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Chapter 5. Constraint Satisfaction Problems (CSPs). Date:2004/3/17 Presenter: Shih, Ya-Ting. Outline. CSPs Backtracking search for CSPs Local search for CSP Problem structure. 1 CSPs. a type of Assignment Problems for problem solving

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Constraint Satisfaction Problems (CSPs)

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Chapter 5

Constraint Satisfaction Problems (CSPs)

Date:2004/3/17Presenter: Shih, Ya-Ting


Outline

  • CSPs

  • Backtracking search for CSPs

  • Local search for CSP

  • Problem structure


1 CSPs

  • a type of Assignment Problems for problem solving

  • states & goal test : a standard, structured, & very simple representation

  • CSP is defined by –a set of variables, X1,X2,…,Xn,with values from domain D1,D2,…,Dn,and a set of constraints, C1,C2,…,Cn,specifying allowance combinations of values for subsets of variables


1 CSPs (cont.)

  • State is defined by an assignment of values to some or all of the variables,{ Xii= vi , Xjj= vj , … }

  • Consistent ( or legal) assignment is an assignment that does not violate any constraints.

  • Complete assignment is one in which every variable is mentioned.

  • Solution is a complete assignment that satisfies all the constraints.


1.1 Example: Map-Coloring

NT

Q

WA

SA

NSW

V

T


1.1 Example: Map-Coloring (cont.)


1.2 Constraint graph


1.3 Varieties of CSPs

  • Discrete variables- finite domains:eg. Mapping coloring、8-Queens puzzle - infinite domains: (integers , strings , etc.)need a constraint language eg. Job scheduling ,StartJob1+5<StartJob3

  • Continuous variables eg. Hubble Space Telescope observation Linear programming problems


1.4 Varieties of Constraints

  • Unary constraint:involves a single variableeg. SA ≠ green

  • Binary constraint:involves pairs of variableseg. SA ≠ WA

  • Higher-order constraint:involves 3 or more variableseg. Cryptarithemetic puzzles (F5.2(a))(p.148)

  • Preference constrainteg. red is better than greenoften representable by a cost for each variable assignment constrained optimization problems


2 Backtracking search for CSPs

  • In all CSPs, variable assignment are commutative. (if…)eg. [ WA = red then NT = green ] same as [ NT = green then WA = red ]

  • Only need to consider assignment to a single value at each node.

  • Backtracking search-- a form of DSF search for CSP with single–variable assignments


2.1 Backtracking Example


2.2 Some Key Questions of Backtracking Search

  • Variable and value orderingwhich variable should be assigned next, and in what order should its values be tried?

  • Propagating information through constraintswhat are the implications of the current variable assignments for the other unassigned variables?

  • Intelligent backtrackingwhen a path fails– that is, a state is reached in which a variable has no legal value – can the search avoid repeating this failure in subsequent paths?


2.2.1 Most constrained variable

  • Minimum remaining variable (MRV) heuristicor Most constrained variable heuristicor Fail-First heuristic-- choose the variable with the fewer legal values


2.2.2 Most constraining variable

  • Degree heuristic-- tie-breaker among most constrained variables-- choose the variable with the most constrains on remaining variables


2.2.3 Least constraining value

  • Least-constraining-value heuristic-- try to leave the maximum flexibility for subsequent variable assignment-- prefer the value that rules out the fewest choices for the neighboring variables in the constraint graph


2.3 Constraint Propagation

  • Propagation the implications of a constraint on one variable onto other variables -- Forward checking-- Arc consistency (more stronger)


2.3.1 Forward Checking

  • Idea:Keep track of remaining legal values for unassigned variables. Terminate search when any variable has no legal values.

  • Whenever a variable X is assigned, the forward checking process looks at each unassigned variable Y that is connected to X by a constraint and deletes from Y’s domain. Any value that is inconsistent with the value chosen for X.


2.3.1 Forward Checking (cont.)

WA

NT

Q

NSW

V

SA

T

Initial domains

After WA=red

After Q=green

After V=blue

Figure 5.6

Partial assignment { WA=red, Q=green V=blue } is inconsistent with the constraint of the problem


2.3.2 Arc Consistency

  • Simplest form of propagation makes each arc consistent.

  • X→Y is consistent iff for every value x of X, there is some allowed y

  • Applying arc consistency has result in early detection of an inconsistency that is not detected by pure forward checking.


2.3.2 Arc Consistency (cont.-1)

Consistent!!


2.3.2 Arc Consistency (cont.-2)

Inconsistent!!


2.3.2 Arc Consistency (cont.-3)


2.4 Intelligent backtracking

  • called chronological backtrackingcause the most recent decision point is revisited-- to go all the way back to one of the set of variables that caused the failure-- conflict set

  • The backjumping method backtracks to the most recent variable in the conflict set

  • Backjumping occurs when every value in a domain is in conflict with the current assignment .


2.4 Intelligent backtracking (cont.)

  • fixed variable ordering Q, NSW, V, T, SA, WA, NT

  • partial assignment {Q=red, NSW=green, V=blue, T=red} (conflict set for SA) every value violates a constraint

7

1

6

2

5

3

4


3 Local Search for CSP

  • min-conflicts heuristic choose value that violates the fewest constraints

  • 8-Queens problem


4 Problem structure


4.1 Tree-structured CSPs


4.2 Nearly tree-structured CSPs


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