<s>
The	O
dual	O
problem	O
is	O
a	O
reformulation	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
expressing	O
each	O
constraint	O
of	O
the	O
original	O
problem	O
as	O
a	O
variable	O
.	O
</s>
<s>
The	O
join	B-Application
graphs	I-Application
and	O
join	O
trees	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
are	O
graphs	O
representing	O
its	O
dual	O
problem	O
or	O
a	O
problem	O
obtained	O
from	O
the	O
dual	O
problem	O
removing	O
some	O
redundant	O
constraints	O
.	O
</s>
<s>
The	O
dual	O
problem	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
contains	O
a	O
variable	O
for	O
each	O
constraint	O
of	O
the	O
original	O
problem	O
.	O
</s>
<s>
A	O
graph	O
obtained	O
from	O
the	O
dual	O
graph	O
by	O
removing	O
some	O
redundant	O
edges	O
is	O
called	O
a	O
join	B-Application
graph	I-Application
.	O
</s>
<s>
The	O
dual	O
problem	O
can	O
be	O
solved	O
from	O
a	O
join	B-Application
graph	I-Application
since	O
all	O
removed	O
edges	O
are	O
redundant	O
.	O
</s>
<s>
In	O
turn	O
,	O
the	O
problem	O
can	O
be	O
solved	O
efficiently	O
if	O
that	O
join	B-Application
graph	I-Application
is	O
a	O
tree	O
,	O
using	O
algorithms	O
tailored	O
for	O
acyclic	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
.	O
</s>
<s>
Another	O
method	O
for	O
finding	O
out	O
whether	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
has	O
a	O
join	O
tree	O
uses	O
the	O
primal	O
graph	O
of	O
the	O
problem	O
,	O
rather	O
than	O
the	O
dual	O
graph	O
.	O
</s>
<s>
The	O
primal	O
graph	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
a	O
graph	O
whose	O
nodes	O
are	O
problem	O
variables	O
and	O
whose	O
edges	O
represent	O
the	O
presence	O
of	O
two	O
variables	O
in	O
the	O
same	O
constraint	O
.	O
</s>
<s>
Not	O
all	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
have	O
a	O
join	O
tree	O
.	O
</s>
<s>
Join-tree	B-Application
clustering	I-Application
is	O
a	O
specific	O
method	O
to	O
modify	O
problems	O
in	O
such	O
a	O
way	O
they	O
acquire	O
a	O
joint	O
tree	O
.	O
</s>
<s>
Decomposition	B-Application
methods	I-Application
generalize	O
join-tree	B-Application
clustering	I-Application
by	O
grouping	O
variables	O
in	O
such	O
a	O
way	O
the	O
resulting	O
problem	O
has	O
a	O
join	O
tree	O
.	O
</s>
<s>
Decomposition	B-Application
methods	I-Application
directly	O
associate	O
a	O
tree	O
with	O
problems	O
;	O
the	O
nodes	O
of	O
this	O
tree	O
are	O
associated	O
variables	O
and/or	O
constraints	O
of	O
the	O
original	O
problem	O
.	O
</s>
