<s>
In	O
constraint	B-Application
satisfaction	I-Application
,	O
a	O
decomposition	B-Application
method	I-Application
translates	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
into	O
another	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
that	O
is	O
binary	O
and	O
acyclic	O
.	O
</s>
<s>
Decomposition	B-Application
methods	I-Application
work	O
by	O
grouping	O
variables	O
into	O
sets	O
,	O
and	O
solving	O
a	O
subproblem	O
for	O
each	O
set	O
.	O
</s>
<s>
Each	O
structural	B-Application
restriction	I-Application
defined	O
a	O
measure	O
of	O
complexity	O
of	O
solving	O
the	O
problem	O
after	O
conversion	O
;	O
this	O
measure	O
is	O
called	O
width	O
.	O
</s>
<s>
Fixing	O
a	O
maximal	O
allowed	O
width	O
is	O
a	O
way	O
for	O
identifying	O
a	O
subclass	O
of	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
.	O
</s>
<s>
Solving	O
problems	O
in	O
this	O
class	O
is	O
polynomial	O
for	O
most	O
decompositions	O
;	O
if	O
this	O
holds	O
for	O
a	O
decomposition	O
,	O
the	O
class	O
of	O
fixed-width	O
problems	O
form	O
a	O
tractable	O
subclass	O
of	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
.	O
</s>
<s>
Decomposition	B-Application
methods	I-Application
translate	O
a	O
problem	O
into	O
a	O
new	O
one	O
that	O
is	O
easier	O
to	O
solve	O
.	O
</s>
<s>
By	O
definition	O
,	O
a	O
decomposition	B-Application
method	I-Application
produces	O
a	O
binary	O
acyclic	O
problem	O
;	O
such	O
problems	O
can	O
be	O
solved	O
in	O
time	O
polynomial	O
in	O
its	O
size	O
.	O
</s>
<s>
The	O
width	O
of	O
a	O
decomposition	B-Application
method	I-Application
is	O
a	O
measure	O
of	O
the	O
size	O
of	O
problem	O
it	O
produced	O
.	O
</s>
<s>
While	O
decompositions	O
of	O
fixed	O
width	O
can	O
be	O
used	O
to	O
efficiently	O
solve	O
a	O
problem	O
,	O
a	O
bound	O
on	O
the	O
width	O
of	O
instances	O
does	O
necessarily	O
produce	O
a	O
tractable	O
structural	B-Application
restriction	I-Application
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
decomposition	B-Application
methods	I-Application
and	O
their	O
associated	O
width	O
are	O
defined	O
in	O
such	O
a	O
way	O
not	O
only	O
solving	O
the	O
problem	O
given	O
a	O
fixed-width	O
decomposition	O
of	O
it	O
is	O
polynomial-time	O
,	O
but	O
also	O
finding	O
a	O
fixed	O
width	O
decomposition	O
of	O
a	O
fixed-width	O
problem	O
is	O
polynomial-time	O
.	O
</s>
<s>
Decomposition	B-Application
methods	I-Application
create	O
a	O
problem	O
that	O
is	O
easy	O
to	O
solve	O
from	O
an	O
arbitrary	O
one	O
.	O
</s>
<s>
In	O
order	O
for	O
the	O
new	O
problem	O
to	O
be	O
solvable	O
efficiently	O
,	O
the	O
primal	B-Application
graph	I-Application
of	O
the	O
new	O
problem	O
is	O
required	O
to	O
be	O
acyclic	O
.	O
</s>
<s>
Image:Tree-decomposition-1-corrected.svg	O
An	O
example	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
;	O
this	O
problem	O
is	O
binary	O
,	O
and	O
the	O
constraints	O
are	O
represented	O
by	O
edges	O
of	O
this	O
graph	O
.	O
</s>
<s>
A	O
decomposition	B-Application
method	I-Application
is	O
usually	O
defined	O
by	O
providing	O
a	O
tree	O
whose	O
nodes	O
are	O
the	O
variables	O
of	O
the	O
new	O
problem	O
;	O
for	O
each	O
node	O
,	O
also	O
provided	O
are	O
the	O
associated	O
set	O
of	O
original	O
variables	O
and	O
possibly	O
a	O
set	O
of	O
original	O
constraints	O
used	O
to	O
build	O
the	O
domain	O
of	O
the	O
variable	O
in	O
the	O
new	O
problem	O
.	O
</s>
<s>
A	O
number	O
of	O
decomposition	B-Application
methods	I-Application
exist	O
.	O
</s>
<s>
The	O
following	O
are	O
the	O
decomposition	B-Application
methods	I-Application
defined	O
for	O
binary	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
.	O
</s>
<s>
Since	O
a	O
problem	O
can	O
be	O
made	O
binary	O
by	O
translating	O
it	O
into	O
its	O
dual	B-Application
problem	I-Application
or	O
using	O
hidden	B-Application
variables	I-Application
,	O
these	O
methods	O
can	O
be	O
indirectly	O
used	O
to	O
provide	O
a	O
tree	O
decomposition	O
for	O
arbitrary	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
.	O
</s>
<s>
If	O
the	O
constraints	O
of	O
a	O
binary	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
are	O
viewed	O
as	O
edges	O
of	O
a	O
graph	O
whose	O
nodes	O
are	O
the	O
variables	O
,	O
this	O
tree	O
is	O
a	O
decomposition	O
of	O
the	O
problem	O
.	O
</s>
<s>
The	O
cycle	O
decomposition	B-Application
method	I-Application
split	O
a	O
problem	O
into	O
a	O
cyclic	O
and	O
an	O
acyclic	O
part	O
.	O
</s>
<s>
While	O
it	O
does	O
not	O
fit	O
in	O
the	O
definition	O
of	O
the	O
other	O
decomposition	B-Application
methods	I-Application
,	O
which	O
produce	O
a	O
tree	O
whose	O
nodes	O
are	O
labeled	O
with	O
sets	O
of	O
nodes	O
,	O
it	O
can	O
be	O
easily	O
reformulated	O
in	O
such	O
terms	O
.	O
</s>
<s>
This	O
decomposition	B-Application
method	I-Application
is	O
based	O
on	O
the	O
idea	O
that	O
,	O
after	O
some	O
variables	O
are	O
given	O
a	O
value	O
,	O
what	O
remains	O
of	O
the	O
problem	O
once	O
these	O
variables	O
have	O
been	O
eliminated	O
may	O
be	O
acyclic	O
.	O
</s>
<s>
A	O
similar	O
definition	O
can	O
be	O
given	O
for	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
using	O
its	O
primal	B-Application
graph	I-Application
.	O
</s>
<s>
However	O
,	O
a	O
tree	O
like	O
those	O
generated	O
by	O
the	O
other	O
decomposition	B-Application
methods	I-Application
can	O
be	O
obtained	O
from	O
the	O
tree	O
that	O
results	O
from	O
removing	O
the	O
cutset	O
;	O
this	O
is	O
done	O
by	O
choosing	O
a	O
root	O
,	O
adding	O
all	O
variables	O
of	O
the	O
cutset	O
to	O
all	O
its	O
nodes	O
,	O
and	O
the	O
variables	O
of	O
each	O
node	O
to	O
all	O
its	O
children	O
.	O
</s>
<s>
The	O
following	O
methods	O
can	O
be	O
used	O
for	O
translating	O
an	O
arbitrary	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
,	O
either	O
binary	O
or	O
otherwise	O
.	O
</s>
<s>
Since	O
they	O
can	O
also	O
be	O
used	O
on	O
binary	O
problems	O
,	O
they	O
can	O
also	O
be	O
used	O
on	O
the	O
result	O
of	O
making	O
constraints	O
binary	O
,	O
either	O
by	O
translating	O
to	O
the	O
dual	B-Application
problem	I-Application
or	O
by	O
using	O
hidden	B-Application
variables	I-Application
.	O
</s>
<s>
The	O
biconnected	O
decomposition	O
of	O
an	O
arbitrary	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
the	O
biconnected	O
decomposition	O
of	O
its	O
primal	B-Application
graph	I-Application
.	O
</s>
<s>
Every	O
constraint	O
can	O
be	O
enforced	O
on	O
a	O
node	O
of	O
the	O
tree	O
because	O
each	O
constraint	O
creates	O
a	O
clique	O
on	O
its	O
variables	O
on	O
the	O
primal	B-Application
graph	I-Application
,	O
and	O
a	O
clique	O
is	O
either	O
a	O
biconnected	O
component	O
or	O
a	O
subset	O
of	O
a	O
biconnected	O
component	O
.	O
</s>
<s>
A	O
tree	O
decomposition	O
of	O
an	O
arbitrary	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
a	O
tree	O
decomposition	O
of	O
its	O
primal	B-Application
graph	I-Application
.	O
</s>
<s>
Every	O
constraint	O
can	O
be	O
enforced	O
on	O
a	O
node	O
of	O
the	O
tree	O
because	O
each	O
constraint	O
creates	O
a	O
clique	O
on	O
its	O
variables	O
on	O
the	O
primal	B-Application
graph	I-Application
and	O
,	O
for	O
every	O
tree	O
decomposition	O
,	O
the	O
variables	O
of	O
a	O
clique	O
are	O
completely	O
contained	O
in	O
the	O
variables	O
of	O
some	O
node	O
.	O
</s>
<s>
A	O
hinge	O
decomposition	O
is	O
based	O
on	O
the	O
sets	O
of	O
variables	O
that	O
are	O
minimal	O
hinges	O
of	O
the	O
hypergraph	O
whose	O
nodes	O
are	O
the	O
variables	O
of	O
the	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
and	O
whose	O
hyperedges	O
are	O
the	O
scopes	O
of	O
its	O
constraints	O
.	O
</s>
<s>
A	O
hinge	O
decomposition	O
is	O
based	O
on	O
the	O
correspondence	O
between	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
and	O
hypergraphs	O
.	O
</s>
<s>
A	O
hinge	O
decomposition	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
a	O
tree	O
whose	O
nodes	O
are	O
some	O
minimal	O
hinges	O
of	O
the	O
hypergraph	O
associated	O
to	O
the	O
problem	O
and	O
such	O
that	O
some	O
other	O
conditions	O
are	O
met	O
.	O
</s>
<s>
Tree	O
clustering	O
or	O
join-tree	B-Application
clustering	I-Application
is	O
based	O
on	O
merging	O
constraints	O
in	O
such	O
a	O
way	O
the	O
resulting	O
problem	O
has	O
a	O
join	O
tree	O
,	O
this	O
join	O
tree	O
is	O
the	O
result	O
of	O
the	O
decomposition	O
.	O
</s>
<s>
A	O
join	O
tree	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
a	O
tree	O
in	O
which	O
each	O
node	O
is	O
associated	O
a	O
constraints	O
(	O
and	O
vice	O
versa	O
)	O
and	O
such	O
that	O
the	O
subtree	O
of	O
nodes	O
whose	O
constraint	O
contains	O
a	O
variable	O
is	O
connected	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>
Tree	O
clustering	O
is	O
based	O
on	O
the	O
fact	O
that	O
a	O
problem	O
has	O
a	O
join	O
tree	O
if	O
and	O
only	O
if	O
its	O
primal	B-Application
graph	I-Application
is	O
chordal	O
and	O
conformant	O
with	O
the	O
problem	O
,	O
the	O
latter	O
meaning	O
that	O
the	O
variables	O
of	O
every	O
maximal	O
clique	O
of	O
the	O
primal	B-Application
graph	I-Application
are	O
the	O
scope	O
of	O
a	O
constraint	O
and	O
vice	O
versa	O
.	O
</s>
<s>
These	O
are	O
binary	O
constraints	O
satisfied	O
by	O
any	O
pair	O
of	O
values	O
,	O
and	O
are	O
used	O
only	O
to	O
add	O
edges	O
to	O
the	O
primal	B-Application
graph	I-Application
of	O
the	O
problem	O
.	O
</s>
<s>
In	O
particular	O
,	O
chordality	O
is	O
obtained	O
by	O
adding	O
edges	O
producing	O
the	O
induced	B-Application
graph	I-Application
of	O
the	O
primal	B-Application
graph	I-Application
according	O
to	O
an	O
arbitrary	O
ordering	O
.	O
</s>
<s>
This	O
procedure	O
is	O
correct	O
because	O
the	O
induced	B-Application
graph	I-Application
is	O
always	O
chordal	O
and	O
is	O
obtained	O
adding	O
edges	O
to	O
the	O
original	O
graph	O
.	O
</s>
<s>
Conformality	O
requires	O
that	O
the	O
maximal	O
cliques	O
of	O
the	O
primal	B-Application
graph	I-Application
are	O
exactly	O
the	O
scope	O
of	O
the	O
constraints	O
.	O
</s>
<s>
While	O
the	O
scope	O
of	O
every	O
original	O
constraint	O
is	O
clique	O
on	O
the	O
primal	B-Application
graph	I-Application
,	O
this	O
clique	O
is	O
not	O
necessarily	O
maximal	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
scope	O
of	O
every	O
original	O
constraint	O
is	O
a	O
clique	O
of	O
the	O
primal	B-Application
graph	I-Application
.	O
</s>
<s>
Image:Join-tree-clustering-1.svg	O
Image:Join-tree-clustering-2.svg	O
Image:Join-tree-clustering-3.svg	O
An	O
example	O
:	O
a	O
binary	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
(	O
join-tree	B-Application
clustering	I-Application
can	O
also	O
be	O
applied	O
to	O
non-binary	O
constraints	O
.	O
)	O
</s>
<s>
Join-tree	B-Application
clustering	I-Application
replaces	O
the	O
constraints	O
on	O
{	O
x3	O
,	O
x4	O
,	O
x5	O
,	O
x6}	O
with	O
two	O
equivalent	O
constraints	O
,	O
one	O
on	O
{	O
x3	O
,	O
x4	O
,	O
x5}	O
and	O
one	O
on	O
{	O
x4	O
,	O
x5	O
,	O
x6}	O
.	O
</s>
<s>
Join-tree	B-Application
clustering	I-Application
can	O
be	O
seen	O
as	O
a	O
decomposition	B-Application
method	I-Application
in	O
which	O
:	O
</s>
<s>
Image:Query-decomposition-1.svg	O
Image:Query-decomposition-2.svg	O
A	O
hypergraph	O
representation	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
:	O
the	O
constraints	O
are	O
given	O
names	O
(	O
P	O
,	O
Q	O
,	O
R	O
,	O
S	O
,	O
T	O
)	O
,	O
and	O
their	O
scopes	O
are	O
shown	O
(	O
P	O
(	O
a	O
,	O
b	O
,	O
c	O
)	O
means	O
that	O
constraint	O
P	O
is	O
on	O
the	O
variables	O
{	O
a	O
,	O
b	O
,	O
c}	O
A	O
query	O
decomposition	O
of	O
the	O
problem	O
.	O
</s>
<s>
Beside	O
the	O
common	O
conditions	O
for	O
a	O
decomposition	B-Application
method	I-Application
(	O
the	O
scope	O
of	O
each	O
constraint	O
is	O
in	O
at	O
least	O
a	O
set	O
of	O
variables	O
associated	O
with	O
a	O
node	O
and	O
the	O
subtree	O
induced	O
by	O
an	O
original	O
variable	O
is	O
connected	O
)	O
,	O
the	O
following	O
two	O
conditions	O
are	O
required	O
to	O
hold	O
:	O
</s>
<s>
The	O
width	O
of	O
instances	O
is	O
a	O
form	O
of	O
efficiency	O
of	O
decomposition	B-Application
methods	I-Application
.	O
</s>
<s>
The	O
comparison	O
between	O
all	O
other	O
decomposition	B-Application
methods	I-Application
is	O
based	O
on	O
generalization	O
and	O
beating	O
.	O
</s>
<s>
Beating	O
means	O
that	O
there	O
are	O
classes	O
of	O
problems	O
that	O
have	O
fixed	O
width	O
according	O
to	O
a	O
decomposition	B-Application
method	I-Application
but	O
not	O
according	O
to	O
another	O
.	O
</s>
<s>
This	O
is	O
a	O
polynomial-time	O
problem	O
,	O
as	O
it	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
using	O
,	O
for	O
example	O
,	O
an	O
algorithm	O
for	O
enforcing	O
directional	B-Application
arc	I-Application
consistency	I-Application
.	O
</s>
<s>
Bounding	O
the	O
width	O
of	O
a	O
decomposition	B-Application
method	I-Application
by	O
a	O
constant	O
creates	O
a	O
structural	B-Application
restriction	I-Application
,	O
that	O
is	O
,	O
it	O
limits	O
the	O
possible	O
scopes	O
of	O
constraints	O
,	O
but	O
not	O
their	O
relations	O
.	O
</s>
<s>
The	O
complementary	O
way	O
for	O
obtaining	O
tractable	O
subclasses	O
of	O
constraint	B-Application
satisfaction	I-Application
is	O
by	O
placing	O
restriction	O
over	O
the	O
relations	O
of	O
constraints	O
;	O
these	O
are	O
called	O
relational	B-Application
restriction	I-Application
,	O
and	O
the	O
set	O
of	O
allowed	O
relations	O
is	O
called	O
constraint	O
language	O
.	O
</s>
<s>
If	O
solving	O
problems	O
having	O
a	O
decomposition	O
width	O
bounded	O
by	O
a	O
constant	O
is	O
in	O
P	O
,	O
the	O
decomposition	O
leads	O
to	O
a	O
tractable	O
structural	B-Application
restriction	I-Application
.	O
</s>
<s>
While	O
most	O
tractable	O
structural	O
restrictions	O
derive	O
from	O
fixing	O
the	O
width	O
of	O
a	O
decomposition	B-Application
method	I-Application
,	O
others	O
have	O
been	O
developed	O
.	O
</s>
<s>
Some	O
can	O
be	O
reformulated	O
in	O
terms	O
of	O
decomposition	B-Application
methods	I-Application
:	O
for	O
example	O
,	O
the	O
restriction	O
to	O
binary	O
acyclic	O
problem	O
can	O
be	O
reformulated	O
as	O
that	O
of	O
problem	O
of	O
treewidth	O
1	O
;	O
the	O
restriction	O
of	O
induced	O
width	O
(	O
which	O
is	O
not	O
defined	O
in	O
terms	O
of	O
a	O
decomposition	O
)	O
can	O
be	O
reformulated	O
as	O
tree	O
clustering	O
.	O
</s>
<s>
An	O
early	O
structural	B-Application
restriction	I-Application
(	O
that	O
later	O
evolved	O
into	O
that	O
based	O
on	O
induced	O
width	O
)	O
is	O
based	O
on	O
the	O
width	O
of	O
the	O
primal	B-Application
graph	I-Application
of	O
the	O
problem	O
.	O
</s>
