G51IAI
Introduction to Artificial
Intelligence
Dr Rong Qu
Constraint Satisfaction Problems
An Example


Constraint satisfaction problems are
everywhere in our everyday life
Map coloring



A region in a map has several sub-regions
Given a set of colors
Assign each sub-region a color so that no
adjacent sub-regions have the same color
G51IAI – Introduction to AI
2 of 31
An Example

Map coloring

Can we use 3 colors to color the map below
(Australia)?
NT
Q
WA
SA
NSW
V
T
G51IAI – Introduction to AI
3 of 31
An Example

Map coloring

The above map can be modelled as a graph
WA
NT
Q
Node: sub-region
WA
 Edge: constraint (adjacency)
SA
 Problem modelled NSW
as graph
coloring: adjacent V
nodes
have different colors
NT
Q

SA
NSW
V
T
G51IAI
– Introduction
to AI
Relationship
between
problem elements  constraints
T
4 of 31
CSP – Definition

A constraint satisfaction problem (CSP)
consists of




a set of variables {x1, x2, … xi};
a finite set of domain D (possible values)
associated with each variable;
a set of constraints C restricting the values that
the variables can simultaneously take
CSP is to assign values to variables so that all
constraints are satisfied
G51IAI – Introduction to AI
5 of 31
CSP - Domain

The domain of a variable is a set of possible
values that can be assigned to the variable




{1 … 500} – finite, integer
{red, green, blue} – symbols
{yes, no} – Boolean
Temperatures – infinite; continuous
G51IAI – Introduction to AI
6 of 31
CSP - Constraints

Examples of constraints




The demand will be more than five thousands
units in August
John prefer to work at only weekends
Schedule these employees to cover all shifts
Optimise the benefit of products
G51IAI – Introduction to AI
7 of 31
CSP - Constraints

Constraint is a set of relations upon domains
of variables


Restriction on the values the variables can
simultaneously take
What values are (dis-)allowed among the variables





Functions
Matrices
Inequalities
x1 ≠ x2 or x1 < x2
3x + 6y = 0, y + x < 7
G51IAI – Introduction to AI
8 of 31
CSP - Constraints
Formal definition

A constraint Ci,j,k… is a subset of possible
relations between values of variables xi, xj, xk,
...; Ci,j,k…is a subset of Di × Dj × Dk × …
G51IAI – Introduction to AI
9 of 31
CSP - Constraints

Types of constraints




Unary: affect only one variable
 x >= 1, WA = green
Binary: affect two variables
 x >= y, WA ≠ NT
High-order: affect more than two variables
 alldifferent(WA, NT, Q)
 x = y = z
Binary CSP

With only unary and binary constraints
G51IAI – Introduction to AI
10 of 31
CSP - Constraints

Simplification of constraints

Replace a constraint by an equivalent constraint
which has a simpler form
Problem is easier to
understand
Information of
original form is more
apparent
G51IAI – Introduction to AI
11 of 31
CSP - Solutions

A solution to a CSP is an assignment of each
variable with a value (within its domain), such
that all constraints C are satisfied



if a CSP has a feasible solution
find one (any) solution
all solutions
G51IAI – Introduction to AI
12 of 31
CSP - Solutions

Often we not only want a CSP solution but
also the best one (optimal solution)



Optimisation problems
Objective function: to measure how good a
solution is
Constraint Optimisation Problems (COP)
G51IAI – Introduction to AI
13 of 31
CSP – Other Definitions

Label


a variable-value pair representing assigning the
value to the variable
Compound label

the finite set of labels representing simultaneous
assignments
G51IAI – Introduction to AI
14 of 31
CSP – Other Definitions

Satisfiable


a CSP (V,D,C) is satisfiable iff there is a compound
label assigning values to all variables V that satisfy
all constraints C
A solution can then be defined as

A compound label for all variables which satisfy all
the constraints
G51IAI – Introduction to AI
15 of 31
CSP – Example

Variables


Domains


{WA, NT, Q, SA, NSW, V, T}
WA
NT
SA
Q
NSW
V
{red, green, blue}
T
Constraints

{not_same(WA,NT), not_same(WA, SA),
not_same(NT,SA), …}
G51IAI – Introduction to AI
16 of 31
CSP – Example
WA
NT
SA
Q
NSW
V
T
The above formulate the map coloring problem into a
constraint satisfaction problem
 Solution
 {WA=red, NT=green, SA=blue, Q=red,
NSW=green, V=red, T=red}
G51IAI – Introduction to AI
17 of 31
CSP – Example


WA = red is a label
The problem is satisfiable.
G51IAI – Introduction to AI
18 of 31
CSP – Example


The above solution using 3 colors
 Is actually an optimal solution (using the least
colors)
There are more than one solution
 For example, given 7 colors, we’ll have 7!
solutions for the problem (at least)
G51IAI – Introduction to AI
19 of 31
CSP – Example

Graph coloring is NP-Hard (Karp, 1972)


Optimal solutions of least colors cannot be
obtained for 90-vertice graph (Johnson, 1991)
Constraint based techniques are the mostly
studied early approaches
G51IAI – Introduction to AI
20 of 31
CSP

So why is graph coloring important?



A typical constraint satisfaction problem
Can be used to model many real problems
CSP has many uses




Scheduling
Timetabling
Other optimisation problems in industrial
…
G51IAI – Introduction to AI
21 of 31
CSP – Search Tree

Depth first search
?
G51IAI – Introduction to AI
22 of 31
CSP – Search Tree

Backtracking
?
G51IAI – Introduction to AI
23 of 31
CSP Techniques

Constraint propagation


Consistency enforcing
 Arc consistency, path consistency
Search strategies



Back and forward checking
Variable and Value ordering
Branch & bound (B&B)
 Constraint optimisation techniques
G51IAI – Introduction to AI
24 of 31
CSP – Example II

Scene labeling problem



The first CSP studied
In vision, scene are captured as images, and
processed as lines to represent images
Lines need to be interpreted into types
G51IAI – Introduction to AI
25 of 31
CSP – Example II

Scene labeling problem

Types of lines
 Convex edges +


Concave edges –


The edge is moving away from the viewer
The edge moves towards the viewer
Occluding edges 

+
+
Face at right can be seen
by the viewer
+
-
G51IAI – Introduction to AI
26 of 31
CSP – Example II

Scene labeling problem defined as CSP


Variables
Domains
E
F
I
G
D
+-
A
H
C
B
G51IAI – Introduction to AI
27 of 31
CSP – Example II

Scene labeling problem defined as CSP

Constraints
G51IAI – Introduction to AI
28 of 31
CSP – Example III

Sudoku



Variables
Domain
Constraints
6
5
7
3
9
6
1
6
3
4
4
7
1
5
9
8
4
1
3
2
G51IAI – Introduction to AI
2
6
8
4
5
29 of 31
CSP – Example IV

N-Queen problem



Variables
Domain
Constraints
G51IAI – Introduction to AI
30 of 31