-Algorithm
.	O
</s>
<s>
One	O
can	O
analogously	O
define	O
-APX-complete	O
classes	O
;	O
some	O
such	O
classes	O
contain	O
well-known	O
optimization	O
problems	O
.	O
</s>
<s>
Log-APX-completeness	O
and	O
poly-APX-completeness	O
are	O
defined	O
in	O
terms	O
of	O
AP-reductions	O
rather	O
than	O
PTAS-reductions	O
;	O
this	O
is	O
because	O
PTAS-reductions	O
are	O
not	O
strong	O
enough	O
to	O
preserve	O
membership	O
in	O
Log-APX	B-Algorithm
and	O
Poly-APX	B-Algorithm
,	O
even	O
though	O
they	O
suffice	O
for	O
APX	B-Algorithm
.	O
</s>
<s>
Log-APX-complete	O
,	O
consisting	O
of	O
the	O
hardest	O
problems	O
that	O
can	O
be	O
approximated	O
efficiently	O
to	O
within	O
a	O
factor	O
logarithmic	O
in	O
the	O
input	O
size	O
,	O
includes	O
min	O
dominating	O
set	O
when	O
degree	O
is	O
unbounded	O
.	O
</s>
<s>
Poly-APX-complete	O
,	O
consisting	O
of	O
the	O
hardest	O
problems	O
that	O
can	O
be	O
approximated	O
efficiently	O
to	O
within	O
a	O
factor	O
polynomial	O
in	O
the	O
input	O
size	O
,	O
includes	O
max	O
independent	O
set	O
in	O
the	O
general	O
case	O
.	O
</s>
<s>
There	O
also	O
exist	O
problems	O
that	O
are	O
exp-APX-complete	O
,	O
where	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
exponential	O
in	O
the	O
input	O
size	O
.	O
</s>
