Constraint Satisfaction Problems
Basic Algorithms
My Thanks to Roman Bartak
(for “stealing” some of his slides)
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
1
Search Algorithms for CSPs

We will study variations of DFS especially for CSPs.

These algorithms are based on backtracking search
Simple or Chronological Backtracking (BT)
 Backjumping (BJ) and Conflict-Based Backjumping
 Forward Checking (FC)
 Maintaining Arc Consistency (MAC)

Also two variations of hill climbing
 Min-conflicts
 Min-conflicts with Random Walk
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
2
Intelligent Backtracking

ΒΤ suffers from thrashing


One way to get rid of the problem is using intelligent backtracking
algorithms


it visits again and again the same regions of the search tree because it
has a very local view of the problem
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
3
BJ vs. BT

We want to color each area in the map with a different color

We have three colors
red, green, blue
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
4
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!
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
5
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
6
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

ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
7
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 Dx all the values that are
not consistent with v
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
8
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
9
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 Dx; delete v from Dx
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 Dx = Ø
return NIL
end
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
10
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 Dy’ do
if (y,v) is incompatible with compound_label with respect
to the constraints between y and the variables of
compound_label
then Dy’  Dy’ – {v}
end
end
return doms’
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
11
FC in operation
NSW
V
SA
T
RGB
RGB
RGB
RGB
RGB
GB
RGB
RGB
RGB
GB
RGB
R
B
G
RB
RGB
B
RGB
R
B
G
R
B
WA
NT
RGB
RGB
after WA=R
R
after Q=G
after V=B
initial domains
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
Q
RGB
12
Consistency Techniques






removing inconsistent values from variables’ domains
graph representation of the CSP
 binary and unary constraints only (relatively easy)
 nodes = variables
A>5
 edges = constraints
node consistency (NC)
A
A<C
arc consistency (AC)
AB
path consistency (PC)
C
(strong) k-consistency
B
B=C
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
13
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>5

A
A<C
AB
If D(A)={0,…,9} node consistency
will remove values 0,…,5
C
B
B=C
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
14
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
15
Arc Consistency (AC)

the most widely used consistency technique (good
simplification/performance ratio)

deals with individual binary constraints
a
a
a
b
b
b
c
c
c
X


Y
Z
repeated revisions of arcs
Directional (one pass) AC
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
16
AC - Example

Problem:
X::{1,2}, Y::{1,2}, Z::{1,2}
X = Y, X  Z, Y > Z
1
1
Y
2
X
X
1 2
1 2
2
Y
1
1 2
2
Z
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
Z
17
Arc Consistency propagation:
Crossword Puzzle example
1
2
3
4
5
….No more changes!
X1
X2
X4
astar
live
load
peal
peel
save
talk
live
load
peal
peel
save
talk
happy
hello
hoses
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
18
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)

ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
19
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 Dx; delete v from Dx
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 Dx = Ø
return NIL
end
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
20
Algorithms for Arc Consistency

Arc consistency can be enforced with Ο(ed2) 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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
21
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
22
Achieving Arc Consistency

Runtime of AC-3: O(ed3) for graph e binary constraints, and maximum
domain size of d

For one constraint, function Revise costs O(d2) and it can be called d
times
 there are e constraints, so the complexity is O(ed3)

AC-2001/3.1 achieves the optimal Ο(ed2) complexity by using a set
of pointers Lastx,a,y

For each value a of a variable, Lastx,a,y points to the most recently
discovered value in the domain of y that supports a
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> Lastx,a,y and (a,b) is consistent
then delete a from the domain of x
else Lastx,a,y = first such value
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
23
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

ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
24
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
25
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
26
Is AC enough?



empty domain => no solution
cardinality of all domains is 1 => solution
Problem:
X
X::{1,2}, Y::{1,2}, Z::{1,2}
X  Y, X  Z, Y  Z
1
1
2
2
Y
Z
1
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
2
27
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
28
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
look ahead
Labelling order
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
29
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
conflict
b
b
b
still conflict
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
30
Look Ahead Methods


preventing future conflicts via consistency checks among not yet
instantiated variables
forward checking (FC)

partial look ahead (PLA)


AC to direct neighbourhood
DAC
(full) look ahead (LA)


Arc Consistency
Path Consistency
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
labelling order

instantiated
variable
31
Look Ahead - Example

Problem:
X::{1,2}, Y::{1,2}, Z::{1,2}
X = Y, X  Z, Y > Z
X
1
Y
Z
{1}
{}
{2}
{1}
2
action
labelling
propagation
labelling
propagation
result
fail
solution
generate & test - 7 steps
backtracking - 5 steps
propagation - 2 steps
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
32
4-queen problem
Q1
1
2
3
4
Q2
Q3
Q4
Place 4 queens so that no two queens are
in attack.
Qi: line number of queen in column i, for 1i4
Q1, Q2, Q3, Q4 
Q1Q2, Q1Q3, Q1Q4,
Q2Q3, Q2Q4,
Q3Q4,
Q1Q2-1, Q1Q2+1, Q1Q3-2, Q1Q3+2,
Q1Q4-3, Q1Q4+3,
Q2Q3-1, Q2Q3+1, Q2Q4-2, Q2Q4+2,
Q3Q4-1, Q3Q4+1
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
33
4-queen problem first solution
Q1
Q2
Q3
Q4
1
2
3
4
There is a total of 256 valuations
GT algorithm will generate
64 valuations with Q1=1;
+
+
=
48 valuations with Q1=2, 1Q23;
3 valuations with Q1=2, Q2=4, Q3=1;
115 valuations to find first solution
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
34
4-queen problem, BT algorithm
Q1
Q2
Q3
Q4
1
2
3
4
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
1
2
3
4
1
2
3
4
1
2
3
4
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
Q1
Q2
Q3
Q4
1
2
3
4
35
4-queen problem, FC algorithm
Q1
Q2
Q3
Q4
1
2
3
4
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
1
2
3
4
1
2
3
4
Q1
Q2
Q3
Q4
1
2
3
4
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
1
2
3
4
Q1
1
2
3
4
Q1
Q2
Q3
Q4
1
2
3
4
36
4-queen problem, MAC algorithm
Q1
1
2
3
4
Q2
Q3
Q4
Value 3 of Q2 is unsupported in Q3,
Value 4 of Q3 is unsupported in Q2,
Value 2 of Q3 is unsupported in Q4,
x
x
x
Q1
Q2
Q3
Q4
1
2
3
4
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
Q1
1
2
3
4
Q2
Q3
Q4
Q1
Q2
Q3
Q4
1
2
3
4
37
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
 …

ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
38
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, FCCBJ 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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
39
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
40
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 ?
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
We always need
experiments!!!
41
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
42
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
43
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?
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1
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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.

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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

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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
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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
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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

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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
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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
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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

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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”)
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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”)
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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

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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”)
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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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