<s>
In	O
constraint	B-Application
satisfaction	I-Application
,	O
local	B-Application
consistency	I-Application
conditions	O
are	O
properties	O
of	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
related	O
to	O
the	O
consistency	B-General_Concept
of	O
subsets	O
of	O
variables	O
or	O
constraints	O
.	O
</s>
<s>
Various	O
kinds	O
of	O
local	B-Application
consistency	I-Application
conditions	O
are	O
leveraged	O
,	O
including	O
node	O
consistency	B-General_Concept
,	O
arc	O
consistency	B-General_Concept
,	O
and	O
path	B-Application
consistency	I-Application
.	O
</s>
<s>
Every	O
local	B-Application
consistency	I-Application
condition	O
can	O
be	O
enforced	O
by	O
a	O
transformation	O
that	O
changes	O
the	O
problem	O
without	O
changing	O
its	O
solutions	O
.	O
</s>
<s>
Such	O
a	O
transformation	O
is	O
called	O
constraint	B-Application
propagation	I-Application
.	O
</s>
<s>
Constraint	B-Application
propagation	I-Application
works	O
by	O
reducing	O
domains	O
of	O
variables	O
,	O
strengthening	O
constraints	O
,	O
or	O
creating	O
new	O
ones	O
.	O
</s>
<s>
Constraint	B-Application
propagation	I-Application
can	O
also	O
be	O
used	O
as	O
an	O
unsatisfiability	O
checker	O
,	O
incomplete	O
in	O
general	O
but	O
complete	O
in	O
some	O
particular	O
cases	O
.	O
</s>
<s>
Local	B-Application
consistency	I-Application
conditions	O
can	O
be	O
grouped	O
into	O
various	O
classes	O
.	O
</s>
<s>
The	O
original	O
local	B-Application
consistency	I-Application
conditions	O
require	O
that	O
every	O
consistent	O
assignment	O
can	O
be	O
consistently	O
extended	O
to	O
another	O
variable	O
.	O
</s>
<s>
Directional	B-Application
consistency	I-Application
only	O
requires	O
this	O
condition	O
to	O
be	O
satisfied	O
when	O
the	O
other	O
variable	O
is	O
higher	O
than	O
the	O
ones	O
in	O
the	O
assignment	O
,	O
according	O
to	O
a	O
given	O
order	O
.	O
</s>
<s>
Relational	B-Application
consistency	I-Application
includes	O
extensions	O
to	O
more	O
than	O
one	O
variable	O
,	O
but	O
this	O
extension	O
is	O
only	O
required	O
to	O
satisfy	O
a	O
given	O
constraint	O
or	O
set	O
of	O
constraints	O
.	O
</s>
<s>
In	O
this	O
article	O
,	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
defined	O
as	O
a	O
set	O
of	O
variables	O
,	O
a	O
set	O
of	O
domains	O
,	O
and	O
a	O
set	O
of	O
constraints	O
.	O
</s>
<s>
The	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
referred	O
to	O
in	O
this	O
article	O
are	O
assumed	O
to	O
be	O
in	O
a	O
special	O
form	O
.	O
</s>
<s>
The	O
"	O
standard	O
"	O
local	B-Application
consistency	I-Application
conditions	O
all	O
require	O
that	O
all	O
consistent	O
partial	O
evaluations	O
can	O
be	O
extended	O
to	O
another	O
variable	O
in	O
such	O
a	O
way	O
that	O
the	O
resulting	O
assignment	O
is	O
consistent	O
.	O
</s>
<s>
Node	O
consistency	B-General_Concept
requires	O
that	O
every	O
unary	O
constraint	O
on	O
a	O
variable	O
is	O
satisfied	O
by	O
all	O
values	O
in	O
the	O
domain	O
of	O
the	O
variable	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
For	O
example	O
,	O
given	O
a	O
variable	O
with	O
a	O
domain	O
of	O
and	O
a	O
constraint	O
,	O
node	O
consistency	B-General_Concept
would	O
restrict	O
the	O
domain	O
to	O
and	O
the	O
constraint	O
could	O
then	O
be	O
discarded	O
.	O
</s>
<s>
A	O
variable	O
of	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
arc-consistent	O
with	O
another	O
one	O
if	O
each	O
of	O
its	O
admissible	O
values	O
are	O
consistent	O
with	O
some	O
admissible	O
value	O
of	O
the	O
second	O
variable	O
.	O
</s>
<s>
Arc	O
consistency	B-General_Concept
can	O
also	O
be	O
defined	O
relative	O
to	O
a	O
specific	O
binary	O
constraint	O
:	O
a	O
binary	O
constraint	O
is	O
arc-consistent	O
if	O
every	O
value	O
of	O
one	O
variable	O
has	O
a	O
value	O
of	O
the	O
second	O
variable	O
such	O
that	O
they	O
satisfy	O
the	O
constraint	O
.	O
</s>
<s>
This	O
definition	O
of	O
arc	O
consistency	B-General_Concept
is	O
similar	O
to	O
the	O
above	O
,	O
but	O
is	O
given	O
specific	O
to	O
a	O
constraint	O
.	O
</s>
<s>
This	O
is	O
the	O
form	O
of	O
constraint	B-Application
propagation	I-Application
that	O
enforces	O
arc	O
consistency	B-General_Concept
:	O
it	O
removes	O
,	O
from	O
the	O
domain	O
of	O
the	O
variable	O
,	O
every	O
value	O
that	O
does	O
not	O
correspond	O
to	O
a	O
value	O
of	O
the	O
other	O
variable	O
.	O
</s>
<s>
Constraint	B-Application
propagation	I-Application
can	O
make	O
the	O
whole	O
problem	O
arc	O
consistent	O
by	O
repeating	O
this	O
removal	O
for	O
all	O
pairs	O
of	O
variables	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
is	O
arc	O
consistent	O
with	O
but	O
the	O
algorithm	O
reduces	O
the	O
domain	O
of	O
,	O
arc	O
consistency	B-General_Concept
of	O
with	O
does	O
not	O
hold	O
any	O
longer	O
,	O
and	O
has	O
to	O
be	O
enforced	O
again	O
.	O
</s>
<s>
A	O
simplistic	O
algorithm	O
would	O
cycle	O
over	O
the	O
pairs	O
of	O
variables	O
,	O
enforcing	O
arc-consistency	B-Application
,	O
repeating	O
the	O
cycle	O
until	O
no	O
domains	O
change	O
for	O
a	O
whole	O
cycle	O
.	O
</s>
<s>
The	O
AC-3	B-Application
algorithm	I-Application
improves	O
over	O
this	O
algorithm	O
by	O
ignoring	O
constraints	O
that	O
have	O
not	O
been	O
modified	O
since	O
they	O
were	O
last	O
analyzed	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
works	O
on	O
a	O
set	O
of	O
constraints	O
that	O
initially	O
contains	O
all	O
of	O
them	O
;	O
at	O
each	O
step	O
,	O
it	O
takes	O
a	O
constraint	O
and	O
enforces	O
arc-consistency	B-Application
;	O
if	O
this	O
operation	O
may	O
have	O
produced	O
a	O
violation	O
of	O
arc-consistency	B-Application
over	O
another	O
constraint	O
,	O
it	O
places	O
it	O
back	O
in	O
the	O
set	O
of	O
constraints	O
to	O
analyze	O
.	O
</s>
<s>
This	O
way	O
,	O
once	O
arc-consistency	B-Application
is	O
enforced	O
on	O
a	O
constraint	O
,	O
this	O
constraint	O
is	O
not	O
considered	O
again	O
unless	O
the	O
domain	O
of	O
one	O
of	O
its	O
variables	O
is	O
changed	O
.	O
</s>
<s>
Path	B-Application
consistency	I-Application
is	O
a	O
property	O
similar	O
to	O
arc	O
consistency	B-General_Concept
,	O
but	O
considers	O
pairs	O
of	O
variables	O
instead	O
of	O
only	O
one	O
.	O
</s>
<s>
The	O
form	O
of	O
constraint	B-Application
propagation	I-Application
that	O
enforces	O
path	B-Application
consistency	I-Application
works	O
by	O
removing	O
some	O
satisfying	O
assignment	O
from	O
a	O
constraint	O
.	O
</s>
<s>
Indeed	O
,	O
path	B-Application
consistency	I-Application
can	O
be	O
enforced	O
by	O
removing	O
from	O
a	O
binary	O
constraint	O
all	O
evaluations	O
that	O
cannot	O
be	O
extended	O
to	O
another	O
variable	O
.	O
</s>
<s>
As	O
for	O
arc	O
consistency	B-General_Concept
,	O
this	O
removal	O
might	O
have	O
to	O
consider	O
a	O
binary	O
constraint	O
more	O
than	O
once	O
.	O
</s>
<s>
As	O
for	O
arc	O
consistency	B-General_Concept
,	O
the	O
resulting	O
problem	O
has	O
the	O
same	O
solutions	O
of	O
the	O
original	O
one	O
,	O
as	O
the	O
removed	O
values	O
are	O
in	O
no	O
solution	O
.	O
</s>
<s>
The	O
form	O
of	O
constraint	B-Application
propagation	I-Application
that	O
enforces	O
path	B-Application
consistency	I-Application
might	O
introduce	O
new	O
constraints	O
.	O
</s>
<s>
However	O
,	O
some	O
pair	O
of	O
values	O
might	O
be	O
removed	O
by	O
constraint	B-Application
propagation	I-Application
.	O
</s>
<s>
The	O
name	O
"	O
path	B-Application
consistency	I-Application
"	O
derives	O
from	O
the	O
original	O
definition	O
,	O
which	O
involved	O
a	O
pair	O
of	O
variables	O
and	O
a	O
path	O
between	O
them	O
,	O
rather	O
than	O
a	O
pair	O
and	O
a	O
single	O
variable	O
.	O
</s>
<s>
Arc	O
and	O
path	B-Application
consistency	I-Application
can	O
be	O
generalized	O
to	O
non-binary	O
constraints	O
using	O
tuples	B-Application
of	O
variables	O
instead	O
of	O
a	O
single	O
one	O
or	O
a	O
pair	O
.	O
</s>
<s>
A	O
tuple	B-Application
of	O
variables	O
is	O
-consistent	O
with	O
another	O
variable	O
if	O
every	O
consistent	O
evaluation	O
of	O
the	O
variables	O
can	O
be	O
extended	O
with	O
a	O
value	O
of	O
the	O
other	O
variable	O
while	O
preserving	O
consistency	B-General_Concept
.	O
</s>
<s>
Strong	O
-consistency	O
is	O
-consistency	O
for	O
all	O
.	O
</s>
<s>
The	O
particular	O
case	O
of	O
2-consistency	O
coincides	O
with	O
arc	O
consistency	B-General_Concept
(	O
all	O
problems	O
are	O
assumed	O
node-consistent	O
in	O
this	O
article	O
)	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
3-consistency	O
coincides	O
with	O
path	B-Application
consistency	I-Application
only	O
if	O
all	O
constraints	O
are	O
binary	O
,	O
because	O
path	B-Application
consistency	I-Application
does	O
not	O
involve	O
ternary	O
constraints	O
while	O
3-consistency	O
does	O
.	O
</s>
<s>
Another	O
way	O
of	O
generalizing	O
arc	O
consistency	B-General_Concept
is	O
hyper-arc	B-Application
consistency	I-Application
or	O
generalized	O
arc	O
consistency	B-General_Concept
,	O
which	O
requires	O
extendibility	O
of	O
a	O
single	O
variable	O
in	O
order	O
to	O
satisfy	O
a	O
constraint	O
.	O
</s>
<s>
Constraint	B-Application
propagation	I-Application
(	O
enforcing	O
a	O
form	O
of	O
local	B-Application
consistency	I-Application
)	O
might	O
produce	O
an	O
empty	O
domain	O
or	O
an	O
unsatisfiable	O
constraint	O
.	O
</s>
<s>
Indeed	O
,	O
local	B-Application
consistency	I-Application
is	O
only	O
relative	O
to	O
the	O
consistency	B-General_Concept
of	O
groups	O
of	O
variables	O
.	O
</s>
<s>
For	O
example	O
,	O
arc	O
consistency	B-General_Concept
guarantees	O
that	O
every	O
consistent	O
evaluation	O
of	O
a	O
variable	O
can	O
be	O
consistently	O
extended	O
to	O
another	O
variable	O
.	O
</s>
<s>
However	O
,	O
constraint	B-Application
propagation	I-Application
can	O
be	O
used	O
to	O
prove	O
satisfiability	O
in	O
some	O
cases	O
.	O
</s>
<s>
A	O
similar	O
condition	O
holds	O
for	O
path	B-Application
consistency	I-Application
.	O
</s>
<s>
The	O
special	O
cases	O
in	O
which	O
satisfiability	O
can	O
be	O
established	O
by	O
enforcing	O
arc	O
consistency	B-General_Concept
and	O
path	B-Application
consistency	I-Application
are	O
the	O
following	O
ones	O
.	O
</s>
<s>
enforcing	O
arc	O
consistency	B-General_Concept
establishes	O
satisfiability	O
of	O
problems	O
made	O
of	O
binary	O
constraints	O
with	O
no	O
cycles	O
(	O
a	O
tree	O
of	O
binary	O
constraints	O
)	O
;	O
</s>
<s>
enforcing	O
path	B-Application
consistency	I-Application
establishes	O
satisfiability	O
for	O
binary	O
constraints	O
(	O
possibly	O
with	O
cycles	O
)	O
with	O
binary	O
domains	O
;	O
</s>
<s>
enforcing	O
strong	O
consistency	B-General_Concept
establishes	O
satisfiability	O
of	O
problems	O
containing	O
variables	O
.	O
</s>
<s>
Some	O
definitions	O
or	O
results	O
about	O
relative	O
consistency	B-General_Concept
hold	O
only	O
in	O
special	O
cases	O
.	O
</s>
<s>
When	O
the	O
domains	O
are	O
composed	O
of	O
integers	O
,	O
bound	B-Application
consistency	I-Application
can	O
be	O
defined	O
.	O
</s>
<s>
This	O
form	O
of	O
consistency	B-General_Concept
is	O
based	O
on	O
the	O
consistency	B-General_Concept
of	O
the	O
extreme	O
values	O
of	O
the	O
domains	O
,	O
that	O
is	O
,	O
the	O
minimum	O
and	O
maximum	O
values	O
a	O
variable	O
can	O
take	O
.	O
</s>
<s>
When	O
constraints	O
are	O
algebraic	O
or	O
Boolean	O
,	O
arc	O
consistency	B-General_Concept
is	O
equivalent	O
to	O
adding	O
new	O
constraint	O
or	O
syntactically	O
modifying	O
an	O
old	O
one	O
,	O
and	O
this	O
can	O
be	O
done	O
by	O
suitably	O
composing	O
constraints	O
.	O
</s>
<s>
Efficient	O
specialized	O
algorithms	O
for	O
enforcing	O
arc	O
consistency	B-General_Concept
on	O
such	O
constraints	O
exist	O
.	O
</s>
<s>
When	O
the	O
domain	O
of	O
a	O
variable	O
is	O
reduced	O
to	O
a	O
single	O
value	O
,	O
this	O
value	O
can	O
be	O
removed	O
from	O
all	O
other	O
domains	O
by	O
constraint	B-Application
propagation	I-Application
when	O
enforcing	O
arc	O
consistency	B-General_Concept
.	O
</s>
<s>
More	O
precisely	O
,	O
after	O
arc	O
consistency	B-General_Concept
is	O
enforced	O
,	O
the	O
number	O
of	O
unassigned	O
variables	O
must	O
not	O
exceed	O
the	O
number	O
of	O
values	O
in	O
the	O
union	O
of	O
their	O
domains	O
.	O
</s>
<s>
This	O
condition	O
can	O
be	O
checked	O
easily	O
on	O
a	O
constraint	O
in	O
the	O
alldifferent	O
form	O
,	O
but	O
does	O
not	O
correspond	O
to	O
arc	O
consistency	B-General_Concept
of	O
the	O
network	O
of	O
disequalities	O
.	O
</s>
<s>
A	O
second	O
property	O
of	O
the	O
single	O
alldifferent	O
constraint	O
is	O
that	O
hyper-arc	B-Application
consistency	I-Application
can	O
be	O
efficiently	O
checked	O
using	O
a	O
bipartite	O
matching	O
algorithm	O
.	O
</s>
<s>
The	O
constraint	O
states	O
that	O
the	O
total	O
available	O
amount	O
of	O
resources	O
is	O
L	O
.	O
Specialized	O
constraint	B-Application
propagation	I-Application
techniques	O
for	O
cumulative	O
constraints	O
exists	O
;	O
different	O
techniques	O
are	O
used	O
depending	O
on	O
which	O
variable	O
domains	O
are	O
already	O
reduced	O
to	O
a	O
single	O
value	O
.	O
</s>
<s>
A	O
third	O
specialized	O
constraint	O
that	O
is	O
used	O
in	O
constraint	B-Application
logic	I-Application
programming	I-Application
is	O
the	O
element	O
one	O
.	O
</s>
<s>
In	O
constraint	B-Application
logic	I-Application
programming	I-Application
,	O
lists	O
are	O
allowed	O
as	O
values	O
of	O
variables	O
.	O
</s>
<s>
Specialized	O
constraint	B-Application
propagation	I-Application
rules	O
for	O
these	O
constraints	O
exist	O
.	O
</s>
<s>
Directional	B-Application
consistency	I-Application
is	O
the	O
variant	O
of	O
arc	O
,	O
path	O
,	O
and	O
-consistency	O
tailored	O
for	O
being	O
used	O
by	O
an	O
algorithm	O
that	O
assigns	O
values	O
to	O
variables	O
following	O
a	O
given	O
order	O
of	O
variables	O
.	O
</s>
<s>
If	O
an	O
algorithm	O
evaluates	O
variables	O
in	O
the	O
order	O
,	O
consistency	B-General_Concept
is	O
only	O
useful	O
when	O
it	O
guarantees	O
that	O
values	O
of	O
lower-index	O
variables	O
are	O
all	O
consistent	O
with	O
values	O
of	O
higher-index	O
ones	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
enforcing	O
consistency	B-General_Concept
with	O
variables	O
that	O
are	O
already	O
evaluated	O
is	O
not	O
necessary	O
:	O
if	O
the	O
algorithm	O
chooses	O
a	O
value	O
that	O
is	O
inconsistent	O
with	O
the	O
current	O
partial	O
evaluation	O
,	O
inconsistency	O
is	O
detected	O
anyway	O
.	O
</s>
<s>
Assuming	O
that	O
the	O
order	O
of	O
evaluation	O
of	O
the	O
variables	O
is	O
,	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
is	O
directionally	O
arc	O
consistent	O
if	O
every	O
variable	O
is	O
arc	O
consistent	O
with	O
any	O
other	O
variable	O
such	O
that	O
.	O
</s>
<s>
Directional	O
path	B-Application
consistency	I-Application
is	O
similar	O
,	O
but	O
two	O
variables	O
have	O
to	O
be	O
path	O
consistent	O
with	O
only	O
if	O
.	O
</s>
<s>
Strong	B-Application
directional	I-Application
path	I-Application
consistency	I-Application
means	O
both	O
directional	O
path	B-Application
consistency	I-Application
and	O
directional	B-Application
arc	I-Application
consistency	I-Application
.	O
</s>
<s>
Similar	O
definitions	O
can	O
be	O
given	O
for	O
the	O
other	O
forms	O
of	O
consistency	B-General_Concept
.	O
</s>
<s>
Constraint	B-Application
propagation	I-Application
enforcing	O
directional	B-Application
arc	I-Application
consistency	I-Application
iterates	O
over	O
variables	O
from	O
the	O
last	O
to	O
the	O
first	O
,	O
enforcing	O
at	O
each	O
step	O
the	O
arc	O
consistency	B-General_Concept
of	O
every	O
variable	O
of	O
lower	O
index	O
with	O
it	O
.	O
</s>
<s>
If	O
the	O
order	O
of	O
the	O
variables	O
is	O
,	O
this	O
algorithm	O
iterates	O
over	O
variables	O
from	O
to	O
;	O
for	O
variable	O
,	O
it	O
enforces	O
arc	O
consistency	B-General_Concept
of	O
every	O
variable	O
of	O
index	O
lower	O
than	O
with	O
.	O
</s>
<s>
Enforcing	O
directional	B-Application
arc	I-Application
consistency	I-Application
starts	O
with	O
,	O
and	O
makes	O
arc	O
consistent	O
with	O
it	O
by	O
removing	O
the	O
value	O
.	O
</s>
<s>
Enforcing	O
directional	B-Application
arc	I-Application
consistency	I-Application
proceeds	O
with	O
.	O
</s>
<s>
Directional	O
path	B-Application
consistency	I-Application
and	O
strong	B-Application
directional	I-Application
path	I-Application
consistency	I-Application
can	O
be	O
enforced	O
by	O
algorithms	O
similar	O
to	O
the	O
one	O
for	O
arc	O
consistency	B-General_Concept
.	O
</s>
<s>
They	O
process	O
variables	O
from	O
to	O
;	O
for	O
every	O
variable	O
two	O
variables	O
with	O
are	O
considered	O
,	O
and	O
path	B-Application
consistency	I-Application
of	O
them	O
with	O
is	O
enforced	O
.	O
</s>
<s>
If	O
constraint	B-Application
propagation	I-Application
reduces	O
its	O
set	O
of	O
satisfying	O
assignments	O
,	O
it	O
effectively	O
create	O
a	O
new	O
non-trivial	O
constraint	O
.	O
</s>
<s>
Constraint	B-Application
propagation	I-Application
enforcing	O
strong	B-Application
directional	I-Application
path	I-Application
consistency	I-Application
is	O
similar	O
,	O
but	O
also	O
enforces	O
arc	O
consistency	B-General_Concept
.	O
</s>
<s>
Directional	B-Application
consistency	I-Application
guarantees	O
that	O
partial	O
solutions	O
satisfying	O
a	O
constraint	O
can	O
be	O
consistently	O
extended	O
to	O
another	O
variable	O
of	O
higher	O
index	O
.	O
</s>
<s>
There	O
are	O
two	O
cases	O
in	O
which	O
this	O
does	O
not	O
happen	O
,	O
and	O
directional	B-Application
consistency	I-Application
guarantees	O
satisfiability	O
if	O
no	O
domain	O
is	O
empty	O
and	O
no	O
constraint	O
is	O
unsatisfiable	O
.	O
</s>
<s>
The	O
first	O
case	O
is	O
that	O
of	O
a	O
binary	O
constraint	O
problem	O
with	O
an	O
ordering	O
of	O
the	O
variables	O
that	O
makes	O
the	O
ordered	B-Application
graph	I-Application
of	O
constraint	O
having	O
width	B-Application
1	O
.	O
</s>
<s>
If	O
this	O
is	O
the	O
case	O
,	O
the	O
width	B-Application
of	O
the	O
graph	O
bounds	O
the	O
maximal	O
number	O
of	O
lower	O
(	O
according	O
to	O
the	O
ordering	O
)	O
nodes	O
a	O
node	O
is	O
joined	O
to	O
.	O
</s>
<s>
Directional	B-Application
arc	I-Application
consistency	I-Application
guarantees	O
that	O
every	O
consistent	O
assignment	O
to	O
a	O
variable	O
can	O
be	O
extended	O
to	O
higher	O
nodes	O
,	O
and	O
width	B-Application
1	O
guarantees	O
that	O
a	O
node	O
is	O
not	O
joined	O
to	O
more	O
than	O
one	O
lower	O
node	O
.	O
</s>
<s>
Indeed	O
,	O
no	O
other	O
lower	O
variable	O
is	O
joined	O
to	O
that	O
higher	O
variable	O
,	O
as	O
the	O
graph	O
has	O
width	B-Application
1	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
if	O
a	O
constraint	O
problem	O
has	O
width	B-Application
1	O
with	O
respect	O
to	O
an	O
ordering	O
of	O
its	O
variables	O
(	O
which	O
implies	O
that	O
its	O
corresponding	O
graph	O
is	O
a	O
tree	O
)	O
and	O
the	O
problem	O
is	O
directionally	O
arc	O
consistent	O
with	O
respect	O
to	O
the	O
same	O
ordering	O
,	O
a	O
solution	O
(	O
if	O
any	O
)	O
can	O
be	O
found	O
by	O
iteratively	O
assigning	O
variables	O
according	O
to	O
the	O
ordering	O
.	O
</s>
<s>
The	O
second	O
case	O
in	O
which	O
directional	B-Application
consistency	I-Application
guarantees	O
satisfiability	O
if	O
no	O
domain	O
is	O
empty	O
and	O
no	O
constraint	O
is	O
unsatisfiable	O
is	O
that	O
of	O
binary	O
constraint	O
problems	O
whose	O
graph	O
has	O
induced	B-Application
width	I-Application
2	O
,	O
using	O
strong	B-Application
directional	I-Application
path	I-Application
consistency	I-Application
.	O
</s>
<s>
Indeed	O
,	O
this	O
form	O
of	O
consistency	B-General_Concept
guarantees	O
that	O
every	O
assignment	O
to	O
a	O
variable	O
or	O
a	O
pair	O
of	O
variables	O
can	O
be	O
extended	O
to	O
a	O
higher	O
variable	O
,	O
and	O
width	B-Application
2	O
guarantees	O
that	O
this	O
variable	O
is	O
not	O
joined	O
to	O
another	O
pair	O
of	O
lower	O
variables	O
.	O
</s>
<s>
The	O
reason	O
why	O
the	O
induced	B-Application
width	I-Application
is	O
considered	O
instead	O
of	O
the	O
width	B-Application
is	O
that	O
enforcing	O
directional	O
path	B-Application
consistency	I-Application
may	O
add	O
constraints	O
.	O
</s>
<s>
Indeed	O
,	O
if	O
two	O
variables	O
are	O
not	O
in	O
the	O
same	O
constraint	O
but	O
are	O
in	O
a	O
constraint	O
with	O
a	O
higher	O
variable	O
,	O
some	O
pairs	O
of	O
their	O
values	O
may	O
violate	O
path	B-Application
consistency	I-Application
.	O
</s>
<s>
As	O
a	O
result	O
,	O
constraint	B-Application
propagation	I-Application
may	O
produce	O
a	O
problem	O
whose	O
graph	O
has	O
more	O
edges	O
than	O
the	O
original	O
one	O
.	O
</s>
<s>
Width	B-Application
2	O
guarantees	O
that	O
every	O
consistent	O
partial	O
evaluation	O
can	O
be	O
extended	O
to	O
a	O
solution	O
,	O
but	O
this	O
width	B-Application
is	O
relative	O
to	O
the	O
generated	O
graph	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
induced	B-Application
width	I-Application
being	O
2	O
is	O
required	O
for	O
strong	B-Application
directional	I-Application
path	I-Application
consistency	I-Application
to	O
guarantee	O
the	O
existence	O
of	O
solutions	O
.	O
</s>
<s>
Directional	O
-consistency	O
is	O
the	O
guarantee	O
that	O
every	O
consistent	O
assignment	O
to	O
variables	O
can	O
be	O
consistently	O
extended	O
to	O
another	O
variable	O
that	O
is	O
higher	O
in	O
the	O
order	O
.	O
</s>
<s>
Strong	O
directional	O
-consistency	O
is	O
defined	O
in	O
a	O
similar	O
way	O
,	O
but	O
all	O
groups	O
of	O
at	O
most	O
variables	O
are	O
considered	O
.	O
</s>
<s>
If	O
a	O
problem	O
is	O
strongly	O
directionally	O
-consistent	O
and	O
has	O
width	B-Application
less	O
than	O
and	O
has	O
no	O
empty	O
domain	O
or	O
unsatisfiable	O
constraint	O
,	O
it	O
has	O
solutions	O
.	O
</s>
<s>
Every	O
problem	O
can	O
be	O
made	O
strongly	O
directionally	O
-consistent	O
,	O
but	O
this	O
operation	O
may	O
increase	O
the	O
width	B-Application
of	O
its	O
corresponding	O
graphs	O
.	O
</s>
<s>
The	O
constraint	B-Application
propagation	I-Application
procedure	O
that	O
enforces	O
directional	B-Application
consistency	I-Application
is	O
similar	O
to	O
that	O
used	O
for	O
directional	B-Application
arc	I-Application
consistency	I-Application
and	O
path	B-Application
consistency	I-Application
.	O
</s>
<s>
Consistency	B-General_Concept
of	O
these	O
variables	O
with	O
is	O
checked	O
and	O
possibly	O
enforced	O
by	O
removing	O
satisfying	O
assignments	O
from	O
the	O
constraint	O
among	O
all	O
these	O
variables	O
(	O
if	O
any	O
,	O
or	O
creating	O
a	O
new	O
one	O
otherwise	O
)	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
even	O
if	O
the	O
width	B-Application
of	O
the	O
original	O
problem	O
is	O
,	O
the	O
width	B-Application
of	O
the	O
resulting	O
instance	O
may	O
be	O
greater	O
.	O
</s>
<s>
If	O
this	O
is	O
the	O
case	O
,	O
directional	O
strong	O
consistency	B-General_Concept
does	O
not	O
imply	O
satisfiability	O
even	O
if	O
no	O
domain	O
is	O
empty	O
and	O
no	O
constraint	O
is	O
unsatisfiable	O
.	O
</s>
<s>
However	O
,	O
constraint	B-Application
propagation	I-Application
only	O
adds	O
constraints	O
to	O
variables	O
that	O
are	O
lower	O
than	O
the	O
one	O
it	O
is	O
currently	O
considering	O
.	O
</s>
<s>
Since	O
this	O
algorithm	O
can	O
be	O
seen	O
as	O
a	O
modification	O
of	O
the	O
previous	O
one	O
with	O
a	O
value	O
that	O
is	O
changed	O
to	O
the	O
number	O
of	O
parents	O
of	O
each	O
node	O
,	O
it	O
is	O
called	O
adaptive	O
consistency	B-General_Concept
.	O
</s>
<s>
This	O
algorithm	O
enforces	O
strongly	O
directional	O
-consistency	O
with	O
equal	O
to	O
the	O
induced	B-Application
width	I-Application
of	O
the	O
problem	O
.	O
</s>
<s>
This	O
algorithm	O
is	O
not	O
always	O
polynomial-time	O
,	O
as	O
the	O
number	O
of	O
constraints	O
introduced	O
by	O
enforcing	O
strong	O
directional	B-Application
consistency	I-Application
may	O
produce	O
an	O
exponential	O
increase	O
of	O
size	O
.	O
</s>
<s>
The	O
problem	O
is	O
however	O
solvable	O
in	O
polynomial	O
time	O
if	O
the	O
enforcing	O
strong	O
directional	B-Application
consistency	I-Application
does	O
not	O
superpolynomially	O
enlarge	O
the	O
instance	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
if	O
an	O
instance	O
has	O
induced	B-Application
width	I-Application
bounded	O
by	O
a	O
constant	O
,	O
it	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
It	O
can	O
be	O
defined	O
as	O
a	O
reformulation	O
of	O
adaptive	O
consistency	B-General_Concept
.	O
</s>
<s>
This	O
algorithm	O
is	O
equivalent	O
to	O
enforcing	O
adaptive	O
consistency	B-General_Concept
.	O
</s>
<s>
Since	O
they	O
both	O
enforce	O
consistency	B-General_Concept
of	O
a	O
variable	O
with	O
all	O
its	O
parents	O
,	O
and	O
since	O
no	O
new	O
constraint	O
is	O
added	O
after	O
a	O
variable	O
is	O
considered	O
,	O
what	O
results	O
is	O
an	O
instance	O
that	O
can	O
be	O
solved	O
without	O
backtracking	B-Algorithm
.	O
</s>
<s>
Since	O
the	O
graph	O
of	O
the	O
instance	O
they	O
produce	O
is	O
a	O
subgraph	O
of	O
the	O
induced	O
graph	O
,	O
if	O
the	O
induced	B-Application
width	I-Application
is	O
bounded	O
by	O
a	O
constant	O
the	O
generated	O
instance	O
is	O
of	O
size	O
polynomial	O
in	O
the	O
size	O
of	O
the	O
original	O
instance	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
if	O
the	O
induced	B-Application
width	I-Application
of	O
an	O
instance	O
is	O
bounded	O
by	O
a	O
constant	O
,	O
solving	O
it	O
can	O
be	O
done	O
in	O
polynomial	O
time	O
by	O
the	O
two	O
algorithms	O
.	O
</s>
<s>
While	O
the	O
previous	O
definitions	O
of	O
consistency	B-General_Concept
are	O
all	O
about	O
consistency	B-General_Concept
of	O
assignments	O
,	O
relational	B-Application
consistency	I-Application
involves	O
satisfaction	O
of	O
a	O
given	O
constraint	O
or	O
set	O
of	O
constraints	O
only	O
.	O
</s>
<s>
More	O
precisely	O
,	O
relational	B-Application
consistency	I-Application
implies	O
that	O
every	O
consistent	O
partial	O
assignment	O
can	O
be	O
extended	O
in	O
such	O
a	O
way	O
that	O
a	O
given	O
constraint	O
or	O
set	O
of	O
constraints	O
is	O
satisfied	O
.	O
</s>
<s>
The	O
difference	O
between	O
"	O
regular	O
"	O
consistency	B-General_Concept
and	O
relational	B-Application
arc	I-Application
consistency	I-Application
is	O
that	O
the	O
latter	O
only	O
requires	O
the	O
extended	O
assignment	O
to	O
satisfy	O
a	O
given	O
constraint	O
,	O
while	O
the	O
former	O
requires	O
it	O
to	O
satisfy	O
all	O
relevant	O
constraints	O
.	O
</s>
<s>
In	O
particular	O
,	O
relational	B-Application
path	I-Application
consistency	I-Application
is	O
similar	O
to	O
relational	O
arc-consistency	B-Application
,	O
but	O
two	O
constraints	O
are	O
used	O
in	O
place	O
of	O
one	O
.	O
</s>
<s>
For	O
more	O
than	O
two	O
constraints	O
,	O
relational	O
-consistency	O
is	O
defined	O
.	O
</s>
<s>
Relational	O
-consistency	O
involves	O
a	O
set	O
of	O
constraints	O
and	O
a	O
variable	O
that	O
is	O
in	O
the	O
scope	O
of	O
all	O
these	O
constraints	O
.	O
</s>
<s>
Strong	O
relational	O
consistency	B-General_Concept
is	O
defined	O
as	O
above	O
:	O
it	O
is	O
the	O
property	O
of	O
being	O
relational	O
-consistent	O
for	O
every	O
.	O
</s>
<s>
Relational	B-Application
consistency	I-Application
can	O
also	O
be	O
defined	O
for	O
more	O
variables	O
,	O
instead	O
of	O
one	O
.	O
</s>
<s>
If	O
an	O
order	O
of	O
the	O
variables	O
is	O
given	O
,	O
relational	B-Application
consistency	I-Application
can	O
be	O
restricted	O
to	O
the	O
cases	O
when	O
the	O
variables(s )	O
the	O
evaluation	O
should	O
be	O
extendable	O
to	O
follow	O
the	O
other	O
variables	O
in	O
the	O
order	O
.	O
</s>
<s>
This	O
modified	O
condition	O
is	O
called	O
directional	O
relational	B-Application
consistency	I-Application
.	O
</s>
<s>
A	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
may	O
be	O
relationally	O
consistent	O
,	O
have	O
no	O
empty	O
domain	O
or	O
unsatisfiable	O
constraint	O
,	O
and	O
yet	O
be	O
unsatisfiable	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
evaluation	O
cannot	O
be	O
extended	O
to	O
satisfy	O
all	O
these	O
-or-less	O
constraints	O
,	O
violating	O
the	O
condition	O
of	O
strong	O
relational	O
-consistency	O
.	O
</s>
<s>
The	O
condition	O
that	O
makes	O
strong	O
relational	B-Application
path	I-Application
consistency	I-Application
equivalent	O
to	O
satisfiability	O
is	O
that	O
of	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
for	O
which	O
there	O
exists	O
an	O
order	O
of	O
the	O
variables	O
that	O
makes	O
all	O
constraint	O
to	O
be	O
represented	O
by	O
row	O
convex	O
matrices	O
.	O
</s>
<s>
Since	O
these	O
row	O
are	O
convex	O
,	O
and	O
they	O
have	O
a	O
common	O
element	O
pairwise	O
because	O
of	O
path	B-Application
consistency	I-Application
,	O
they	O
also	O
have	O
a	O
shared	O
common	O
element	O
,	O
which	O
represents	O
a	O
value	O
of	O
the	O
last	O
variable	O
that	O
is	O
consistent	O
with	O
the	O
other	O
ones	O
.	O
</s>
<s>
All	O
forms	O
of	O
local	B-Application
consistency	I-Application
can	O
be	O
enforced	O
by	O
constraint	B-Application
propagation	I-Application
,	O
which	O
may	O
reduce	O
the	O
domains	O
of	O
variables	O
and	O
the	O
sets	O
of	O
assignments	O
satisfying	O
a	O
constraint	O
and	O
may	O
introduce	O
new	O
constraints	O
.	O
</s>
<s>
Whenever	O
constraint	B-Application
propagation	I-Application
produces	O
an	O
empty	O
domain	O
or	O
an	O
unsatisfiable	O
constraint	O
,	O
the	O
original	O
problem	O
is	O
unsatisfiable	O
.	O
</s>
<s>
Therefore	O
,	O
all	O
forms	O
of	O
local	B-Application
consistency	I-Application
can	O
be	O
used	O
as	O
approximations	O
of	O
satisfiability	O
.	O
</s>
<s>
Such	O
approximated	O
algorithms	O
can	O
be	O
used	O
by	O
search	O
algorithms	O
(	O
backtracking	B-Algorithm
,	O
backjumping	B-Application
,	O
local	B-Application
search	I-Application
,	O
etc	O
.	O
)	O
</s>
<s>
Even	O
if	O
constraint	B-Application
propagation	I-Application
does	O
not	O
produce	O
an	O
empty	O
domain	O
or	O
an	O
unsatisfiable	O
constraint	O
,	O
it	O
may	O
nevertheless	O
reduce	O
the	O
domains	O
or	O
strengthen	O
the	O
constraints	O
.	O
</s>
<s>
Local	B-Application
consistency	I-Application
proves	O
satisfiability	O
in	O
some	O
restricted	O
cases	O
(	O
see	O
Complexity	O
of	O
constraint	B-Application
satisfaction	I-Application
#Restrictions	O
)	O
.	O
</s>
<s>
This	O
is	O
the	O
case	O
for	O
some	O
special	O
kind	O
of	O
problems	O
and/or	O
for	O
some	O
kinds	O
of	O
local	B-Application
consistency	I-Application
.	O
</s>
<s>
For	O
example	O
,	O
enforcing	O
arc	O
consistency	B-General_Concept
on	O
binary	O
acyclic	O
problems	O
allows	O
for	O
telling	O
whether	O
the	O
problem	O
is	O
satisfiable	O
.	O
</s>
<s>
Enforcing	O
strong	O
directional	O
-consistency	O
allows	O
telling	O
the	O
satisfiability	O
of	O
problems	O
that	O
have	O
induced	B-Application
width	I-Application
according	O
to	O
the	O
same	O
order	O
.	O
</s>
<s>
Adaptive	O
directional	B-Application
consistency	I-Application
allows	O
telling	O
the	O
satisfiability	O
of	O
an	O
arbitrary	O
problem	O
.	O
</s>
