Constraint Processing and Programming Introductory Exemple

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Constraint Processing and ProgrammingIntroductory
Exemple

• Javier Larrosa

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Combinatorial Problem Solving

• Science and Engineering are full of
  Combinatorial Problems
• Solving algorithms are exponentially
  expensive
• A lot of tricks can be applied to improve
  performance
• Naive Approach
• For each problem, design and implement an
  efficient algorithm

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• Place numbers 1 through 8 on nodes
• each number appears exactly once
• no connected nodes have consecutive numbers

Note the graph symmetry
Acknowledgement Patrick Prosser
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Heuristic Search
Guess a value, but be prepared to backtrack
Which nodes are hardest to number?
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Heuristic Search
Which nodes are hardest to number?
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Heuristic Search
Which are the least constraining values to use?
Values 1 8
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Heuristic Search
Symmetry means we dont need to consider 8 1
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Inference/propagation
1,2,3,4,5,6,7,8
We can now eliminate many values for other nodes
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Inference/propagation
3,4,5,6
3,4,5,6
By symmetry
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Inference/propagation
3,4,5,6
1,2,3,4,5,6,7,8
3,4,5,6
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Inference/propagation
3,4,5,6
3,4,5,6
3,4,5,6
3,4,5,6
By symmetry
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Inference/propagation
3,4,5,6
3,4,5,6
3,4,5,6,7
2,3,4,5,6
3,4,5,6
3,4,5,6
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Inference/propagation
3,4,5,6
3,4,5,6
3,4,5,6
3,4,5,6
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Inference/propagation
3,4,5,6
3,4,5,6
3,4,5,6
3,4,5,6
And propagate
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Inference/propagation
3,4,5
4,5,6
3,4,5
4,5,6
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Inference/propagation
4,5,6
3,4,5
4,5,6
By symmetry
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Inference/propagation
4,5,6
3,4,5
4,5,6
And propagate
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Inference/propagation
5,6
4,5
4,5,6
More propagation?
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Inference/propagation
A solution
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The Declarative Approach to Combinatorial Problem
Solving

• Declarative Approach
• 1. Model the problem using a well-defined
  language
• 2. Solve the problem using general-purpose
  techniques
• Constraint Programming and SAT solving follow
  this approach.

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Constraint programming methodology

• Model problem
• Solve model
• Verify and analyze solution

• specify in terms of constraints on acceptable
  solutions
• define/choose constraint model
• variables, domains, constraints

• define/choose algorithm
• define/choose heuristics

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Constraint programming methodology

• Model problem
• Solve model
• Verify and analyze solution

• specify in terms of constraints on acceptable
  solutions
• define/choose constraint model
• variables, domains, constraints

• define/choose algorithm
• define/choose heuristics

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

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Given a CSP

• Determine whether it has a solution or not
• Find any solution
• Find all solutions
• Find an optimal solution, given some cost function

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Constraint model for puzzle
variables v1, , v8 domains 1, , 8
constraints v1 v2 ? 1 v1 v3 ? 1
v7 v8 ? 1 alldifferent(v1, , v8)
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Example instruction scheduling
(a b) c
Given a basic-block of code and a single-issue
pipelined processor, find the minimum schedule
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Example evaluate (a b) c
instructions A r1 ? a B r2 ? b C r3
? c D r1 ? r1 r2 E r1 ? r1 r3
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Example evaluate (a b) c
non-optimal schedule A r1 ? a B r2 ?
b nop nop D r1 ? r1 r2 C r3 ?
c nop nop E r1 ? r1 r3
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Example evaluate (a b) c
optimal schedule A r1 ? a B r2 ? b C r3 ?
c nop D r1 ? r1 r2 E r1 ? r1 r3
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Constraint model
variables A, B, C, D, E domains 1, , m
constraints D ? A 3 D ? B 3 E ? C 3 E
? D 1 alldifferent(A, B, C, D, E)
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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
v1
v2
v3
v4
v5
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Example 3-coloring
A solution v1 ? 1 v2 ? 2 v3 ? 2
v4 ? 1 v5 ? 3
v1
v2
v3
v4
v5
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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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Constraint model
x1
x2
x3
x4
variables x1, x2 , x3 , x4 domains 1, 2,
3, 4 constraints xi ? xj and xi - xj
? i - j
1
2
3
4
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Example 4-queens
x1
x2
x3
x4
A solution x1 ? 2 x2 ? 4 x3 ? 1 x4 ? 3
Q
1
Q
2
Q
3
Q
4
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Constraint programming methodology

• Model problem
• Solve model
• Verify and analyze solution

• specify in terms of constraints on acceptable
  solutions
• define/choose constraint model
• variables, domains, constraints

• define/choose algorithm
• define/choose heuristics

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Constraint programming methodology

• Model problem
• Solve model
• Verify and analyze solution

• specify in terms of constraints on acceptable
  solutions
• define/choose constraint model
• variables, domains, constraints

• define/choose algorithm
• define/choose heuristics

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Application areas

• Paradigm of choice for many hard combinatorial
  problems
• scheduling
• planning
• vehicle routing
• configuration
• bioinformatics

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Commercial applications