Constraint Satisfaction
Problems (CSP)
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General search versus CSP
General search
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Blackbox State
External Child-generator
State-space can be infinite
External goal test
Goals can occur at any depth
Goals can have different costs
All the search algorithms we
discussed until now are
appropriate.
Heuristics are aimed at estimating
the cost to goal node
CSP
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State is made-up of state variables
Children generation involves
assigning values to more variables
State space is finite
A state is a goal state if all
variables are assigned and no
constraints are violated
All goals occur at the same depth
In the basic formulation, all goals
have the same cost
Only the Depth-first search makes
sense!
Heuristics are aimed at picking the
right variable to assign next, and
deciding the right value to assign
to it
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Also called most constrained variable
Usually lead to speedup by a factor of
1,000 or more
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Unassigned variables
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Try to leave the maximum flexibility for subsequent variable assignments
Does the ordering of values matter if we want to find ALL (instead of A) solutions?
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Fail-first versus fail-last
 Variable:
fail-first
 Minimize the number of nodes in the search
tree, i.e., minimize the number of possible
assignments to check
 Value: fail-last
 We only need ONE solution, why not look at the
most likely values first?
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NT
WA SA
Q
NSW
V
T
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NT Q
WA SA
NSW
V
T
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NT Q
WA SA
NSW
V
T
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NT
WA SA
Q
NSW
V
T
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NT Q
WA
SA
NSW
V
T
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NT Q
WA
SA
NSW
V
T
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Constraint propagation
 Node
consistency: all values in the variable’s domain
satisfy the variable’s unary constraint
 Arc consistency: Xi is arc-consistent with Xj if for every
value in the current domain Di, there is some value in the
domain Dj that satisfy the binary constraint on the arc
(Xi, Xj)
 Path consistency: involve three variables
 K-consistency: involve K variables
×
Xi is arc-consistent with Xj = Xj is arc-consistent with Xi ??
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Constraint propagation in Sudoku
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Summary
 Formulation
of CSP
 Backtracking for CSP
 Most constrained variable
 Least constrained value
 Degree heuristic for tie-breaking
 Forward checking
 Constraint propagation
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