An optimal Arc Consistency algorithm for a chain of Atmost - PowerPoint PPT Presentation

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work An optimal Arc Consistency algorithm for a chain of Atmost constraints with cardinality Mohamed Siala , Emmanuel Hebrard, and Marie-Jos´ e Huguet Toulouse, France Mohamed SIALA October 2012 CP 2012 1 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Outline The AtMostSeqCard constraint Filtering the domains Experimental results Conclusion & Future work Mohamed SIALA October 2012 CP 2012 2 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Definition AtMostSeqCard ( u , q , d , [ x 1 , . . . , x n ]) ⇔ n − q q n � � � ( x i + l ≤ u ) ∧ ( x i = d ) i =0 i =1 l =1 Mohamed SIALA October 2012 CP 2012 3 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Definition AtMostSeqCard ( u , q , d , [ x 1 , . . . , x n ]) ⇔ n − q q n � � � ( x i + l ≤ u ) ∧ ( x i = d ) i =0 i =1 l =1 Example AtMostSeqCard (2 , 4 , 4 , [ x 1 , . . . , x 7 ]) 0 1 1 0 1 1 0 1 1 0 0 1 0 1 — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — Mohamed SIALA October 2012 CP 2012 3 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Context Sequence Constraints • Let [ x 1 , . . . , x n ] be a sequence of integer variables. Mohamed SIALA October 2012 CP 2012 4 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Context Sequence Constraints • Let [ x 1 , . . . , x n ] be a sequence of integer variables. • The Among constraint ensures that the number of occurrences of values in { v 1 .. v k } in a subsequence [ x i 1 , . . . , x iq ] is bounded between l and u . Mohamed SIALA October 2012 CP 2012 4 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Context Sequence Constraints • Let [ x 1 , . . . , x n ] be a sequence of integer variables. • The Among constraint ensures that the number of occurrences of values in { v 1 .. v k } in a subsequence [ x i 1 , . . . , x iq ] is bounded between l and u . • AmongSeq : the conjunction of all n − q + 1 Among on q consecutive variables (i.e. � n − q i =0 Among ([ x i +1 , . . . , x i + q ])). Mohamed SIALA October 2012 CP 2012 4 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Context Sequence Constraints • Let [ x 1 , . . . , x n ] be a sequence of integer variables. • The Among constraint ensures that the number of occurrences of values in { v 1 .. v k } in a subsequence [ x i 1 , . . . , x iq ] is bounded between l and u . • AmongSeq : the conjunction of all n − q + 1 Among on q consecutive variables (i.e. � n − q i =0 Among ([ x i +1 , . . . , x i + q ])). • Gen-Sequence : Conjunction of (consecutive) Among Mohamed SIALA October 2012 CP 2012 4 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Context Sequence Constraints • Let [ x 1 , . . . , x n ] be a sequence of integer variables. • The Among constraint ensures that the number of occurrences of values in { v 1 .. v k } in a subsequence [ x i 1 , . . . , x iq ] is bounded between l and u . • AmongSeq : the conjunction of all n − q + 1 Among on q consecutive variables (i.e. � n − q i =0 Among ([ x i +1 , . . . , x i + q ])). • Gen-Sequence : Conjunction of (consecutive) Among • AtMostSeqCard ≡ AmongSeq ⊕ a cardinality constraint Mohamed SIALA October 2012 CP 2012 4 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Context Sequence Constraints • Let [ x 1 , . . . , x n ] be a sequence of integer variables. • The Among constraint ensures that the number of occurrences of values in { v 1 .. v k } in a subsequence [ x i 1 , . . . , x iq ] is bounded between l and u . • AmongSeq : the conjunction of all n − q + 1 Among on q consecutive variables (i.e. � n − q i =0 Among ([ x i +1 , . . . , x i + q ])). • Gen-Sequence : Conjunction of (consecutive) Among • AtMostSeqCard ≡ AmongSeq ⊕ a cardinality constraint → AtMostSeqCard can be encoded with a Gen-Sequence Mohamed SIALA October 2012 CP 2012 4 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Context Sequence Constraints • Let [ x 1 , . . . , x n ] be a sequence of integer variables. • The Among constraint ensures that the number of occurrences of values in { v 1 .. v k } in a subsequence [ x i 1 , . . . , x iq ] is bounded between l and u . • AmongSeq : the conjunction of all n − q + 1 Among on q consecutive variables (i.e. � n − q i =0 Among ([ x i +1 , . . . , x i + q ])). • Gen-Sequence : Conjunction of (consecutive) Among • AtMostSeqCard ≡ AmongSeq ⊕ a cardinality constraint → AtMostSeqCard can be encoded with a Gen-Sequence → AtMostSeqCard can be encoded with a Global Sequencing Constraint ( Gsc ) Mohamed SIALA October 2012 CP 2012 4 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Existing complexities Gen-Sequence • cost-Regular encoding: O (2 q n ) [Van Hoeve et al, 2009] • Gen-Sequence: O ( n 3 ) [Van Hoeve et al, 2009] • Flow-based Algorithm: O ( n 2 ) [Maher et al, 2008] Gsc • Gcc encoding, Not AC, NP-Hard [Puget and R´ egin, 1997] Mohamed SIALA October 2012 CP 2012 5 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Why the AtMostSeqCard constraint? [1] Figure: The car-sequencing problem Mohamed SIALA October 2012 CP 2012 6 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work Why the AtMostSeqCard constraint? [2] 7 days, 4 employees, 3 periods, 40h per week, Atmost(1,3) D E N D E N D E N D E N D E N D E N D E N d emp 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 5 emp 2 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 5 emp 3 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 5 emp 4 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 5 Table: Crew-rostering problem Mohamed SIALA October 2012 CP 2012 7 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work The proposed algorithm • Let ( x 1 , . . . , x n ) be a boolean sequence subject to AtMostSeqCard ( u , q , d , [ x 1 , . . . , x n ]) • Our filtering algorithm is based on a greedy procedure (denoted by leftmost ). • leftmost : computes an assignment w maximizing the cardinality of the sequence with respect to the AtMost constraints. Mohamed SIALA October 2012 CP 2012 8 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work − → w = leftmost ( u = 2, q = 4) c x i w max 1 2 3 4 . 0 0 0 . 0 1 1 . 0 . 0 . 0 0 0 . 0 0 0 1 1 . 0 . 0 1 1 . 0 . 0 Mohamed SIALA October 2012 CP 2012 9 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work − → w = leftmost ( u = 2, q = 4) c x i w max 1 2 3 4 → . — 0 0 0 0 . 0 1 1 . 0 . 0 . 0 0 0 . 0 0 0 1 1 . 0 . 0 1 1 . 0 . 0 Mohamed SIALA October 2012 CP 2012 9 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work − → w = leftmost ( u = 2, q = 4) c x i w max 1 2 3 4 → . — 0 0 0 0 — 0 . 0 1 1 . 0 . 0 . 0 0 0 . 0 0 0 1 1 . 0 . 0 1 1 . 0 . 0 Mohamed SIALA October 2012 CP 2012 9 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work − → w = leftmost ( u = 2, q = 4) c x i w max 1 2 3 4 → . — 0 0 0 0 0 — 0 . — 0 1 1 . 0 . 0 . 0 0 0 . 0 0 0 1 1 . 0 . 0 1 1 . 0 . 0 Mohamed SIALA October 2012 CP 2012 9 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work − → w = leftmost ( u = 2, q = 4) c x i w max 1 2 3 4 → . — 0 0 0 0 1 0 — 0 . — 0 1 — 1 . 0 . 0 . 0 0 0 . 0 0 0 1 1 . 0 . 0 1 1 . 0 . 0 Mohamed SIALA October 2012 CP 2012 9 / 23

The AtMostSeqCard constraint Filtering the domains Experimental results LAAS-CNRS Conclusion & Future work − → w = leftmost ( u = 2, q = 4) c x i w max 1 2 3 4 . 0 0 0 0 1 1 0 0 . 0 1 1 . 0 . 0 . 0 0 0 . 0 0 0 1 1 . 0 . 0 1 1 . 0 . 0 Mohamed SIALA October 2012 CP 2012 9 / 23