ID2204: Constraint Programming Introduction & Overview Lecture 01, 2018-03-19 Christian Schulte cschulte@kth.se Software and Computer Systems School of Electrical Engineering and Computer Science KTH Royal Institute of Technology Sweden
Sudoku 2 5 9 7 3 2 9 6 2 4 9 7 6 9 1 8 4 1 6 3 8 6 8 � Assign blank fields digits such that: digits distinct per rows, columns, blocks 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 4
Propagation 2 5 9 7 3 2 9 6 2 4 9 7 1,2,3,4,5,6,7,8,9 6 9 1 8 4 1 6 3 8 6 8 � Prune digits from fields such that: digits distinct per rows, columns, blocks 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 10
Propagation 2 5 9 7 3 2 9 6 2 4 9 7 1,3,5,6,7,8 6 9 1 8 4 1 6 3 8 6 8 � Prune digits from fields such that: digits distinct per rows , columns, blocks 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 11
Propagation 2 5 9 7 3 2 9 6 2 4 9 7 1,3,6,7 6 9 1 8 4 1 6 3 8 6 8 � Prune digits from fields such that: digits distinct per rows, columns , blocks 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 12
Propagation 2 5 9 7 3 2 9 6 2 4 9 7 1,3,6 6 9 1 8 4 1 6 3 8 6 8 � Prune digits from fields such that: digits distinct per rows, columns, blocks 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 13
Iterated Propagation 2 5 9 7 3 2 9 6 2 4 9 7 6 9 1 8 4 1 6 3 8 6 8 � Iterate propagation for rows, columns, blocks � What if no assignment: search... later 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 14
Running Example: SMM � Find distinct digits for letters, such that SEND + MORE = MONEY 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 19
Constraint Model for SMM � Variables: S,E,N,D,M,O,R,Y � {0,…,9} � Constraints: distinct(S,E,N,D,M,O,R,Y) 1000 × S+100 × E+10 × N+D + 1000 × M+100 × O+10 × R+E = 10000 × M+1000 × O+100 × N+10 × E+Y S � 0 M � 0 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 20
Finding a Solution � Compute with possible values � rather than enumerating assignments � Prune inconsistent values � constraint propagation � Search � branch: define search tree � explore: explore search tree for solution 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 22
Constraint Propagation
Constraint Store finite domain constraints x � {3,4,5} y � {3,4,5} � Maps variables to possible values � Others: finite sets, intervals, trees, ... 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 26
Propagators � Implement (non-basic) constraints distinct(x 1 ,…,x n ) x + 2*y = z 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 27
Propagators x � y y>3 x � {3,4,5} y � {3,4,5} � Amplify store by constraint propagation 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 29
Propagators x � y y>3 x � {3,4,5} y � {4,5} � Amplify store by constraint propagation 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 30
Propagators x � y y>3 x � {3,4,5} y � {4,5} � Amplify store by constraint propagation 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 31
Propagators x � y y>3 x � {4,5} y � {4,5} � Amplify store by constraint propagation 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 32
Propagators x � y y>3 x � {4,5} y � {4,5} � Amplify store by constraint propagation � Disappear when done (subsumed, entailed) no more propagation possible � 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 33
Propagators x � y x � {4,5} y � {4,5} � Amplify store by constraint propagation � Disappear when done (subsumed, entailed) no more propagation possible � 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 34
Propagation for SMM � Results in store S � {9} E � {4,…,7} N � {5,…,8} D � {2,…,8} M � {1} O � {0} R � {2,…,8} Y � {2,…,8} � Propagation alone not sufficient! � create simpler sub-problems � branching 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 35
Constraints and Propagators � Constraints state relations among variables � which value combinations satisfy constraint � Propagators implement constraints � prune values in conflict with constraint � Constraint propagation drives propagators for several constraints 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 36
Search
Search: Branching x � y x � {4,5} y � {4,5} x � 4 x=4 x � y x � y x � {4} y � {4} x � {5} y � {4,5} � Create subproblems with additional information enable further constraint propagation � 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 39
Example Branching Strategy � Pick variable x with at least two values � Pick value n from domain of x � Branch with x � n x=n and � Part of model 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 40
Search: Exploration � Iterate propagation and branching � Orthogonal: branching � exploration � Nodes: � Unsolved � Failed � Succeeded 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 41
SMM: Unique Solution SEND + MORE = MONEY 9567 + 1085 = 10652 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 42
Heuristics for Branching � Which variable � least possible values (first-fail) � application dependent heuristic � Which value � minimum, median, maximum x � m x=m or � split with median m x � m x<m or � Problem specific 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 43
SMM: Solution With First-fail SEND + MORE = MONEY 9567 + 1085 = 10652 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 44
Send Most Money (SMM++) � Find distinct digits for letters, such that SEND + MOST = MONEY and MONEY maximal 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 45
Best Solution Search � Naïve approach: � compute all solutions � choose best � Branch-and-bound approach: � compute first solution � add “betterness” constraint to open nodes � next solution will be “better” � prunes search space 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 46
Branch-and-bound Search � Find first solution 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 47
Branch-and-bound Search � Explore with additional constraint 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 49
Branch-and-bound Search � Guarantees better solutions 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 51
Branch-and-bound Search � Last solution best 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 52
Branch-and-bound Search � Proof of optimality 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 53
Modelling SMM++ � Constraints and branching as before � Order among solutions with constraints � so-far-best solution S,E,N,D,M,O,T,Y � current node S,E,N,D,M,O,T,Y � constraint added 10000 × M +1000 × O +100 × N +10 × E + Y < 10000 × M +1000 × O +100 × N +10 × E + Y 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 54
SMM++: Branch-and-bound SEND + MOST = MONEY 9782 + 1094 = 10876 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 55
SMM: Strong Propagation SEND + MORE = MONEY 9567 + 1085 = 10652 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 63
Acknowledgments � I am grateful to Pierre Flener for helpful comments and bugreports on these slides 2018-03-19 ID2204, L01, Christian Schulte, EECS, KTH 97
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