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Constraint Satisfaction Problems Russell and Norvig: Parts of Chapter 5 Slides adapted from: robotics.stanford.edu/~latombe/cs121/2004/home.htm Prof: Dekang Lin. http://www.cs.ualberta.ca/~lindek/366

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Intro Example 1: 4 Queens 1 3 2 4 3241

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1 3 2 4 3241 1 3 2 4 32411 3 2 4 3241 1 3 2 4 3241

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1 3 2 4 3241 X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4} X1 X2 X3 X4

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Intro Example 1: 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4} X1 X2 X3 X4

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Intro Example 1: 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {1,2,,4} X4 {1,2,3,4} X2 {,,3,4} X1 X2 X3 X4

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Intro Example 1: 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {,2,, } X4 {1,,,4} X2 {1,2,3, } X1 X2 X3 X4

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Intro Example 1: 4 Queens 1 3 2 4 3241 X1 {,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4} X1 X2 X3 X4

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Intro Example 1: 4 Queens X1 {1,2,3,4} X3 {1,,3, } X4 {1,,3,4} X2 {1,2,3,4} 1 3 2 4 3241 X1 X2 X3 X4

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Intro Example 1: 4 Queens 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,,4} X4 {,,3,4} X2 {,,,4} X1 X2 X3 X4

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Intro Example 1: 4 Queens 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,,4} X4 {,,3,4} X2 {,,,4} X1 X2 X3 X4

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Intro Example 1: 4 Queens 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,,4} X4 {,,3, } X2 {,,,4} X1 X2 X3 X4

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Intro Example 1: 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4} X1 X2 X3 X4

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Intro Example 2: 8-Queens

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66555566655556 5 5 5 5 5 6 7

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Intro Example 2: 8-Queens 344335344335 4 3 3 3 4 5

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Intro Example 2: 8-Queens 4323443234 3 3 3 4 3

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Intro Example 2: 8-Queens 4221342213 3 3 3 1

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Intro Example 2: 8-Queens 221221 2 2 1

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Intro Example 2: 8-Queens

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What is Needed? Not just a successor function and goal test But also a means to propagate the constraints imposed by one queen on the others and an early failure test Explicit representation of constraints and constraint manipulation algorithms

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Constraint Satisfaction Problem Set of variables {X 1, X 2, …, X n } Each variable X i has a domain D i of possible values. Usually, D i is discrete and finite Set of constraints {C 1, C 2, …, C p } Each constraint C k involves a subset of variables and specifies the allowable combinations of values of these variables Assign a value to every variable such that all constraints are satisfied

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Example: 8-Queens Problem 64 variables X ij, i = 1 to 8, j = 1 to 8 Domain for each variable {1,0} Constraints are of the forms: X ij = 1 X ik = 0 for all k = 1 to 8, k j X ij = 1 X kj = 0 for all k = 1 to 8, k i Similar constraints for diagonals X ij = 8

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Example: 8-Queens Problem 8 variables X i, i = 1 to 8 Domain for each variable {1,2,…,8} Constraints are of the forms: X i = k X j k for all j = 1 to 8, j i Similar constraints for diagonals

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CSP as a Search Problem Initial state: empty assignment Successor function: a value is assigned to any unassigned variable, which does not conflict with the currently assigned variables Goal test: the assignment is complete Path cost: irrelevant n variables of domain size d O(d n ) distinct complete assignments

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Compare!! Most-constraining-variable heuristic: Select the variable that is involved in the largest number of constraints on other unassigned variables. (Rationale: maximize the number of values eliminated, hence future branching factors) Least-constraining-value heuristic: Prefer the value that leaves the largest subset of legal values for other unassigned variables Each variable will eventually have a value assigned to it (so, pick first the most-constraining one) but a value does not have to be assigned (so, pick first the least-constraining one)

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Local Search for CSP 1 2 3 3 2 2 3 2 2 2 2 2 0 2 Pick initial complete assignment (at random) Repeat Pick a conflicted variable var (at random) Set the new value of var to minimize the number of conflicts If the new assignment is not conflicting then return it (min-conflicts heuristics)

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Why Does It Work? Local search with min-conflict heuristic works extremely well for million-queen problems The reason: Solutions are densely distributed in the O(n n ) space, which means that on the average a solution is only a few steps away from a randomly picked assignment

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Applications CSP techniques have allowed solving very complex problems Numerous applications, e.g.: Crew assignments to flights Management of transportation fleet Flight/rail schedules Task scheduling in port operations Design Brain surgery

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Stereotaxic Brain Surgery

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2000 < Tumor < 2200 2000 < B2 + B4 < 2200 2000 < B4 < 2200 2000 < B3 + B4 < 2200 2000 < B3 < 2200 2000 < B1 + B3 + B4 < 2200 2000 < B1 + B4 < 2200 2000 < B1 + B2 + B4 < 2200 2000 < B1 < 2200 2000 < B1 + B2 < 2200 0 < Critical < 500 0 < B2 < 500 T C B1 B2 B3 B4 T

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Stereotaxic Brain Surgery

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Constraint Programming “Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it.” Eugene C. Freuder, Constraints, April 1997

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