Revisiting the cardinality Operator and Introducing the cardinality-path Constraint Family Nicolas Beldiceanu and Mats Carlsson S I C S Lägerhyddsvägen 18 75237 Uppsala {nicolas,matsc}@sics.se ICLP 2001, Paphos, Cyprus
Outline of the Presentation The cardinality Operator The cardinality-path Family • Situation of the cardinality-path Family • Instances of the cardinality-path Family Filtering Algorithms for cardinality-path • Required Basic Services • Lower Bound of the Minimum Number of Constraints that Hold • Pruning According to the Bounds • Example of Pruning Conclusion and Perspectives
The cardinality Operator The cardinality-path Family • Situation of the cardinality-path Family • Instances of the cardinality-path Family Filtering Algorithms for cardinality-path • Required Basic Services • Lower Bound of the Minimum Number of Constraints that Hold • Pruning According to the Bounds • Example of Pruning Conclusion and Perspectives
The cardinality Operator The cardinality operator , Van Hentenryck, P., Deville, Y. (ICLP 1991) : n : C = ∑ # CTR j (V 1 ,.., V m ) • Definition j j=1 • Pruning based on: counting C ≤ 5 - 1 CTR 1 CTR 2 CTR 3 CTR 4 CTR 5 entailment If C ≥ 1+3 then CTR 1 CTR 2 CTR 3 CTR 4 CTR 5 CTR 2 , CTR 4 , CTR 5 hold
The Pruning Perfect if the constraints are independant , but weak if the constraints share some variables: X<Y Y<Z Z<X C ≤ 5 - 1 - 1 CTR 1 CTR 2 CTR 3 CTR 4 CTR 5 Generalize existing deduction rules of the cardinality operator
Estimate the maximum number of constraints which hold Partition the constraints in sets of the following types: ... : the conjunction of constraints does not hold ... : the conjunction of constraints allway holds ... : the conjunction of constraints may hold Number of conjunctions of constraints which do not hold C ≤ 5 - 1 CTR 1 CTR 2 CTR 3 CTR 4 CTR 5
Enforcing conjunctions of constraints Enforcing constraints according to min(C): If C ≥ 5 - 1 then CTR 1 CTR 2 CTR 3 CTR 4 CTR 5 CTR 4 , CTR 5 hold
Pruning Values Pruning a variable according to min(C): CTR 1 ∧ CTR 2 ⇒ CTR 4 ∧ CTR 5 ⇒ CTR 6 ∧ CTR 7 ⇒ val ∉ dom(Var ) val ∉ dom(Var ) val ∉ dom(Var ) If C > 7 - 1 - 1 - 1 CTR 1 CTR 2 CTR 3 CTR 4 CTR 5 CTR 6 CTR 7 then val ∉ dom(Var ) then
The cardinality Operator The cardinality-path Family • Situation of the cardinality-path Family • Instances of the cardinality-path Family Filtering Algorithms for cardinality-path • Required Basic Services • Lower Bound of the Minimum Number of Constraints that Hold • Pruning According to the Bounds • Example of Pruning Conclusion and Perspectives
The cardinality-path Family n-k+1 : C = ∑ # CTR (V i ,..,V i+k-1 ) • Definition i=1 considers interaction between constraints • Pruning : (shared variables) V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8
Situation of the cardinality-path Family Inside the Classification • Constraint parameters + • Graph generator + • Elementary constraint + • Graph properties A complete description of the global constraint is explicitly provided !
cyclic_change( NCHANGE , L , VARIABLES ) • A R G U M E N T : NCHANGE : dvar L : int VARIABLES : collection( var -dvar) • R E S T R I C T I O N (S) : NCHANGE ≥ 0 NCHANGE ≤ | VARIABLES | L > 0 required( VARIABLES.var ) • V E R T E X I N P U T : VARIABLES • V E R T E X G E N E R A T O R : IDENTITY • E D G E I N P U T : VARIABLES • E D G E G E N E R A T O R : PATH • E D G E A R I T Y : 2 • E D G E C O N S T R A I N T : ( VARIABLES.var [1]+1) mod L ≠ VARIABLES.var [2] • G R A P H P R O P E R T Y : NEDGE = NCHANGE cyclic_change(2,4,{ var -3, var -0, var -2, var -3, var -1})
Graph Generation for cyclic_change( NCHANGE , L , VARIABLES ) : VARIABLES • V E R T E X I N P U T • V E R T E X G E N E R A T O R : IDENTITY Collections of items: VARIABLES Vertices generator: IDENTITY V 1 V 2 V 3 V 4 V 5
Graph Generation for cyclic_change( NCHANGE , L , VARIABLES ) : VARIABLES • E D G E I N P U T • E D G E G E N E R A T O R : PATH • E D G E A R I T Y : 2 V 1 V 2 V 3 V 4 V 5 Edge generator: PATH V 1 V 2 V 3 V 4 V 5
Elementary Constraint for cyclic_change( NCHANGE , L , VARIABLES ) • E D G E C O N S T R A I N T : ( VARIABLES.var [1]+1) mod L ≠ VARIABLES.var [2] V 1 V 2 V 4 V 5 V 3 ( V 1 +1) mod L ( V 2 +1) mod L ( V 3 +1) mod L ( V 4 +1) mod L ≠ ≠ ≠ ≠ V 2 V 3 V 4 V 5
Graph Property for cyclic_change( NCHANGE , L , VARIABLES ) • G R A P H P R O P E R T Y : NEDGE = NCHANGE 3 0 3 1 2 (3+1) mod 4 (0+1) mod 4 (2+1) mod 4 (3+1) mod 4 ≠ ≠ ≠ ≠ 0 2 3 1 ( false ) ( true ) ( false ) ( true ) cyclic_change( 2 ,4,{ var -3, var -0, var -2, var -3, var -1})
”Signature” of the cardinality-path family • V E R T E X G E N E R A T O R : IDENTITY • E D G E G E N E R A T O R : PATH • E D G E A R I T Y : k > 1 • C O N S T R A I N T P R O P E R T Y : - • G R A P H P R O P E R T Y : NEDGE = var ctr ctr ctr ctr Constraint pattern
The cardinality Operator The cardinality-path Family • Situation of the cardinality-path Family • Instances of the cardinality-path Family Filtering Algorithms for cardinality-path • Required Basic Services • Lower Bound of the Minimum Number of Constraints that Hold • Pruning According to the Bounds • Example of Pruning Conclusion and Perspectives
Bounds for C and Member and example of Arity constraint CTR solution C : 0.. n − 1 change( C , { V 1 ,.., V n }, ≠ ) 2 V i ≠ V i +1 change( 3 ,{ 4,4, 3, 4, 1 }, ≠ ) 2 C : 0.. n − 1 cyclic_change( C , L , { V 1 ,.., V n }) ( V i +1)mod L ≠ V i +1 cyclic_change( 2 ,5, { 2,3,4,0, 2,3, 1 }) C : 0.. n − 1 2 smooth( C , T , { V 1 ,.., V n }) |V i − V i +1 | > T smooth( 1 ,2, { 1,3,4,5, 2 }) C : 0.. n − 2 3 number_of_rest( C , { V 1 ,.., V n }) V i =0 ∧ V i +1 =0 ∧ V i +2 ≠ 0 number_of_rest( 2 , { 2,0, 0,1,1,0,2,0, 0,1,2 }) k C : a .. b relaxed_sliding_sum( a,b,low,up,k ,{ V 1 ,.., V n }) i + k − 1 low ≤ Σ V i ≤ up relaxed_sliding_sum( 3 , 4 ,3,7,4, { 2,4, 2, 0, 0,3,4 }) j = i
The cardinality Operator The cardinality-path Family • Situation of the cardinality-path Family • Instances of the cardinality-path Family Filtering Algorithms for cardinality-path • Required Basic Services • Lower Bound of the Minimum Number of Constraints that Hold • Pruning According to the Bounds • Example of Pruning Conclusion and Perspectives
Required Services • enforce_CTR(V i ,..,V i+k − 1 ) MAIN • enforce_NOT_CTR(V i ,..,V i+k − 1 ) PRUNING ALGORITHM • create_choice_point • backtrack ASSUMPTION : For constraint CTR or its negation, failure detection should be independent from the order in which the constraints are posted
Organisation of the Algorithms cardinality_path( C , {V 1 ,..,V n }, CTR ) FILTERING ALGORITHMS find BOUNDS PROPAGATE from C to {V 1 ,..,V n } for C
The cardinality Operator The cardinality-path Family • Situation of the cardinality-path Family • Instances of the cardinality-path Family Filtering Algorithms for cardinality-path • Required Basic Services • Lower Bound of the Minimum Number of Constraints that Hold • Pruning According to the Bounds • Example of Pruning Conclusion and Perspectives
Lower Bound of the Number of Constraints that Hold cyclic_change( C , 5, {V1,V2,V3,V4,V5,V6,V7,V8,V9}) CTR: (V i +1) mod 5 ≠ V i+1 V1: 0,3 V1: 0,3 (V1+1)mod 5=V2 V1: 3 V2: 4 V2: 2,3,4 (V2+1)mod 5=V3 V1: 3 V2: 4 V3: 0 V3: 0,4 (V3+1)mod 5=V4 V1: 3 V2: 4 V3: 0 V4: 1 (V4+1)mod 5=V5 V1: 3 V2: 4 V3: 0 V4: 1 V5: 2 V4: 0,1,2,3,4 (V5+1)mod 5=V6 contradiction V5: 0,1,2,3 V6: 0,2,4 V6: 0,2,4 (V6+1)mod 5=V7 V6: 0,4 V7: 0,1 V7: 0,1,2 (V7+1)mod 5=V8 V6: 0,4 V7: 0,1 V8: 1,2 V8: 0,1,2 (V8+1)mod 5=V9 contradiction V9: 0,4 V9: 0,4 min(C) ≥ 2
The cardinality Operator The cardinality-path Family • Situation of the cardinality-path Family • Instances of the cardinality-path Family Filtering Algorithms for cardinality-path • Required Basic Services • Lower Bound of the Minimum Number of Constraints that Hold • Pruning According to the Bounds • Example of Pruning Conclusion and Perspectives
Propagation from C to {V 1 ,..,V n } Set min_break to 0. before[ i ] (1 ≤ i ≤ n - k +1): Set choice to 1. Create_choice_point. min.number of constraints that hold in for i =1 to n - k +1 do { CTR(V 1 ,.., V k ) ,.., CTR(V i-1 ,.., V i+k-2 } Set before[ i ] to min_break . if choice =0 then Set choice to 1. Create_choice_point. if enforce ¬ CTR(V i ,..,V i+k-1 ) fails then Backtrack. Set choice to 0. vbefore[ i ] ( k ≤ i ≤ n ): Set min_break to min_break +1. state of dom(V i ) after stating constraint CTR(V i-k+1 ,.., V i ) Set vbefore[ i + k -1] to dom(V i+k-1 ) .
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