<s>
MAX-3SAT	B-Application
is	O
a	O
problem	O
in	O
the	O
computational	O
complexity	O
subfield	O
of	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
It	O
generalises	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
(	O
SAT	B-Algorithm
)	O
which	O
is	O
a	O
decision	O
problem	O
considered	O
in	O
complexity	O
theory	O
.	O
</s>
<s>
Given	O
a	O
3-CNF	B-Application
formula	O
Φ	O
(	O
i.e.	O
</s>
<s>
MAX-3SAT	B-Application
is	O
a	O
canonical	O
complete	O
problem	O
for	O
the	O
complexity	O
class	O
MAXSNP	O
(	O
shown	O
complete	O
in	O
Papadimitriou	O
pg	O
.	O
</s>
<s>
The	O
decision	O
version	O
of	O
MAX-3SAT	B-Application
is	O
NP-complete	O
.	O
</s>
<s>
The	O
Karloff-Zwick	B-Algorithm
algorithm	I-Algorithm
runs	O
in	O
polynomial-time	O
and	O
satisfies	O
≥	O
7/8	O
of	O
the	O
clauses	O
.	O
</s>
<s>
The	O
PCP	O
theorem	O
implies	O
that	O
there	O
exists	O
an	O
ε	O
>	O
0	O
such	O
that	O
(	O
1-ε	O
)	O
-approximation	O
of	O
MAX-3SAT	B-Application
is	O
NP-hard	O
.	O
</s>
<s>
For	O
every	O
R	O
,	O
add	O
clauses	O
representing	O
fR( xi1	O
,...,	O
xiq	O
)	O
using	O
2q	O
SAT	B-Algorithm
clauses	O
.	O
</s>
<s>
This	O
requires	O
a	O
maximum	O
of	O
q2q	O
3-SAT	O
clauses	O
.	O
</s>
<s>
He	O
constructs	O
a	O
PCP	O
Verifier	O
for	O
3-SAT	O
that	O
reads	O
only	O
3	O
bits	O
from	O
the	O
Proof	O
.	O
</s>
<s>
For	O
every	O
ε	O
>	O
0	O
,	O
there	O
is	O
a	O
PCP-verifier	O
M	O
for	O
3-SAT	O
that	O
reads	O
a	O
random	O
string	O
r	O
of	O
length	O
and	O
computes	O
query	O
positions	O
ir	O
,	O
jr	O
,	O
kr	O
in	O
the	O
proof	O
π	O
and	O
a	O
bit	O
br	O
.	O
</s>
<s>
MAX-3SAT(B )	O
is	O
the	O
restricted	O
special	O
case	O
of	O
MAX-3SAT	B-Application
where	O
every	O
variable	O
occurs	O
in	O
at	O
most	O
B	O
clauses	O
.	O
</s>
<s>
This	O
is	O
useful	O
because	O
MAX-3SAT(B )	O
can	O
often	O
be	O
used	O
to	O
obtain	O
a	O
PTAS-preserving	O
reduction	O
in	O
a	O
way	O
that	O
MAX-3SAT	B-Application
cannot	O
.	O
</s>
<s>
Moreover	O
,	O
although	O
the	O
decision	O
problem	O
2SAT	B-Application
is	O
solvable	O
in	O
polynomial	O
time	O
,	O
MAX-2SAT(3 )	O
is	O
also	O
APX-hard	O
.	O
</s>
<s>
The	O
best	O
possible	O
approximation	O
ratio	O
for	O
MAX-3SAT(B )	O
,	O
as	O
a	O
function	O
of	O
B	O
,	O
is	O
at	O
least	O
and	O
at	O
most	O
,	O
Luca	O
Trevisan	O
.	O
</s>
<s>
P	O
.	O
Berman	O
,	O
M	O
.	O
Karpinski	O
and	O
A	O
.	O
D	O
.	O
Scott	O
,	O
Approximation	O
Hardness	O
and	O
Satisfiability	O
of	O
Bounded	O
Occurrence	O
Instances	O
of	O
SAT	B-Algorithm
,	O
</s>
<s>
Berman	O
,	O
Karpinski	O
and	O
Scott	O
proved	O
that	O
for	O
the	O
"	O
critical	O
"	O
instances	O
of	O
MAX-3SAT	B-Application
in	O
which	O
each	O
literal	O
occurs	O
exactly	O
twice	O
,	O
and	O
each	O
clause	O
is	O
exactly	O
of	O
size	O
3	O
,	O
the	O
problem	O
is	O
approximation	O
hard	O
for	O
some	O
constant	O
factor.P.	O
</s>
<s>
Berman	O
,	O
M	O
.	O
Karpinski	O
and	O
A	O
.	O
D	O
.	O
Scott	O
,	O
Approximation	O
Hardness	O
of	O
Short	O
Symmetric	O
Instances	O
of	O
MAX-3SAT	B-Application
,	O
</s>
<s>
MAX-EkSAT	O
is	O
a	O
parameterized	O
version	O
of	O
MAX-3SAT	B-Application
where	O
every	O
clause	O
has	O
exactly	O
literals	O
,	O
for	O
k''	O
≥	O
3	O
.	O
</s>
<s>
It	O
can	O
be	O
efficiently	O
approximated	O
with	O
approximation	O
ratio	O
using	O
ideas	O
from	O
coding	B-Error_Name
theory	I-Error_Name
.	O
</s>
<s>
It	O
has	O
been	O
proved	O
that	O
random	O
instances	O
of	O
MAX-3SAT	B-Application
can	O
be	O
approximated	O
to	O
within	O
factor	O
.	O
</s>
