<s>
In	O
artificial	B-Application
intelligence	I-Application
and	O
operations	O
research	O
,	O
a	O
Weighted	B-Application
Constraint	I-Application
Satisfaction	I-Application
Problem	I-Application
(	O
WCSP	O
)	O
is	O
a	O
generalization	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
(	O
CSP	O
)	O
where	O
some	O
of	O
the	O
constraints	O
can	O
be	O
violated	O
(	O
according	O
to	O
a	O
violation	O
degree	O
)	O
and	O
in	O
which	O
preferences	O
among	O
solutions	O
can	O
be	O
expressed	O
.	O
</s>
<s>
This	O
generalization	O
makes	O
it	O
possible	O
to	O
represent	O
more	O
real-world	O
problems	O
,	O
in	O
particular	O
those	O
that	O
are	O
over-constrained	O
(	O
no	O
solution	O
can	O
be	O
found	O
without	O
violating	O
at	O
least	O
one	O
constraint	B-Application
)	O
,	O
or	O
those	O
where	O
we	O
want	O
to	O
find	O
a	O
minimal-cost	O
solution	O
(	O
according	O
to	O
a	O
cost	O
function	O
)	O
among	O
multiple	O
possible	O
solutions	O
.	O
</s>
<s>
A	O
Weighted	O
Constraint	B-Application
Network	O
(	O
WCN	O
)	O
,	O
aka	O
Cost	O
Function	O
Network	O
(	O
CFN	O
)	O
,	O
is	O
a	O
triplet	O
where	O
is	O
a	O
finite	O
set	O
of	O
discrete	O
variables	O
,	O
is	O
a	O
finite	O
set	O
of	O
soft	O
constraints	O
and	O
is	O
either	O
a	O
natural	O
integer	O
or	O
.	O
</s>
<s>
Each	O
soft	O
constraint	B-Application
involves	O
an	O
ordered	O
set	O
of	O
variables	O
,	O
called	O
its	O
scope	O
,	O
and	O
is	O
defined	O
as	O
a	O
cost	O
function	O
from	O
to	O
where	O
is	O
the	O
set	O
of	O
possible	O
instantiations	O
of	O
.	O
</s>
<s>
Without	O
any	O
loss	O
of	O
generality	O
,	O
the	O
existence	O
of	O
a	O
nullary	O
constraint	B-Application
(	O
a	O
cost	O
)	O
as	O
well	O
as	O
the	O
presence	O
of	O
a	O
unary	O
constraint	B-Application
for	O
every	O
variable	O
is	O
assumed	O
.	O
</s>
<s>
The	O
total	O
cost	O
of	O
a	O
complete	O
instantiation	O
is	O
the	O
bounded	O
sum	O
of	O
the	O
cost	O
of	O
on	O
for	O
all	O
soft	O
constraint	B-Application
,	O
including	O
the	O
nullary	O
cost	O
and	O
the	O
unary	O
costs	O
for	O
of	O
the	O
variables	O
in	O
.	O
</s>
<s>
Node	O
consistency	O
(	O
NC	O
)	O
and	O
Arc	O
consistency	O
(	O
AC	O
)	O
,	O
introduced	O
for	O
the	O
Constraint	B-Application
Satisfaction	I-Application
Problem	I-Application
(	O
CSP	O
)	O
,	O
have	O
been	O
studied	O
later	O
in	O
the	O
context	O
of	O
WCSP	O
.	O
</s>
<s>
The	O
goal	O
of	O
Equivalence	O
Preserving	O
Transformations	O
is	O
to	O
concentrate	O
costs	O
on	O
the	O
nullary	O
constraint	B-Application
and	O
remove	O
efficiently	O
instantiations	O
and	O
values	O
with	O
a	O
cost	O
,	O
added	O
to	O
,	O
that	O
is	O
greater	O
than	O
or	O
equal	O
to	O
the	O
forbidden	O
cost	O
or	O
the	O
cost	O
of	O
the	O
best	O
solution	O
found	O
so	O
far	O
.	O
</s>
