<s>
Constraint	B-Application
satisfaction	I-Application
problems	I-Application
(	O
CSPs	O
)	O
are	O
mathematical	O
questions	O
defined	O
as	O
a	O
set	O
of	O
objects	O
whose	O
state	B-Application
must	O
satisfy	O
a	O
number	O
of	O
constraints	B-Application
or	O
limitations	B-Algorithm
.	O
</s>
<s>
CSPs	O
represent	O
the	O
entities	O
in	O
a	O
problem	O
as	O
a	O
homogeneous	O
collection	O
of	O
finite	O
constraints	B-Application
over	O
variables	O
,	O
which	O
is	O
solved	O
by	O
constraint	B-Application
satisfaction	I-Application
methods	O
.	O
</s>
<s>
CSPs	O
are	O
the	O
subject	O
of	O
research	O
in	O
both	O
artificial	B-Application
intelligence	I-Application
and	O
operations	O
research	O
,	O
since	O
the	O
regularity	O
in	O
their	O
formulation	O
provides	O
a	O
common	O
basis	O
to	O
analyze	O
and	O
solve	O
problems	O
of	O
many	O
seemingly	O
unrelated	O
families	O
.	O
</s>
<s>
CSPs	B-Application
often	I-Application
exhibit	I-Application
high	I-Application
complexity	I-Application
,	O
requiring	O
a	O
combination	O
of	O
heuristics	B-Algorithm
and	O
combinatorial	B-Algorithm
search	I-Algorithm
methods	O
to	O
be	O
solved	O
in	O
a	O
reasonable	O
time	O
.	O
</s>
<s>
Constraint	B-Application
programming	I-Application
(	O
CP	B-Application
)	O
is	O
the	O
field	O
of	O
research	O
that	O
specifically	O
focuses	O
on	O
tackling	O
these	O
kinds	O
of	O
problems	O
.	O
</s>
<s>
Additionally	O
,	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
(	O
SAT	O
)	O
,	O
the	O
satisfiability	B-Application
modulo	I-Application
theories	I-Application
(	O
SMT	O
)	O
,	O
mixed	O
integer	O
programming	O
(	O
MIP	O
)	O
and	O
answer	B-Application
set	I-Application
programming	I-Application
(	O
ASP	O
)	O
are	O
all	O
fields	O
of	O
research	O
focusing	O
on	O
the	O
resolution	O
of	O
particular	O
forms	O
of	O
the	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
.	O
</s>
<s>
Examples	O
of	O
problems	O
that	O
can	O
be	O
modeled	O
as	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
include	O
:	O
</s>
<s>
These	O
are	O
often	O
provided	O
with	O
tutorials	O
of	O
CP	B-Application
,	O
ASP	O
,	O
Boolean	B-Algorithm
SAT	I-Algorithm
and	O
SMT	B-Application
solvers	I-Application
.	O
</s>
<s>
In	O
the	O
general	O
case	O
,	O
constraint	B-Application
problems	O
can	O
be	O
much	O
harder	O
,	O
and	O
may	O
not	O
be	O
expressible	O
in	O
some	O
of	O
these	O
simpler	O
systems	O
.	O
</s>
<s>
"	O
Real	O
life	O
"	O
examples	O
include	O
automated	B-Application
planning	I-Application
,	O
lexical	B-General_Concept
disambiguation	I-General_Concept
,	O
musicology	O
,	O
product	B-Application
configuration	I-Application
and	O
resource	B-Algorithm
allocation	I-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
decided	O
by	O
finding	O
a	O
solution	O
,	O
or	O
failing	O
to	O
find	O
a	O
solution	O
after	O
exhaustive	O
search	B-Application
(	O
stochastic	O
algorithms	O
typically	O
never	O
reach	O
an	O
exhaustive	O
conclusion	O
,	O
while	O
directed	O
searches	O
often	O
do	O
,	O
on	O
sufficiently	O
small	O
problems	O
)	O
.	O
</s>
<s>
is	O
a	O
set	O
of	O
constraints	B-Application
.	O
</s>
<s>
Every	O
constraint	B-Application
is	O
in	O
turn	O
a	O
pair	O
,	O
where	O
is	O
a	O
subset	O
of	O
variables	O
and	O
is	O
a	O
-ary	O
relation	B-Algorithm
on	O
the	O
corresponding	O
subset	O
of	O
domains	O
.	O
</s>
<s>
An	O
evaluation	O
satisfies	O
a	O
constraint	B-Application
if	O
the	O
values	O
assigned	O
to	O
the	O
variables	O
satisfies	O
the	O
relation	B-Algorithm
.	O
</s>
<s>
An	O
evaluation	O
is	O
consistent	O
if	O
it	O
does	O
not	O
violate	O
any	O
of	O
the	O
constraints	B-Application
.	O
</s>
<s>
An	O
evaluation	O
is	O
a	O
solution	O
if	O
it	O
is	O
consistent	O
and	O
complete	O
;	O
such	O
an	O
evaluation	O
is	O
said	O
to	O
solve	O
the	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
.	O
</s>
<s>
Constraint	B-Application
satisfaction	I-Application
problems	I-Application
on	O
finite	O
domains	O
are	O
typically	O
solved	O
using	O
a	O
form	O
of	O
search	B-Application
.	O
</s>
<s>
The	O
most	O
used	O
techniques	O
are	O
variants	O
of	O
backtracking	B-Algorithm
,	O
constraint	B-Application
propagation	I-Application
,	O
and	O
local	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
These	O
techniques	O
are	O
also	O
often	O
combined	O
,	O
as	O
in	O
the	O
VLNS	B-Algorithm
method	O
,	O
and	O
current	O
research	O
involves	O
other	O
technologies	O
such	O
as	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Backtracking	B-Algorithm
is	O
a	O
recursive	O
algorithm	O
.	O
</s>
<s>
For	O
each	O
value	O
,	O
the	O
consistency	O
of	O
the	O
partial	O
assignment	O
with	O
the	O
constraints	B-Application
is	O
checked	O
;	O
in	O
case	O
of	O
consistency	O
,	O
a	O
recursive	O
call	O
is	O
performed	O
.	O
</s>
<s>
In	O
this	O
basic	O
backtracking	B-Algorithm
algorithm	I-Algorithm
,	O
consistency	O
is	O
defined	O
as	O
the	O
satisfaction	O
of	O
all	O
constraints	B-Application
whose	O
variables	O
are	O
all	O
assigned	O
.	O
</s>
<s>
Several	O
variants	O
of	O
backtracking	B-Algorithm
exist	O
.	O
</s>
<s>
Backmarking	B-Application
improves	O
the	O
efficiency	O
of	O
checking	O
consistency	O
.	O
</s>
<s>
Backjumping	B-Application
allows	O
saving	O
part	O
of	O
the	O
search	B-Application
by	O
backtracking	B-Algorithm
"	O
more	O
than	O
one	O
variable	O
"	O
in	O
some	O
cases	O
.	O
</s>
<s>
Constraint	B-Application
learning	I-Application
infers	O
and	O
saves	O
new	O
constraints	B-Application
that	O
can	O
be	O
later	O
used	O
to	O
avoid	O
part	O
of	O
the	O
search	B-Application
.	O
</s>
<s>
Look-ahead	B-Application
is	O
also	O
often	O
used	O
in	O
backtracking	B-Algorithm
to	O
attempt	O
to	O
foresee	O
the	O
effects	O
of	O
choosing	O
a	O
variable	O
or	O
a	O
value	O
,	O
thus	O
sometimes	O
determining	O
in	O
advance	O
when	O
a	O
subproblem	O
is	O
satisfiable	O
or	O
unsatisfiable	O
.	O
</s>
<s>
Constraint	B-Application
propagation	I-Application
techniques	O
are	O
methods	O
used	O
to	O
modify	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
.	O
</s>
<s>
More	O
precisely	O
,	O
they	O
are	O
methods	O
that	O
enforce	O
a	O
form	O
of	O
local	B-Application
consistency	I-Application
,	O
which	O
are	O
conditions	O
related	O
to	O
the	O
consistency	O
of	O
a	O
group	O
of	O
variables	O
and/or	O
constraints	B-Application
.	O
</s>
<s>
Constraint	B-Application
propagation	I-Application
has	O
various	O
uses	O
.	O
</s>
<s>
This	O
is	O
not	O
guaranteed	O
to	O
happen	O
in	O
general	O
;	O
however	O
,	O
it	O
always	O
happens	O
for	O
some	O
forms	O
of	O
constraint	B-Application
propagation	I-Application
and/or	O
for	O
certain	O
kinds	O
of	O
problems	O
.	O
</s>
<s>
The	O
most	O
known	O
and	O
used	O
forms	O
of	O
local	B-Application
consistency	I-Application
are	O
arc	O
consistency	O
,	O
hyper-arc	B-Application
consistency	I-Application
,	O
and	O
path	B-Application
consistency	I-Application
.	O
</s>
<s>
The	O
most	O
popular	O
constraint	B-Application
propagation	I-Application
method	O
is	O
the	O
AC-3	B-Application
algorithm	I-Application
,	O
which	O
enforces	O
arc	O
consistency	O
.	O
</s>
<s>
Local	B-Algorithm
search	I-Algorithm
methods	O
are	O
incomplete	O
satisfiability	O
algorithms	O
.	O
</s>
<s>
At	O
each	O
step	O
,	O
a	O
small	O
number	O
of	O
variables	O
are	O
changed	O
in	O
value	O
,	O
with	O
the	O
overall	O
aim	O
of	O
increasing	O
the	O
number	O
of	O
constraints	B-Application
satisfied	O
by	O
this	O
assignment	O
.	O
</s>
<s>
The	O
min-conflicts	B-Application
algorithm	I-Application
is	O
a	O
local	B-Algorithm
search	I-Algorithm
algorithm	I-Algorithm
specific	O
for	O
CSPs	O
and	O
is	O
based	O
on	O
that	O
principle	O
.	O
</s>
<s>
In	O
practice	O
,	O
local	B-Algorithm
search	I-Algorithm
appears	O
to	O
work	O
well	O
when	O
these	O
changes	O
are	O
also	O
affected	O
by	O
random	O
choices	O
.	O
</s>
<s>
An	O
integration	O
of	O
search	B-Application
with	O
local	B-Algorithm
search	I-Algorithm
has	O
been	O
developed	O
,	O
leading	O
to	O
hybrid	B-Application
algorithms	I-Application
.	O
</s>
<s>
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
handles	O
the	O
case	O
when	O
all	O
the	O
available	O
relations	O
are	O
Boolean	O
operators	O
,	O
that	O
is	O
,	O
for	O
domain	O
size	O
2	O
.	O
</s>
<s>
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
was	O
recently	O
generalized	O
to	O
a	O
larger	O
class	O
of	O
relations	O
.	O
</s>
<s>
Most	O
classes	O
of	O
CSPs	O
that	O
are	O
known	O
to	O
be	O
tractable	O
are	O
those	O
where	O
the	O
hypergraph	O
of	O
constraints	B-Application
has	O
bounded	O
treewidth	O
(	O
and	O
there	O
are	O
no	O
restrictions	O
on	O
the	O
set	O
of	O
constraint	B-Application
relations	O
)	O
,	O
or	O
where	O
the	O
constraints	B-Application
have	O
arbitrary	O
form	O
but	O
there	O
exist	O
essentially	O
non-unary	O
polymorphisms	O
of	O
the	O
set	O
of	O
constraint	B-Application
relations	O
.	O
</s>
<s>
The	O
classic	O
model	O
of	O
Constraint	B-Application
Satisfaction	I-Application
Problem	I-Application
defines	O
a	O
model	O
of	O
static	O
,	O
inflexible	O
constraints	B-Application
.	O
</s>
<s>
Dynamic	O
CSPs	O
(	O
DCSPs	O
)	O
are	O
useful	O
when	O
the	O
original	O
formulation	O
of	O
a	O
problem	O
is	O
altered	O
in	O
some	O
way	O
,	O
typically	O
because	O
the	O
set	O
of	O
constraints	B-Application
to	O
consider	O
evolves	O
because	O
of	O
the	O
environment	O
.	O
</s>
<s>
DCSPs	O
are	O
viewed	O
as	O
a	O
sequence	O
of	O
static	O
CSPs	O
,	O
each	O
one	O
a	O
transformation	O
of	O
the	O
previous	O
one	O
in	O
which	O
variables	O
and	O
constraints	B-Application
can	O
be	O
added	O
(	O
restriction	O
)	O
or	O
removed	O
(	O
relaxation	O
)	O
.	O
</s>
<s>
Oracles	O
:	O
the	O
solution	O
found	O
to	O
previous	O
CSPs	O
in	O
the	O
sequence	O
are	O
used	O
as	O
heuristics	B-Algorithm
to	O
guide	O
the	O
resolution	O
of	O
the	O
current	O
CSP	O
from	O
scratch	O
.	O
</s>
<s>
Local	O
repair	O
:	O
each	O
CSP	O
is	O
calculated	O
starting	O
from	O
the	O
partial	O
solution	O
of	O
the	O
previous	O
one	O
and	O
repairing	O
the	O
inconsistent	O
constraints	B-Application
with	O
local	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Constraint	B-Application
recording	O
:	O
new	O
constraints	B-Application
are	O
defined	O
in	O
each	O
stage	O
of	O
the	O
search	B-Application
to	O
represent	O
the	O
learning	O
of	O
inconsistent	O
group	O
of	O
decisions	O
.	O
</s>
<s>
Those	O
constraints	B-Application
are	O
carried	O
over	O
to	O
the	O
new	O
CSP	O
problems	O
.	O
</s>
<s>
Classic	O
CSPs	O
treat	O
constraints	B-Application
as	O
hard	O
,	O
meaning	O
that	O
they	O
are	O
imperative	O
(	O
each	O
solution	O
must	O
satisfy	O
all	O
of	O
them	O
)	O
and	O
inflexible	O
(	O
in	O
the	O
sense	O
that	O
they	O
must	O
be	O
completely	O
satisfied	O
or	O
else	O
they	O
are	O
completely	O
violated	O
)	O
.	O
</s>
<s>
Flexible	O
CSPs	O
relax	O
those	O
assumptions	O
,	O
partially	O
relaxing	O
the	O
constraints	B-Application
and	O
allowing	O
the	O
solution	O
to	O
not	O
comply	O
with	O
all	O
of	O
them	O
.	O
</s>
<s>
This	O
is	O
similar	O
to	O
preferences	O
in	O
preference-based	B-Application
planning	I-Application
.	O
</s>
<s>
MAX-CSP	O
,	O
where	O
a	O
number	O
of	O
constraints	B-Application
are	O
allowed	O
to	O
be	O
violated	O
,	O
and	O
the	O
quality	O
of	O
a	O
solution	O
is	O
measured	O
by	O
the	O
number	O
of	O
satisfied	O
constraints	B-Application
.	O
</s>
<s>
Weighted	B-Application
CSP	I-Application
,	O
a	O
MAX-CSP	O
in	O
which	O
each	O
violation	O
of	O
a	O
constraint	B-Application
is	O
weighted	O
according	O
to	O
a	O
predefined	O
preference	O
.	O
</s>
<s>
Thus	O
satisfying	O
constraint	B-Application
with	O
more	O
weight	O
is	O
preferred	O
.	O
</s>
<s>
Fuzzy	O
CSP	O
model	O
constraints	B-Application
as	O
fuzzy	O
relations	O
in	O
which	O
the	O
satisfaction	O
of	O
a	O
constraint	B-Application
is	O
a	O
continuous	O
function	O
of	O
its	O
variables	O
 '	O
values	O
,	O
going	O
from	O
fully	O
satisfied	O
to	O
fully	O
violated	O
.	O
</s>
<s>
In	O
DCSPs	O
each	O
constraint	B-Application
variable	O
is	O
thought	O
of	O
as	O
having	O
a	O
separate	O
geographic	O
location	O
.	O
</s>
<s>
Strong	O
constraints	B-Application
are	O
placed	O
on	O
information	O
exchange	O
between	O
variables	O
,	O
requiring	O
the	O
use	O
of	O
fully	O
distributed	O
algorithms	O
to	O
solve	O
the	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
.	O
</s>
