Constraint Programming

1
Constraint Programming

• Peter van Beek
• University of Waterloo

2
Applications

• Reasoning tasks
• abductive, diagnostic, temporal, spatial
• Cognitive tasks
• machine vision, natural language processing
• Combinatorial tasks
• scheduling, sequencing, planning

3
Constraint Programming

• CP solve problems by specifying
  
  
  
  constraints on acceptable solutions
• Why CP?
• constraints often a natural part of problems
• once problem is modeled using constraints, wide
  selection of solution techniques available

4
Constraint-based problem solving

• Model problem
• specify in terms of constraints on acceptable
  solutions
• define variables (denotations) and domains
• define constraints in some language
• Solve model
• define search space / choose algorithm
• incremental assignment / backtracking search
• complete assignments / stochastic search
• design/choose heuristics
• Verify and analyze solution

5
Constraint-based problem solving

• Model problem
• specify in terms of constraints on acceptable
  solutions
• define variables (denotations) and domains
• define constraints in some language
• Solve model
• define search space / choose algorithm
• incremental assignment / backtracking search
• complete assignments / stochastic search
• design/choose heuristics
• Verify and analyze solution

Constraint Satisfaction Problem
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Constraint satisfaction problem

• A CSP is defined by
• a set of variables
• a domain of values for each variable
• a set of constraints between variables
• A solution is
• an assignment of a value to each variable that
  satisfies the constraints

7
Constraint-based problem solving

• Model problem
• specify in terms of constraints on acceptable
  solutions
• define variables (denotations) and domains
• define constraints in some language
• Solve model
• define search space / choose algorithm
• incremental assignment / backtracking search
• complete assignments / stochastic search
• design/choose heuristics
• Verify and analyze solution

8
Example Assembly line sequencing

• What order should the cars be manufactured?
• Constraints
• even distributions
• changes in colors
• run length constraints

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Example Scheduling
Four students, Algy, Bertie, Charlie, and Digby
share a flat. Four newspapers are delivered. Each
student reads the newspapers in a particular
order and for a specified amount of
time. Algy arises at 830, Bertie and Charlie at
845, Digby at 930.
What is the earliest that they can all set of for
school?
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Schedule
Sun
Express
Guardian
FT
8 am
9
10
11
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Example Graph coloring
Given k colors, does there exist a coloring of
the nodes such that adjacent nodes are assigned
different colors
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Example 3-coloring
Variables v1, v2 , v3 , v4 , v5 Domains
1, 2, 3 Constraints vi ? vj if vi and vj
are adjacent
v2
v1
v3
v4
v5
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Example 3-coloring
One solution v1 ? 1 v2 ? 2 v3 ? 2 v4 ? 1 v5 ? 3
v2
v1
v3
v4
v5
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Example Boolean satisfiability
Given a Boolean formula, does there exist a
satisfying assignment (an assignment of true or
false to each variable such that the formula
evaluates to true)
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Example 3-SAT
Variables x1, x2 , x3 , x4 , x5 Domains
True, False Constraints (x1 ? x2 ? x4),
(x2 ? x4 ? ?x5), (x3 ? ?x4 ? ?x5)
(x1 ? x2 ? x4) ? (x2 ? x4 ? ?x5) ? (x3 ? ?x4 ?
?x5)
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Example 3-SAT
One solution x1 ? False x2 ? False x3 ? False x4
? True x5 ? False
(x1 ? x2 ? x4) ? (x2 ? x4 ? ?x5) ? (x3 ? ?x4 ?
?x5)
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Example n-queens
Place n-queens on an n ? n board so that no pair
of queens attacks each other
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Example 4-queens
Variables x1, x2 , x3 , x4 Domains 1, 2,
3, 4 Constraints xi ? xj and xi - xj
? i - j
x1
x2
x3
x4
1
2
3
4
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Example 4-queens
x1
x2
x3
x4
One solution x1 ? 2 x2 ? 4 x3 ? 1 x4 ? 3
1
Q
2
Q
3
Q
4
Q
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Search tree for 4-queens
x1
1
2
3
4
x2
x3
x4
(1,1,1,1)
(4,4,4,4)
(2,4,1,3)
(3,1,4,2)
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Specification of forward checking
Invariant
1 ? i ? c,
c ? j ? n,
(xj, xi) is arc-consistent
x1
xc-1
xc
xc1
xn
current
past
future
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Forward checking
4
3
2
1
5
6
x1
Q
1,4,6
x4
x2
Q
1
1
1
x3
Q
1
2
2
1
2
5
3
1,3,4
x4
1
1
2
3
x5
x1
x3
x2
x5
1
1
2
3
3
3,4,6
x6
1
2
2
3
3
x6
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Enforcing arc-consistency
lt
a, b, c
a, b, c
xj
xi
24
Enforcing path-consistency
lt
lt
lt
25
Search graph for 4-queens
(1,1,1,2)
(1,1,4,2)
4
(1,1,1,1)
1
6
(1,1,1,3)
3
(3,1,4,2)
0
(1,1,1,4)
(1,4,1,3)
4
1
(4,1,4,2)
1
(2,4,1,3)
0
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Stochastic search
Initial assignment
x1
Q
Q
x2
Q
x3
Q
x4
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Tractability
NP
NP-Complete
(SAT, TSP, ILP, CSP, )
P
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Reducibility
NP-Complete
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Options

• CSP, binary CSP, SAT, 3-SAT, ILP, ...
• Model and solve in one of these languages
• Model in one language, translate into another to
  solve