<s>
In	O
computational	O
complexity	O
theory	O
,	O
the	O
maximum	B-Application
satisfiability	I-Application
problem	I-Application
(	O
MAX-SAT	B-Application
)	O
is	O
the	O
problem	O
of	O
determining	O
the	O
maximum	O
number	O
of	O
clauses	O
,	O
of	O
a	O
given	O
Boolean	O
formula	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
,	O
that	O
can	O
be	O
made	O
true	O
by	O
an	O
assignment	O
of	O
truth	O
values	O
to	O
the	O
variables	O
of	O
the	O
formula	O
.	O
</s>
<s>
It	O
is	O
a	O
generalization	O
of	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
,	O
which	O
asks	O
whether	O
there	O
exists	O
a	O
truth	O
assignment	O
that	O
makes	O
all	O
clauses	O
true	O
.	O
</s>
<s>
Therefore	O
,	O
if	O
this	O
formula	O
is	O
given	O
as	O
an	O
instance	O
of	O
the	O
MAX-SAT	B-Application
problem	O
,	O
the	O
solution	O
to	O
the	O
problem	O
is	O
the	O
number	O
three	O
.	O
</s>
<s>
The	O
MAX-SAT	B-Application
problem	O
is	O
OptP-complete	O
,	O
and	O
thus	O
NP-hard	O
,	O
since	O
its	O
solution	O
easily	O
leads	O
to	O
the	O
solution	O
of	O
the	O
boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
,	O
which	O
is	O
NP-complete	O
.	O
</s>
<s>
It	O
is	O
also	O
difficult	O
to	O
find	O
an	O
approximate	B-Algorithm
solution	O
of	O
the	O
problem	O
,	O
that	O
satisfies	O
a	O
number	O
of	O
clauses	O
within	O
a	O
guaranteed	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
the	O
optimal	O
solution	O
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
problem	O
is	O
APX-complete	B-Algorithm
,	O
and	O
thus	O
does	O
not	O
admit	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
More	O
generally	O
,	O
one	O
can	O
define	O
a	O
weighted	O
version	O
of	O
MAX-SAT	B-Application
as	O
follows	O
:	O
given	O
a	O
conjunctive	B-Application
normal	I-Application
form	I-Application
formula	O
with	O
non-negative	O
weights	O
assigned	O
to	O
each	O
clause	O
,	O
find	O
truth	O
values	O
for	O
its	O
variables	O
that	O
maximize	O
the	O
combined	O
weight	O
of	O
the	O
satisfied	O
clauses	O
.	O
</s>
<s>
The	O
MAX-SAT	B-Application
problem	O
is	O
an	O
instance	O
of	O
weighted	O
MAX-SAT	B-Application
where	O
all	O
weights	O
are	O
1	O
.	O
</s>
<s>
This	O
algorithm	O
can	O
be	O
derandomized	O
using	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
.	O
</s>
<s>
MAX-SAT	B-Application
can	O
also	O
be	O
expressed	O
using	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
(	O
ILP	O
)	O
.	O
</s>
<s>
Fix	O
a	O
conjunctive	B-Application
normal	I-Application
form	I-Application
formula	O
with	O
variables	O
1	O
,	O
2	O
,	O
...	O
,	O
n	O
,	O
and	O
let	O
denote	O
the	O
clauses	O
of	O
.	O
</s>
<s>
The	O
above	O
program	O
can	O
be	O
relaxed	B-Algorithm
to	O
the	O
following	O
linear	O
program	O
:	O
</s>
<s>
This	O
algorithm	O
can	O
also	O
be	O
derandomized	O
using	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
.	O
</s>
<s>
Many	O
exact	O
solvers	O
for	O
MAX-SAT	B-Application
have	O
been	O
developed	O
during	O
recent	O
years	O
,	O
and	O
many	O
of	O
them	O
were	O
presented	O
in	O
the	O
well-known	O
conference	O
on	O
the	O
boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
and	O
related	O
problems	O
,	O
the	O
SAT	O
Conference	O
.	O
</s>
<s>
In	O
2006	O
the	O
SAT	O
Conference	O
hosted	O
the	O
first	O
MAX-SAT	B-Application
evaluation	O
comparing	O
performance	O
of	O
practical	O
solvers	O
for	O
MAX-SAT	B-Application
,	O
as	O
it	O
has	O
done	O
in	O
the	O
past	O
for	O
the	O
pseudo-boolean	O
satisfiability	O
problem	O
and	O
the	O
quantified	O
boolean	O
formula	O
problem	O
.	O
</s>
<s>
There	O
are	O
several	O
solvers	O
submitted	O
to	O
the	O
last	O
Max-SAT	B-Application
Evaluations	O
:	O
</s>
<s>
Branch	B-Algorithm
and	I-Algorithm
Bound	I-Algorithm
based	O
:	O
Clone	O
,	O
MaxSatz	O
(	O
based	O
on	O
Satz	B-Application
)	O
,	O
IncMaxSatz	O
,	O
IUT_MaxSatz	O
,	O
WBO	O
,	O
GIDSHSat	O
.	O
</s>
<s>
MAX-SAT	B-Application
is	O
one	O
of	O
the	O
optimization	O
extensions	O
of	O
the	O
boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
,	O
which	O
is	O
the	O
problem	O
of	O
determining	O
whether	O
the	O
variables	O
of	O
a	O
given	O
Boolean	O
formula	O
can	O
be	O
assigned	O
in	O
such	O
a	O
way	O
as	O
to	O
make	O
the	O
formula	O
evaluate	O
to	O
TRUE	O
.	O
</s>
<s>
If	O
the	O
clauses	O
are	O
restricted	O
to	O
have	O
at	O
most	O
2	O
literals	O
,	O
as	O
in	O
2-satisfiability	B-Application
,	O
we	O
get	O
the	O
MAX-2SAT	O
problem	O
.	O
</s>
<s>
If	O
they	O
are	O
restricted	O
to	O
at	O
most	O
3	O
literals	O
per	O
clause	O
,	O
as	O
in	O
3-satisfiability	O
,	O
we	O
get	O
the	O
MAX-3SAT	B-Application
problem	O
.	O
</s>
<s>
There	O
are	O
many	O
problems	O
related	O
to	O
the	O
satisfiability	O
of	O
conjunctive	B-Application
normal	I-Application
form	I-Application
Boolean	O
formulas	O
.	O
</s>
<s>
MAX-SAT	B-Application
,	O
where	O
each	O
clause	O
has	O
exactly	O
variables	O
:	O
</s>
<s>
The	O
partial	O
maximum	B-Application
satisfiability	I-Application
problem	I-Application
(	O
PMAX-SAT	O
)	O
asks	O
for	O
the	O
maximum	O
number	O
of	O
clauses	O
which	O
can	O
be	O
satisfied	O
by	O
any	O
assignment	O
of	O
a	O
given	O
subset	O
of	O
clauses	O
.	O
</s>
<s>
The	O
soft	O
satisfiability	B-Algorithm
problem	I-Algorithm
(	O
soft-SAT	O
)	O
,	O
given	O
a	O
set	O
of	O
SAT	B-Algorithm
problems	I-Algorithm
,	O
asks	O
for	O
the	O
maximum	O
number	O
of	O
those	O
problems	O
which	O
can	O
be	O
satisfied	O
by	O
any	O
assignment	O
.	O
</s>
<s>
The	O
minimum	O
satisfiability	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
MAX-SAT	B-Application
problem	O
can	O
be	O
extended	O
to	O
the	O
case	O
where	O
the	O
variables	O
of	O
the	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
belong	O
to	O
the	O
set	O
of	O
reals	O
.	O
</s>
