<s>
The	O
constraint	B-Application
composite	I-Application
graph	I-Application
is	O
a	O
node-weighted	O
undirected	O
graph	O
associated	O
with	O
a	O
given	O
combinatorial	O
optimization	O
problem	O
posed	O
as	O
a	O
weighted	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
.	O
</s>
<s>
Developed	O
and	O
introduced	O
by	O
Satish	O
Kumar	O
Thittamaranahalli	O
(	O
T	O
.	O
K	O
.	O
Satish	O
Kumar	O
)	O
,	O
the	O
idea	O
of	O
the	O
constraint	B-Application
composite	I-Application
graph	I-Application
is	O
a	O
big	O
step	O
towards	O
unifying	O
different	O
approaches	O
for	O
exploiting	O
"	O
structure	O
"	O
in	O
weighted	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
.	O
</s>
<s>
A	O
weighted	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
(	O
WCSP	O
)	O
is	O
a	O
generalization	B-Algorithm
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
in	O
which	O
the	O
constraints	O
are	O
no	O
longer	O
"	O
hard	O
,	O
"	O
but	O
are	O
extended	O
to	O
specify	O
non-negative	O
costs	O
associated	O
with	O
the	O
tuples	B-Application
.	O
</s>
<s>
Weighted	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
find	O
innumerable	O
applications	O
in	O
artificial	B-Application
intelligence	I-Application
and	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
While	O
weighted	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
are	O
NP-hard	O
to	O
solve	O
in	O
general	O
,	O
several	O
subclasses	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
when	O
their	O
weighted	O
constraints	O
exhibit	O
specific	O
kinds	O
of	O
numerical	O
structure	O
.	O
</s>
<s>
Specifically	O
,	O
a	O
weighted	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
can	O
be	O
solved	O
in	O
time	O
exponential	O
only	O
in	O
the	O
treewidth	O
of	O
its	O
variable-interaction	O
graph	O
(	O
constraint	O
network	O
)	O
.	O
</s>
<s>
Unlike	O
the	O
constraint	O
network	O
,	O
the	O
constraint	B-Application
composite	I-Application
graph	I-Application
provides	O
a	O
unifying	O
framework	O
for	O
representing	O
both	O
the	O
graphical	O
structure	O
of	O
the	O
variable-interactions	O
as	O
well	O
as	O
the	O
numerical	O
structure	O
of	O
the	O
weighted	O
constraints	O
.	O
</s>
<s>
It	O
can	O
be	O
constructed	O
using	O
a	O
simple	O
polynomial-time	O
procedure	O
;	O
and	O
a	O
given	O
weighted	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
reducible	O
to	O
the	O
problem	O
of	O
computing	O
the	O
minimum	O
weighted	O
vertex	O
cover	O
for	O
its	O
associated	O
constraint	B-Application
composite	I-Application
graph	I-Application
.	O
</s>
<s>
The	O
"	O
hybrid	O
"	O
computational	O
properties	O
of	O
the	O
constraint	B-Application
composite	I-Application
graph	I-Application
are	O
reflected	O
in	O
the	O
following	O
two	O
important	O
results	O
:	O
</s>
<s>
(	O
Result	O
1	O
)	O
The	O
constraint	B-Application
composite	I-Application
graph	I-Application
of	O
a	O
given	O
weighted	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
has	O
the	O
same	O
treewidth	O
as	O
its	O
associated	O
constraint	O
network	O
.	O
</s>
<s>
(	O
Result	O
2	O
)	O
Many	O
subclasses	O
of	O
weighted	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
that	O
are	O
tractable	O
by	O
virtue	O
of	O
the	O
numerical	O
structure	O
of	O
their	O
weighted	O
constraints	O
have	O
associated	O
constraint	B-Application
composite	I-Application
graphs	I-Application
that	O
are	O
bipartite	O
in	O
nature	O
.	O
</s>
<s>
Result	O
1	O
shows	O
that	O
the	O
constraint	B-Application
composite	I-Application
graph	I-Application
can	O
be	O
used	O
to	O
capture	O
the	O
graphical	O
structure	O
of	O
the	O
variable-interactions	O
(	O
since	O
a	O
minimum	O
weighted	O
vertex	O
cover	O
for	O
any	O
graph	O
can	O
be	O
computed	O
in	O
time	O
exponential	O
only	O
in	O
the	O
treewidth	O
of	O
that	O
graph	O
)	O
.	O
</s>
<s>
Result	O
2	O
shows	O
that	O
the	O
constraint	B-Application
composite	I-Application
graph	I-Application
can	O
also	O
be	O
used	O
to	O
capture	O
the	O
numerical	O
structure	O
of	O
the	O
weighted	O
constraints	O
(	O
since	O
a	O
minimum	O
weighted	O
vertex	O
cover	O
can	O
be	O
computed	O
in	O
polynomial	O
time	O
for	O
bipartite	O
graphs	O
)	O
.	O
</s>
<s>
Empirically	O
,	O
when	O
solving	O
a	O
WCSP	O
,	O
it	O
has	O
been	O
shown	O
that	O
it	O
is	O
more	O
advantageous	O
to	O
apply	O
message	O
passing	O
algorithms	O
and	O
integer	O
linear	O
programming	O
on	O
the	O
WCSP	O
's	O
constraint	B-Application
composite	I-Application
graph	I-Application
than	O
on	O
the	O
WCSP	O
directly	O
.	O
</s>
