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H-const-prop - Constraint Propagation(Where a better...

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1 Constraint Propagation (Where a better exploitation of the constraints further reduces the need to make decisions) R&N: Chap. 5 + Chap. 24, p. 881-884 1 Constraint Propagation … … is the process of determining how the constraints and the possible values of one variable affect the possible values of other variables It is an important form of “least-commitment” reasoning 2 Forward checking is only on simple form of constraint propagation When a pair (X Å v) is added to assignment A do: For each variable Y not in A do: For every constraint C relating Y to variables in A do: Remove all values from Y’s domain that do not satisfy C 3 n = number of variables d = size of initial domains s = maximum number of constraints involving a given variable (s n-1) Forward checking takes O(nsd) time Forward Checking in Map Coloring Empty set: the current assignment {(WA Å R), (Q Å G), (V Å B)} does not lead to a solution WA NT Q NSW V SA T R G B R G B R G B R G B R G B R G B R G B R R G B R G B R G B R G B R G B R G B R G B G R G B R G B G B R G B R B G R B B B R G B 4 Forward Checking in Map Coloring T WA NT SA Q NSW V Contradiction that forward checking did not detect WA NT Q NSW V SA T R G B R G B R G B R G B R G B R G B R G B R R G B R G B R G B R G B R G B R G B R G B G R G B R G B G B R G B R B G R B B B R G B 5 Forward Checking in Map Coloring WA NT Q Contradiction that forward checking did not detect Detecting this contradiction requires a more powerful constraint propagation technique WA NT Q NSW V SA T R G B R G B R G B R G B R G B R G B R G B R R G B R G B R G B R G B R G B R G B R G B G R G B R G B G B R G B R B G R B B B R G B 6
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2 Constraint Propagation for Binary Constraints REMOVE-VALUES( X , Y ) 1. removed Å false 2. For every value v in the domain of Y do If there is no value u in the domain of X such that the constraint on ( X , Y ) is satisfied then a. Remove v from Y ‘s domain b. removed Å true 3. Return removed 7 Constraint Propagation for Binary Constraints AC3 1. Initialize queue Q with all variables (not yet instantiated) 2. While Q ≠ ∅ do a. X Å Remove( Q ) b. For every (not yet instantiated) variable Y related to X by a (binary) constraint do If REMOVE-VALUES( X , Y ) then i. If Y ’s domain = then exit ii. Insert( Y , Q ) 8 Edge Labeling We consider an image of a scene composed of polyhedral objects such that each vertex is the endpoint of exactly three edges R&N: Chap. 24, pages 881-884
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