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Constraint Satisfaction Problems. Contents. Representations Solving with Tree Search and Heuristics Constraint Propagation Tree Clustering. Posing a CSP. A set of variables V 1 , …, V n A domain over each variable D 1 ,…,D n

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## Constraint Satisfaction Problems

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**Contents**• Representations • Solving with Tree Search and Heuristics • Constraint Propagation • Tree Clustering**Posing a CSP**• A set of variables V1, …, Vn • A domain over each variable D1,…,Dn • A set of constraint relations C1,…,Cm between variables which indicate permitted combinations • Goal is to find an assignment to each variable such than none of the constraints are violated**A**A B B C C D D Constraint Graphs • Nodes = variables • Edges = constraints • Example: map coloring**X**X Y Z Y Z Hyper graph Primal constraint graph N-ary Constraint Graphs • Example: • Variables: X=[1,2] Y=[3,4] Z=[5,6] • Constraints: X + Y = Z (Roman Barták, 1998 )**Making a Binary CSP**• Can convert n-ary constraint C into a unary constraint on new variable Vc • Dc = cartesian product of vars in C • Can convert n-ary CSP into a binary CSP • Create var Vc for each constraint C (as above) • Domain Dc = cartesian product – tuples that violate C • Add binary equivalence constraints between new variables Vc, Vc’:C,C’ share var X Vc,Vc’ must agree on X**X**XYZ Y Z XYZ= [(1,3,5),(1,3,6), (1,4,5),(1,4,6), (2,3,5),(2,3,6) (2,4,5),(2,4,6)] Making a Unary Constraint • Variables: X=[1,2] Y=[3,4] Z=[5,6] • Constraints: X + Y = Z (Roman Barták, 1998 )**Making a Unary Constraint**• Variables: X=[1,2] Y=[3,4] Z=[5,6] • Constraints: X + Y = Z X XYZ Y Z XYZ= [(1,4,5), (2,3,5), (2,4,6)] (Roman Barták, 1998 )**X**XYZ Y Y Z WY Dual constraint graph W Making a Binary CSP • Variables: X=[1,2] Y=[3,4] Z=[5,6] W=[1,3] • Constraints: X + Y = Z, W<Y (Roman Barták, 1998 )**XYZ**Y WY Dual constraint graph Making a Binary CSP • Variables: X=[1,2] Y=[3,4] Z=[5,6] W=[1,3] • Constraints: X + Y = Z, W<Y X Y Z W (Roman Barták, 1998 )**Contents**• Representations • Solving with Tree Search and Heuristics • Constraint Propagation • Tree Clustering**Generate and Test**• Generate each possible assignment to the variables and test if constraints are satisfied • Exponential possibilities: O(d n) • Simple but extremely wasteful!**DFS and Backtracking**• Depth first search • Levels represent variables • Branches off nodes represent a possible instantiations of variables • Test against constraints after every variable instantiation and backtrack if violation • Incrementally attempts to extend partial solution • Whole subtrees eliminated at once**Example**V1 red green blue V2 V3 red red green (*,*,*)**Example**V1 red green blue V2 V3 red red green (*,*,*) (r,*,*)**Example**V1 red green blue V2 V3 red red green (*,*,*) (r,*,*) (r,r,*)**Example**V1 red green blue V2 V3 red red green (*,*,*) (r,*,*) (g,*,*) (r,r,*) (g,r,*)**Example**V1 red green blue V2 V3 red red green (*,*,*) (r,*,*) (g,*,*) (r,r,*) (g,r,*) (g,r,r) (g,r,g)**Example**V1 red green blue V2 V3 red red green (*,*,*) (r,*,*) (g,*,*) (b,*,*) (r,r,*) (g,r,*) (b,r,*) (g,r,r) (g,r,g) (b,r,r) (b,r,g)**Forward Checking**• Backtracking is still wasteful • A lot of time is spent searching in areas where no solution remains • Ex. setting V4 to value X1 eliminates all possible values for V8 under the given constraints • Can cause thrashing • Forward checking removes restricted values from the domains of all uninstantiated variables • If a domain becomes empty backtracking is done immediately**Heuristics**• The search can usually be sped up by searching intelligently: • Most-constrained variable: Expand subtree of variables that have the fewest possible values within their domain first • Most-constraining variable: Expand subtree of variables which most restrict others first • Least-constraining value: Choose values that allow the most options for the remaining variables first**Contents**• Representations • Solving with Tree Search and Heuristics • Constraint Propagation • Tree Clustering**Constraint Propagation**• A preprocessing step to shrink the CSP • Constraints are used to gradually narrow down the possible values from the domains of the variables • A singleton may result • If the domains of each variable contain a single value we do not need to search**Arc Consistency**• Arc (Vi,Vj) in a constraint graph is arc consistent if for every value of Vi there is some value that is permitted for Vj • Algorithm: • Complexity O(ed3) do foreach edge (i,j) delete values from Di that cause Arc(Vi,Vj) to fail while deletions**Example**V1 green V2 V3 red green blue green blue Consider edge (1,3)**Example**V1 green V2 V3 red green blue green blue Consider edge (3,1)**Example**V1 green V2 V3 red green blue green blue Consider edge (2,1)**Example**V1 green V2 V3 red green blue green blue Consider edge (2,3)**Example**V1 green V2 V3 red green blue green blue Consistent and a singleton!**Levels of Consistency**• Algorithms we have seen before are combinations of tree search and arc consistency: Generate and Test Backtracking Forward Checking Partial Lookahead Full Lookahead Really Full Lookahead TS BT = TS + AC 1/5 FC = TS + AC 1/4 PL = FC + AC 1/3 FL = FC + AC 1/2 RFL = FC + AC (Nadel, 1988)**Backtracking**• Given: • check(i,Xi,j,Xj): true if Vi = Xi and Vj = Xj is permitted by constraints • revise(i,j): true if Di is empty after making Arc(Vi,Vj) = true function BT(i,var) for(var[i]=Di) CONSISTENT = true for(j=1:i-1) CONISITENT = check(i,var[i],j,var[j]) end if CONSISTENT if i==n disp(var) else BT(i+1,var) end function BT(i) EMPTY_DOMAIN = check_backward(i) if ~EMPTY_DOMAIN for(var[i]=Di) Di = var[i] if i==n disp(var) else BT(i+1) end function check_backward(i) for(j=1:i-1) if revise(i,j) return true end return false**Forward Checking**function FC(i) EMPTY_DOMAIN = check_forward(i) if ~EMPTY_DOMAIN for(var[i]=Di) Di = var[i] if i==n disp(var) else FC(i+1) end function check_forward(i) if i>1 for(j=i:n) if revise(j,i-1) return true end return false • Similar to backtracking except more arc-consistency**Levels of Consistency**Generate and Test Backtracking Forward Checking Partial Lookahead Full Lookahead Really Full Lookahead TS BT = TS + AC 1/5 FC = TS + AC 1/4 PL = FC + AC 1/3 FL = FC + AC 1/2 RFL = FC + AC (Nadel, 1988)**A Stronger Degree of Consistency**• A graph is K-consistent if we can choose values for any K-1 variables that satisfy all the constraints, then for any Kth variable be able to assign it a value that satisfies the constraints • A graph is strongly K-consistent if J-consistent for all J < K • Node consistency is equivalent to strong 1-consistency • Arc consistency is equivalent to strong 2-consistency**Towards Backtrack Free Search**• A graph that has strong n-consistency requires no search • Acquiring strong n-consistency is exponential in the number of variables (Cooper, 1989) • For a general graph that is strongly k-consistent (where k < n) backtracking cannot be avoided**(*,*,*)**(r,*,*) (r,r,*) … Example V1 red green V2 V3 red green green blue Arc consistent, yet a search will backtrack!**Constraint Graph Width**V1 V1 V2 V2 V3 V3 V1 V2 V3 V1 V3 V1 V2 • The nodes of a constraint graph can be ordered V2 V3 V3 V2 V3 V1 V2 V1 1 1 1 2 1 2 1 • The width of a node in an ordered graph is equal to the number of incoming arcs from higher up nodes • The width of an ordered graph is the max width of its vertices • The width of a constraint graph is the min width of each of its orderings**Backtrack Free Search**• Theorem: If a constraint graph is strongly K-consistent, and K is greater than its width, then there exists a search order that is backtrack free • K>2 consistency algorithms add arcs requiring even greater consistency • If a graph has width 1 we can use node and arc consistency to get strong 2-consistency without adding arcs • All tree structured constraint graphs have width 1 (Freuder 1988)**Contents**• Representations • Solving with Tree Search and Heuristics • Constraint Propagation • Tree Clustering**Tree Clustering Motivation**• Tree structured constraint graphs can be solved without backtracking • We would like to turn non-tree graphs into trees by grouping variables • The grouped variables themselves become smaller CSP’s • Solving a CSP is exponential in the worst case so reducing the number of variables we consider at once is also important • If we want the CSP for many queries it is worth investing more time in restructuring it (Dechter, 1988)**ABC**AEF AC AE ACE CDE CE Join graph/tree Redundancy • Constraints in the dual graph are equalities • Variables: A, B, C, D, E, F • Constraints: (ABC), (AEF), (CDE), (ACE) A ABC AEF C E AC AE ACE CDE CE**Tree Clustering**• If the dual graph cannot be reduced to a join tree we can still make it acyclic: • Condition for acyclicity: A CSP is acyclic iff its primal graph is chordal and conformal • Given a primal graph its dual can be made acyclic: • Triangulate graph to make it chordal • The maximal cliques are constraints/nodes in the new dual graph (Beeri, 1983)**Triangulation**• Use maximum cardinality search (m-ordering) to order the nodes • Add an edge between any two nonadjacent nodes that are connected by nodes higher in the ordering (Tarjan, 1984)**The Algorithm**• Build the primal graph for the CSP • O(n2) • Triangulate • O(n2) • Use maximal cliques as new nodes in dual graph • O(n) • Remove any redundancies in the new graph • O(n)**BD**BD D D D D D AD DE AD DE A A E E C C AC CE AC CE Example • Variables: A, B, C, D, E • Constraints: (A,C), (A,D), (B,D), (C,E), (D,E) Still cyclic!**B**E A E C D C D C A B A B D E Order: E, D, C, A, B Example • Variables: A, B, C, D, E • Constraints: (A,C), (A,D), (B,D), (C,E), (D,E)**E**BD BD D D D C D ACD CDE ACD CDE A B CD CD Example • Variables: A, B, C, D, E • Constraints: (A,C), (A,D), (B,D), (C,E), (D,E) Acyclic!**Solving the CSP**• Solve each node of the tree as a separate small CSP • This can be done in parallel • The solutions to each node constitute the domain of that node in the tree • O(d m) • Use arc consistency to reduce the domains of each node • Solve the entire CSP without backtracking**Example CSP’s**• N-queens • Map coloring • Cryptoarithmatic • Wireless network base station placement • Object recognition from image features

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