Constraint Satisfaction Problem Solving

1
Constraint SatisfactionProblem Solving

• Chapter 5

2
Example problems

• Illustrative
• Map coloring
• Cryptarithmetic
• N-queens
• Real
• Class scheduling
• Scheduled Hubble telescope
• Discrete Optimization problems

3
Goal solve the problem

• Find bindings for all the variables that satisfy
  all the constraints.
• Corresponds to boolean satisfiability, except
  that now variables take on values from discrete
  sets.

4
Formal Model

• Finite set of variables X1,Xn
• Variable Xi has values in finite domain Di.
• Constraints C1Cm. A constraint specifies legal
  combinations of the values. Tables.
• Assignment selection of values for all
  variables.
• Consistent assignment satisfies all constraints.

5
Note Constraints can be made binary

• Constraints table of legal values
• One trinary table between(a,b,c).
• Between(2,4,5)
• Replaced by conjunction of two binary relations
• Less(2,4) Less(4,5)

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Simple Mapcolors red, blue, greenGoal assign
colors so touching countries have different colors
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Model

• Variables X1, X2, X3
• Domains Di red,blue,green.
• X1 !X2 Table with 6 entries

red blue
red green
blue red
blue green
green red
green blue
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Naïve BackTracking Solution

• X1 red, X2 red, X3 red, X1!X2 Fail
• Short hand for table lookup (not in table)
• X1 red, X2red, X3blue, X1!X2 Fail
• X1 red, X2red, X3green.Fail
• X1 red, X2blue,.
• Continue to Solution..
• UGH. But it is the way Prolog works.

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Selections in Backtracking

• Selecting a variable to bind
• Selecting a value to assign to the variable
• Does Order Matter?
• Yes and no
• No search is complete any ordering will find a
  solution if one exists.
• Yes search cost changes.

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Controlling Backtracking

• Choosing a variable to bind
• Minimum remaining values (MRV)
• Intuition need to solve eventually so do hardest
  case first
• Degree Heuristic
• choose variable involved in largest number of
  constraints. (tie-breaker)
• Choosing a binding/value for variable
• Least constraining variable
• Intuition maximize search options

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Forward Checking

• Constraint Propagation
• If variable X is bound, find all variables Y
  that are connected to it by a constraint.
  Eliminate values that are inconsistent.
• Can yield dramatic improvement.
• Provably better than backtracking
• Guaranteed to do less work

12
Relook at Map-coloringUsing Forward Checking

• X1 red remove red from X2 and X3 domains
• X2 blue remove blue from X3s domain
• X3 green.
• Done!

13
Arc-Consistency (3)

• Binary CSP (can always force this)
• Arc edge between variables that are in same
  constraint.
• Let X-Y be arc.
• for each x in Domain X if there is no value y
  in domain Y that allows constraint to be met,
  then delete x from domain X.
• Just look in the tables
• Can be done as preprocessing or during search.

14
AC3 Example

• TWOTWO FOUR
• edge between T and F gives
• (TTCarry )/10 F we see that only
• T in 5,6,7,8,9 can work for F 1.
• Edge between F and R says R \ 1.
• Edge between (OO)10 R and O\R
• gives O \ 0.