1
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Constraint Satisfaction Problems
Basic Algorithms
My Thanks to
Roman Bartak
(for “stealing” some of his slides)
2
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Search Algorithms for
CSPs
Simple or Chronological Backtracking (BT)
Backjumping (BJ) and Conflict

Based Backjumping
Forward Checking (FC)
Maintaining Arc Consistency (MAC)
We will study variations of
DFS especially for
CSPs.
These algorithms are based on backtracking search
Also two variations of hill climbing
Min

conflicts
Min

conflicts with Random Walk
3
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Intelligent Backtracking
ΒΤ
suffers from
thrashing
it visits again and again the same regions of the search tree because
it
has a very local view of the problem
One way to get rid of the problem is using
intelligent backtracking
algorithms
BJ, CBJ, DB, Graph

based BJ, Learning
Backjumping
(
BJ
) is different from
ΒΤ
in the following
:
When BJ reaches a dead

end it does not backtrack to the immediately
preceding variables
.
It backtracks to the deepest variable in the search
tree which is in conflict with the current variable
4
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
BJ vs. BT
We want to color each area in the map with a different color
We have three colors
red
,
green
,
blue
5
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
BJ vs. BT
Let’s consider what
ΒΤ
does in the map coloring problem
Assume that variables are assigned in the order
Q
,
NSW
,
V
,
T
,
SA
,
WA
,
NT
Assume that we have reached the partial assignment
Q
=
red
,
NSW
=
green
,
V
=
blue
,
T
=
red
When we try to give a value to the next variable
SA
, we find out that all
possible values violate constraints
Dead end
!
BT will backtrack to try a new value for variable
Τ
!
Not a good idea!
6
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
BJ vs. BT
BJ has a smarter approach to backtracking
It tells us to go back to one of the variables which are responsible for the
dead

end
The set of these variables is called a
conflict set
The conflict set for
SA is
{
Q, NSW, V}
BJ backjumps to the deepest variable in the
conflict set of the variable
where the dead

end occurred
deepest
=
the one we visited most recently
CBJ
,
DB
,
Graph

based BJ
,
Learning, Backmarking
7
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Conflict

based Backjumping (CBJ)
Conflict

based Backjumping
is a
look

back algorithm that performs
intelligent backtracking at dead

ends
In contrast to
BJ which
backjumps only from leaf dead

ends
,
CBJ can
also backjump from dead

ends at inner nodes
for each variable
x
we have a conflict set
when an assignment
(
x,a
) fails because of a constraint violation with a
previous variable
y
,
y
is added to the conflict set of
x
if there are no values left in the domain of the current variable
x
,
CBJ
backjumps to the deepest variable
w
in the conflict set of
x
(as BJ)
and the conflict set of
x
is added to the conflict set of
w
then a further
backjump can occur from
w
8
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Forward Checking
Forward Checking
(FC) belongs to the family of backtracking
algorithms called
lookahead
algorithms
The basic idea of
lookahead is that when you assign a value to a variable
the problem is reduced through
constraint propagation
constraint propagation is defined in a different way for each look

ahead
algorithm
FC does the following
:
When a variable
x
takes a value
v
,
for each future variabe
y
which
appears in a constraint with
x
we remove from
D
x
all the values that are
not consistent with
v
9
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Forward Checking
If the domain of some variable becomes empty then
value
v
is rejected
and we try the next value of
x
The operation of
FC means that the following holds for each step of
the search
:
All values of each
future variable
are compatible with all the values that
have been assigned to past variables
FC maintains a restricted form of
arc consistency
10
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Forward Checking
procedure
FORWARD_CHECKING (
vars
,
doms
,
cons
)
solution
FC (
vars
,
Ø
,
doms
,
cons
)
function
FC (
unlabelled,compound_label,doms,cons
)
returns
a solution or NIL
if
unlabelled
=
Ø
then return
compound_label
else
pick a variable
x
from
unlabelled
repeat
pick a value
v
from
D
x
;
delete
v
from
D
x
doms’
UPDATE(
unlabelled

{x},doms,cons,
compound_label
+ {(
x,v
)})
if
no domain in
doms’
is empty
then
result
FC(
unlabelled

{
x
},
compound_label
+
{(
x,v
)},
doms’
,
cons
)
if
result
NIL
then return
result
end
until
D
x
=
Ø
return
NIL
end
11
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Forward Checking
function
UPDATE (
unlab_vars,doms,cons,compound_label
)
returns
an updated set of domains
for
each variable
y
in
unlab_vars
do
for
each value
v
in
D
y
’
do
if
(
y
,
v
) is incompatible with
compound_label
with respect
to the constraints between
y
and the variables of
compound_label
then
D
y
’
D
y
’
–
{
v
}
end
end
return
doms
’
12
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
FC in operation
R
G
B
R
G
B
R
G
B
R
G
B
R
G
B
R
G
B
R
G
B
R
G
B
R
G
B
R
G
B
R
G
B
G
B
R
G
B
R
B
G
R
B
R
G
B
B
R
G
B
R
B
G
R
B
R
G
B
WA
NT
Q
NSW
V
T
SA
initial domains
after WA=
R
after Q=
G
after V=
B
13
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Consistency Techniques
removing inconsistent values from variables’ domains
graph representation of the CSP
binary and unary constraints only (relatively easy)
nodes = variables
edges = constraints
node consistency (NC)
arc consistency (AC)
path consistency (PC)
(strong) k

consistency
A
B
C
A>5
A
B
A
<C
B=C
14
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Node Consistency
A variable
X
is
node consistent
iff each value a of
X
satisfies all
the unary constraints on
X
Node consistency can be applied as a preprocessing step before
starting search to remove all the node inconsistent values
A
B
C
A>5
A
B
A
<C
B=C
If D(A)={0,…,9} node consistency
will remove values 0,…,5
15
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Arc Consistency
Definition
:
A variable
X
is
arc consistent
iff for each other variable
Y
the
following holds
:
For each value
a
of
Χ
there is at least one value
b
of
Υ
such that
a
and
b
are compatible
Then we say that
a
supports
b
An algorithm that applies
arc consistency deletes values from
the domain
of a variable when they are not supported by any
value in the domain of another variable
16
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Arc Consistency (AC)
the most widely used consistency technique
(good
simplification/performance ratio)
deals with individual binary constraints
repeated revisions of arcs
Directional (one pass) AC
a
b
c
a
b
c
X
Y
a
b
c
Z
17
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
AC

Example
Problem:
X::{1,2}, Y::{1,2}, Z::{1,2}
X = Y,
X
Z, Y > Z
1
2
1
2
1
2
1 2
1 2
1 2
X
Y
Z
X
Y
Z
18
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Arc Consistency propagation:
Crossword Puzzle example
1
2
3
4
5
X1
X2
X4
astar
happy
hello
hoses
live
load
peal
peel
save
talk
live
load
peal
peel
save
talk
….No more changes!
19
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Arc Consistency
We apply arc consistency
:
As a
(
preprocessing
) step before we start search
in that way we can reduce the size of the search tree
and in some cases discover inconsistent problems
While searching after an assignment of a value to a variable
constraint propagation
fast discovery of dead ends
The search algorithm which applies
arc consistency is called
MAC
(maintaining arc consistency)
20
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
MAC
procedure
Maintaining Arc Consistency (
vars
,
doms
,
cons
)
solution
MAC (
vars
,
Ø
,
doms
,
cons
)
function
MAC (
unlabelled,compound_label,doms,cons
)
returns
a solution or NIL
if
unlabelled
=
Ø
then return
compound_label
else
pick a variable
x
from
unlabelled
repeat
pick a value
v
from
D
x
;
delete
v
from
D
x
doms’
AC(
unlabelled

{x},doms,cons,
compound_label
+ {(
x,v
)})
if
no domain in
doms’
is empty
then
result
MAC(
unlabelled

{
x
},
compound_label
+
{(
x,v
)},
doms’
,
cons
)
if
result
NIL
then return
result
end
until
D
x
=
Ø
return
NIL
end
21
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Algorithms for
Arc Consistency
Arc consistency can be enforced with
Ο(
ed
2
)
optimal worst case time
complexity
AC

4, AC

6, AC

7,
AC

2001
AC

3
:
non

optimal,
but simple
AC algorithm
AC

3
and
AC

2001 use:
a queue (or stack) where the variables that are checked for arc
consistency are inserted
a routine
Revise which deletes values that are not supported
AC

4, AC

6, AC

7 use more complex data structures
support lists
22
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Achieving Arc Consistency
From Mackworth (1977a):
procedure
AC

3
(G)
Let Q be the set of (directed) arcs of G (not self

cyclic)
while
Q not empty
do
select and remove any arc (x,y) from Q;
REVISE(x,y)
if
REVISE(x,y) changed the domain of x
then
add
to Q
the set of all arcs of G
(z,x)
that go into x;
procedure
REVISE
(x,y)
for
each
value a
in domain of x
do
if
there is no
value b
in the domain of y such that (
a
,
b
) is consistent
then
delete
a
from the domain of x
23
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Achieving Arc Consistency
AC

2001/3.1
achieves
the optimal
Ο(
ed
2
)
complexity by using a set
of pointers Last
x,a,y
For each value
a
of a variable, Last
x,a,y
points to the most recently
discovered value in the domain of
y
that supports
a
Runtime
of
AC

3
: O(
e
d
3
) for graph
e
binary constraints, and
maximum
domain size of
d
For one constraint, function Revise costs O(
d
2
) and it can be called
d
times
there are
e
constraints, so the complexity is
O(
e
d
3
)
procedure
REVISE

2001/3.1
(x,y)
for
each
value a
in domain of x
do
if
there is no
value b
in the domain of y such that
b> Last
x,a,y
and
(
a
,
b
) is consistent
then
delete
a
from the domain of x
else
Last
x,a,y
= first such value
24
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Algorithms for Arc Consistency
In some cases we can exploit the semantics of certain binary
constraints to achieve an even better complexity
functional, anti

functional, monotonic, piecewise functional, etc.
algorithm AC

5
What the complexity of AC processing for a constraint of the
following types?
x = y
x
≠ y
x < y
x > y
25
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Directional Arc Consistency (DAC)
Observation:
AC has to repeat arc revisions; the total
number of
revisions depends on the number of arcs but
also on the size of
domains (while cycle)
Is it possible to weaken AC in such a way that every arc is
revised just
once?
Definition
:
A
CSP is
directional arc consistent
using a given
order of
variables iff every arc (i,j) such that i<j is arc
consistent
Again, every arc has to be revised, but revision in one
direction is
enough now
26
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Arc Consistency as a Solution Method
Question:
Are there cases where we can guarantee that solubility (or insolubility)
will be determined by applying arc consistency?
Answer
(Freuder 1982):
When the constraint graph of the problem is a tree
In this case, a solution can be found (if one exists) in a backtrack

free
manner by first applying directional arc consistency
A case of polynomially solvable CSPs
Many other such cases exist depending on the structure of the
constraint graph and the nature of the constraints
27
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Is AC enough?
empty domain => no solution
cardinality of all domains is 1 => solution
Problem:
X::{1,2}, Y::{1,2}, Z::{1,2}
X
Y,
X
Z, Y
Z
1
2
1
2
1
2
X
Y
Z
28
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Stronger Levels of
Consistency
Beyond
arc consistency there are numerous other levels of
consistency
path consistency
singleton arc consistency
neighborhood inverse consistency
…
These are stronger than
arc consistency
(
i.e. they delete more
inconsistent values when they are applied
)
But they are more expensive
(
higher time complexity
)
We will review some of them in the next lecture
29
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Constraint Propagation
systematic search only => no efficient
consistency only => no complete
combination of search (backtracking) with consistency techniques
methods:
look back (restoring from conflicts)
look ahead (preventing conflicts)
look back
Labelling order
look ahead
30
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Look Back Methods
intelligent backtracking
consistency checks among instantiated variables
backjumping
backtracks to the conflicting variable
backchecking and backmarking
avoids redundant constraint checking
by remembering conflicting level
for each value
jump
here
a
b
b
b
conflict
still conflict
31
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Look Ahead Methods
preventing future conflicts via consistency checks among not yet
instantiated variables
forward checking
(FC)
AC to direct neighbourhood
partial look ahead
(PLA)
DAC
(full) look ahead
(LA)
Arc Consistency
Path Consistency
labelling order
instantiated
variable
32
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Look Ahead

Example
Problem:
X::{1,2}, Y::{1,2}, Z::{1,2}
X = Y,
X
Z, Y > Z
generate & test

7 steps
backtracking

5 steps
propagation

2 steps
33
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
4

queen problem
Place 4 queens so that no two queens are
in attack.
1
2
3
4
Q
1
Q
2
Q
3
Q
4
Q
i
: line number of queen in column i, for 1
i
4
Q
1
, Q
2
, Q
3
, Q
4
Q
1
Q
2
, Q
1
Q
3
, Q
1
Q
4
,
Q
2
Q
3
, Q
2
Q
4
,
Q
3
Q
4
,
Q
1
Q
2

1, Q
1
Q
2
+1, Q
1
Q
3

2, Q
1
Q
3
+2,
Q
1
Q
4

3, Q
1
Q
4
+3,
Q
2
Q
3

1, Q
2
Q
3
+1, Q
2
Q
4

2, Q
2
Q
4
+2,
Q
3
Q
4

1, Q
3
Q
4
+1
34
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
4

queen problem first solution
1
2
3
4
Q
1
Q
2
Q
3
Q
4
There is a total of 256 valuations
GT algorithm will generate
64
valuations with Q
1
=1;
+
48
valuations with Q
1
=2, 1
Q
2
3;
+
3
valuations with Q
1
=2, Q
2
=4, Q
3
=1;
=
115
valuations to find first solution
35
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
4

queen problem, BT algorithm
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
36
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
4

queen problem, FC algorithm
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
37
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
4

queen problem, MAC algorithm
Value 3 of Q
2
is unsupported in Q
3
,
Value 4 of Q
3
is unsupported in Q
2
,
Value 2 of Q
3
is unsupported in Q
4
,
1
2
3
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
Q
1
Q
2
Q
3
Q
4
Q
1
Q
2
Q
3
Q
4
1
2
3
4
x
1
2
3
4
Q
1
Q
2
Q
3
Q
4
x
x
38
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Hybrid Algorithms
We can combine the operations of various backtracking algorithms to
design
hybrid algorithms
For example
we can combine the
lookahead function of
forward
checking and the
lookback function of
BJ
FC

BJ
FC

CBJ
MAC

BJ
MAC

CBJ
…
39
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
FC

CBJ
Forward Checking with Conflict

based Backjumping
FC

CBJ combines the look

ahead of FC and the intelligent backjumping
of CBJ
each variable is associated with a conflict set
when the forward checking of an assignment (
x,a
) results in a value
deletion from the domain of a variable
y
,
x
is added to the conflict set of
y
if after the forward checking of an assignment (
x,a
) the domain of a
variable
y
is wiped out, the variables in the conflict set of
y
are added to the
conflict set of
x
why is this done?
if there are no more values left in the domain of the current variable
x
, FC

CBJ backjumps to the deepest variable
w
in the conflict set of
x
the conflict set of
x
is added to the conflict set of
w
40
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Evaluation of Backtracking Algorithms
How can we compare backtracking algorithms for
CSPs ?
Time
/
Space Complexity
not very useful
.
They all have exponential time complexity
!
cpu times
number of nodes they visit in the search tree
amount of
consistency checks they perform
amount of backtracks they perform
41
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Evaluation of Backtracking Algorithms
Some theoretical results
:
Search tree nodes visited
FC

CBJ
FC

BJ
FC
BJ
BT
CBJ
BJ
Number of consistency checks
CBJ
BJ
ΒΤ
FC

CBJ
FC

BJ
FC
CPU times
?
We always need
experiments
!!!
42
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Heuristic Methods for
CSPs
Search algorithms must take decisions
:
1)
Which will be the next variable to assign
?
2)
Which value should I give it
?
3)
Which constraint should I check
?
The decisions that the algorithm takes at each step have a drastic
effect on the search space
(
and the efficiency of the algorithm
)
Especially decision
(1)
Heuristics help the algorithms take correct decisions
fail first principle
43
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Heuristic Methods for
CSPs
Variable ordering heuristics
static heuristics
MaxDegree, Bandwidth, …
dynamic heuristics
MRV, Brelaz, dom/deg, dom/wdeg…
Value ordering heuristics
Geelen’s promise, least

constraining…
Heuristics for constraint ordering
based on the cost of propagation
44
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Variable Ordering Heuristics
Minimum Width
The width of a variable
x
is the number of variables that are before
x
,
according to a given ordering, and are constrained with
x
The width of an ordering is the maximum width of all the variables
under that ordering
The width of a constraint graph is the minimum width of all possible
orderings
Variables are ordered in descending width
useful when the degree of the nodes varies significantly
Problem:
How many are the possible orderings?
45
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Variable Ordering Heuristics
Maximum Degree
Variables are ordered in decreasing order of their degree in the
constraint graph
degree is the number of adjacent variables in the graph
Heuristic to find a minimum width ordering
Maximum Cardinality
Selects the first variable arbitrarily
Then, at each stage, selects the variable that is adjacent to the largest set
of already selected variables.
46
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Variable Ordering Heuristics
Minimum Bandwidth
The bandwidth of a variable
x
, according to a given ordering, is the
maximum distance between
x
and any other variable which is adjacent to
x
The bandwidth of an ordering is the maximum bandwidth of all the
variables under that ordering
The bandwidth of a constraint graph is the minimum bandwidth of all
possible orderings
Idea:
The closer the variables involved in a constraint are placed to each
other the less backtracking will be required
Problem:
Computing the minimum bandwidth is NP

complete
47
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Dynamic Variable Ordering Heuristics
Minimum Remaining Values (MRV) or Smallest Domain (SD)
At each stage of search select the variable with the smallest domain size
How do we break ties?
Select a variable randomly
Select the variable with the highest degree in the original graph
Select the variable with the highest future degree (i.e. the one involved
in the maximum number of constraints with future variables). This is
called the
Brelaz
heuristic
Many variations have been proposed
dom/deg, dom/fdeg
48
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
State

of

the

art Dynamic Variable Ordering Heuristics
Weighted degree heuristics
each constraint is associated with a weight initially set to 1
each time a constraint
c
removes the last value from a domain (i.e.
causes a domain wipeout

DWO) its weight is incremented by 1
the
weighted degree
of a variable
x
is the sum of the weights of the
constraints that include x
wdeg heuristic
selects the variable with maximum weighted degree
dom/wdeg heuristic
selects the variable with minimum ration of domain size to weighted degree
What is the rationale behind these heuristics?
they use information gathered throughout search
–
not just from the
current search state like
dom/fdeg
49
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Value Ordering Heuristics
Min

Conflicts
Associate with each value
a
the total number of values in future
variables that are incompatible with
a
Select the value with lowest sum
Alternative:
Divide the number of incompatible values of future variable
x with the domain size of x
Geelen’s Promise
For each value
a
count the total number of values in each future variable
that are compatible with
a
Take the product of the counts. This is called the
promise
of value
a
Select the value with the maximum promise
50
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Constraint Ordering Heuristics
Is this issue important?
not very much when maintaining arc consistency
but there exist heuristics for ordering the constraints in the propagation
queue. Can you think of such a heuristic?
but very important in modern advanced solvers that use propagators
for the various (global) constraints
the idea here is to propagate the less expensive constraints first
51
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Stochastic and Local Search Methods
local search

chooses best neighbouring configuration
hill climbing
neighbourhood = value of one variable changed
min

conflicts
neighbourhood = value of selected conflicting variable
changed
can we avoid local optima?
restarts
if at a local optimum, start procedure from scratch
random

walk
sometimes picks neighbouring configuration randomly
tabu search
few last configurations are forbidden for next step
local search does not guarantee completeness
52
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
The Min

Conflicts Algorithm
Start with a random assignment of values to variables
or a seemingly good one according to some heuristic
some constraints will be violated
Try to repair it
change the value assignment that resolves the greatest numbers of
constraints
local optima can be escaped using
random restarts
Otherwise:
Simulated annealing
Tabu search
Random walk
53
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Min

Conflicts (version 1)
procedure
Min_Conflicts(
P
,
maxTries
,
maxChanges
)
for
i :=1 to
maxTries
do
A
:= initial complete assignment of the variables in
P
for
j:=1 to
maxChanges
do
if
A
satisfies
P
then
return (
A
)
else
x
:= randomly chosen variable whose assignment is in conflict
(x,a)
:= alternative assignment of
x
which satisfies
the maximum number of constraints under the current
assignment
A
if
by making assignment
(x,a)
you get a cost
≤ current cost
then
make the assignment
endif
endfor
endfor
return
(“No solution found”)
54
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Min

Conflicts (version 2)
procedure
Min_Conflicts(
P
,
maxTries
,
maxChanges
)
for
i :=1 to
maxTries
do
A
:= initial complete assignment of the variables in
P
for
j:=1 to
maxChanges
do
if
A
satisfies
P
then
return (
A
)
else
(x,a)
:= the alternative assignment of a variable
x
which minimizes
the number of constraint violations under the current
assignment
A
if
by making assignment
(x,a)
you get a cost
≤ current cost
then
make the assignment
else break
endif
endfor
endfor
return
(“No solution found”)
55
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Min

Conflicts with Random Walk
How
can we
leave the local optimum without a restart
(i.e. via a local step)?
By adding some “noise” to the algorithm!
Random walk
a state from the neighbourhood is selected randomly
(e.g., the value is
chosen randomly)
such technique can hardly find a solution
so it needs some guide
Random walk can be combined with the heuristic guiding
the
search via probability distribution:
p

probability of using the random walk
(1

p)

probability of using the heuristic guide
56
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Min

Conflicts with Random Walk (version 1)
procedure
Min_Conflicts(
P
,
maxChanges,p
)
A
:= initial complete assignment of the variables in
P
for
j:=1 to
maxChanges
do
if
A
satisfies
P
then
return (
A
)
else
if
probability
p
verified
x
:= randomly chosen variable whose assignment is in conflict
(x,a)
:= randomly chosen alternative assignment of
x
else
(x,a)
:= the alternative assignment of a variable
x
which minimizes
the number of constraint violations under the current
assignment
A
make the assignment
(x,a)
endif
endfor
return
(“No solution found”)
57
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ

Lecture 1
Min

Conflicts with Random Walk (version 2)
procedure
Min_Conflicts(
P
,
maxChanges,p
)
A
:= initial complete assignment of the variables in
P
for
j:=1 to
maxChanges
do
if
A
satisfies
P
then
return (
A
)
else
x
:= randomly chosen variable whose assignment is in conflict
if
probability
p
verified
(x,a)
:= randomly chosen alternative assignment of
x
else
(x,a)
:= the alternative assignment of
x
which satisfies
the maximum number of constraints under the current assignment
A
make the assignment
(x,a)
endif
endfor
return
(“No solution found”)
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