1. G51IAI Introduction to Artificial Intelligence
Dr Rong Qu
Constraint Satisfaction Problems

2. 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
everywhere in our life start from a very simple example.
probably the mostly used example in introducing CSPs. problem is very simple, but will lead to some important concepts.
G51IAI – Introduction to AI

3. An Example Map coloring
Can we use 3 colors to color the map below (Australia)?
rather simple problem - can easily find out a solution in seconds.
a map of 100 regions - won’t be that easy to find a coloring satisfying the constraint.
G51IAI – Introduction to AI

4. An Example Map coloring The above map can be modelled as a graphWA Q SA
NSW V T NT
Node: sub-region
Edge: constraint (adjacency)
Problem modelled as graph coloring: adjacent nodes have different colors
to solve bigger and more complex map coloring problem - need to firstly model the problem.
G51IAI – Introduction to AI
Relationship between problem elements  constraints

5. 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
standard definition of CSP - all the examples we introduce later can be defined using this standard form.
G51IAI – Introduction to AI

6. 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

7. 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

8. 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
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9. 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

10. CSP - Constraints Types of constraints Binary CSP
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

11. 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

12. 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

13. 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

14. 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

15. 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

16. CSP – Example Variables Domains Constraints {WA, NT, Q, SA, NSW, V, T}
{red, green, blue}
Constraints
{not_same(WA,NT), not_same(WA, SA), not_same(NT,SA), …}
G51IAI – Introduction to AI

17. CSP – Example WA Q SA
NSW V T NT
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

18. CSP – Example WA = red is a label The problem is satisfiable.
G51IAI – Introduction to AI

19. 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

20. 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

21. CSP So why is graph coloring important? CSP has many uses
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

22. CSP – Search Tree
Depth first search ?
G51IAI – Introduction to AI

23. CSP – Search Tree
Backtracking ?
G51IAI – Introduction to AI

24. CSP Techniques Constraint propagation Search strategies
Consistency enforcing
Arc consistency, path consistency
Search strategies
Back and forward checking
Variable and Value ordering
Branch & bound (B&B)
Constraint optimisation techniques
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25. 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
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26. CSP – Example II Scene labeling problem Types of lines Convex edges +
The edge is moving away from the viewer
Concave edges –
The edge moves towards the viewer
Occluding edges 
Face at right can be seen by the viewer + -
Convex: moving away from the viewer
Concave: moving towards the viewer
area on the right: face of the object which can be seen as the viewer
G51IAI – Introduction to AI

27. CSP – Example II Scene labeling problem defined as CSP Variables
Domains G H I B C A D F E
Convex: moving away from the viewer
Concave: moving towards the viewer
area on the right: face of the object which can be seen as the viewer
+ -  
G51IAI – Introduction to AI

28. CSP – Example II Scene labeling problem defined as CSP Constraints
G51IAI – Introduction to AI

29. CSP – Example III Sudoku Variables Domain Constraints 6 5 2 3 9 7 1 48
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30. CSP – Example IV N-Queen problem Variables Domain Constraints
G51IAI – Introduction to AI