Constraint Satisfaction: An Overview

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Principles of Intelligent SystemsConstraint
Satisfaction Local Search
Written by Dr John ThorntonSchool of IT,
Griffith University Gold Coast
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Outline

• The Constraint Satisfaction Paradigm
• Variables, Domains and Constraints
• Example Problem Domains
• Constraint Satisfaction Algorithms
• Local Search Algorithms
• Summary

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The Constraint Satisfaction Paradigm

• Knowledge represented by constraints
• What is allowed or disallowed.
• Transforms broad range of problems into a
  computable form
• Standard representation means general purpose
  algorithms can be used
• solvers know how to solve problems
• issue becomes how to model a problem

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Variables, Domains and Constraints

• A Constraint Satisfaction Problem (CSP) is
• a set of variables each with
• a set of domain of values, and
• a set of constraints that define which
  combinations of variable values are allowed
• the task is to find a domain value for each
  variable such that all constraints are
  simultaneously satisfied

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Problem Domains

• N-Queens
• Satisfiability Problems
• Binary Constraint Satisfaction
• Nurse Rostering
• University Timetabling

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Local Search Algorithms

• Getting stuck
• Variable and Constraint-based Search
• Random Walk
• Tabu Search
• NOVELTY and RNOVELTY
• Constraint Weighting

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Satisfiability Problems

• Conjunctive Normal Form Example
• ?x Ú ?y Ú ?z Ù x Ú y Ú ?z Ù ?x Ú y Ù ?y Ú
  z
• To Solve
• All clauses evaluate true
• e.g. x false, y true, z true
• As a CSP
• Each clause is a constraint, each literal is a
  variable with a true, false domain

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Predicate to Conjunctive Form

• Predicate ?(X) (dog (X) ? animal (X))
• Clause (? dog(X) Ú animal(X))
• Clause is false only if ? dog(X) is false and
  animal(X) is false, i.e. if X is a dog and X is
  not an animal - in other words if X is a dog then
  X is an animal

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Binary Constraint Satisfaction

• Two variable constraint representation
• Non-binary to binary transformation
• Phase transition
• p2crit 1 - m-2/p1(n - 1)
• where p1 constraint density
• p2 constraint tightness
• m uniform domain size
• n number of variables

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Nurse Scheduling
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Binary Constraint Representation
Variable 1
0 1 2 3 4
0
1
Variable 2
2
3
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Random Walk Local Search

• if randomly generated probability lt p
• Take best cost local move
• else
• Take randomly selected local move

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Tabu Search

• if cost(best local move) lt best cost yet
• Take best local move
• else
• Take best cost local move from the set of domain
  values that have not been used in the last n moves

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The Eight Queens Problem
Domain
Variable
Constraint
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Constraint Satisfaction Algorithms

• Backtracking
• Algorithm
• 8-Queens Example
• 8-Queens Performance
• Local Search
• Algorithm
• 8-Queens Example
• 8-Queens Performance

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Summary

• Constraint Satisfaction
• practical knowledge representation
• allows general purpose approaches
• Algorithms
• Backtracking
• Local Search

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Backtracking Algorithm
V list of all variables vi (i 1 to n) U
empty TryVariable() select next vi from V and
move to U for each domain value di of vi if di
satisfies all constraints with U Value(vi)
di if(V is empty) SaveSolution() else
TryVariable() move vi from U back to V
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Basic Local Search Algorithm
assign a domain value di to each variable
vi while no solution and not stuck and not timed
out bestCost ? ? bestList ? ? for each
variable vi Cost(Value(vi)) gt 0 for each
domain value di of vi if Cost(di) lt bestCost
bestCost ? Cost(di) bestList ? di
else if Cost(di) bestCost bestList ?
bestList ? di Take a randomly selected move from
bestList
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Backtracking Performance
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Local Search Performance
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Local Search Diagram
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Eight Queens using Backtracking
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Eight Queens using Local Search
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Variable/Constraint-based Search

• Variable-based search
• Randomly select variable in constraint violation
• Try all domain values for that variable
• Constraint-based search
• Randomly select a violated constraint
• Try all domain values for all variables in the
  constraint that improve the constraint

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NOVELTY

• select best and second best cost moves that
  improve a violated constraint
• if best cost move not most recent or p gt random
  value
• select best cost move
• else
• select second best cost move

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RNOVELTY

• Same as NOVELTY except when best cost move is
  also most recent, then
• if second best cost - best cost lt c or p lt
  random value
• select second best cost move
• else
• select best cost move

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Constraint Weighting

• Add weight to all constraints violated at a local
  minimum graph colouring example
• Use these weights to calculate cost of subsequent
  moves
• Adding weights changes shape of cost surface
  fills in the minimum

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Constraint Weighting Example

• Graph Colouring

(a) Local minimum, cost 2w (b) After
Weighting, cost 3w (c) Final solution,
cost w