Introduction & Overviewuser.it.uu.se/~pierref/courses/COCP/slides/T12b-CP... · 2018. 10. 24. · ID2204: Constraint Programming Introduction & Overview Lecture 01, 2018-03-19 Christian
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ID2204: Constraint Programming
Introduction & OverviewLecture 01, 2018-03-19
Christian Schultecschulte@kth.se
Software and Computer SystemsSchool of Electrical Engineering and Computer ScienceKTH Royal Institute of TechnologySweden
2018-03-19
Sudoku
� Assign blank fields digits such that:digits distinct per rows, columns, blocks
9
2
2 5
9
7 3
6
2
6 9
7
4 9
1
8
6 3
4
6 8
1
8
ID2204, L01, Christian Schulte, EECS, KTH 4
2018-03-19
Propagation
� Prune digits from fields such that:digits distinct per rows, columns, blocks
9
2
2 5
9
7 3
6
2
6 9
7
4 9
1
8
6 3
4
6 8
1
8
1,2,3,4,5,6,7,8,9
ID2204, L01, Christian Schulte, EECS, KTH 10
2018-03-19
Propagation
� Prune digits from fields such that:digits distinct per rows, columns, blocks
9
2
2 5
9
7 3
6
96
2
7
1
94
8
6 3
4
6 8
1
8
1,3,5,6,7,8
ID2204, L01, Christian Schulte, EECS, KTH 11
2018-03-19
Propagation
� Prune digits from fields such that:digits distinct per rows, columns, blocks
9
2 9
52
7 3
6
2
6 9
7
4 9
1
8
6 3
86
4 1
8
1,3,6,7
ID2204, L01, Christian Schulte, EECS, KTH 12
2018-03-19
Propagation
� Prune digits from fields such that:digits distinct per rows, columns, blocks
9
2
2 5
9
7 3
6
2
6 9
7
4 9
1
8
6 3
4
6 8
1
8
1,3,6
ID2204, L01, Christian Schulte, EECS, KTH 13
2018-03-19
Iterated Propagation
� Iterate propagation for rows, columns, blocks� What if no assignment: search... later
9
2
2 5
9
7 3
6
2
6 9
7
4 9
1
8
6 3
4
6 8
1
8
ID2204, L01, Christian Schulte, EECS, KTH 14
2018-03-19
Running Example: SMM
� Find distinct digits for letters, such that
SEND
+ MORE
= MONEY
ID2204, L01, Christian Schulte, EECS, KTH 19
2018-03-19
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+YSz0 Mz0
ID2204, L01, Christian Schulte, EECS, KTH 20
2018-03-19
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
ID2204, L01, Christian Schulte, EECS, KTH 22
Constraint Propagation
2018-03-19
Constraint Store
� Maps variables to possible values� Others: finite sets, intervals, trees, ...
x�{3,4,5} y�{3,4,5}
finite domain constraints
ID2204, L01, Christian Schulte, EECS, KTH 26
2018-03-19
Propagators
� Implement (non-basic) constraints
distinct(x1,…,xn)
x + 2*y = z
ID2204, L01, Christian Schulte, EECS, KTH 27
2018-03-19
Propagators
� Amplify store by constraint propagation
x�{3,4,5} y�{3,4,5}
xty y>3
ID2204, L01, Christian Schulte, EECS, KTH 29
2018-03-19
Propagators
� Amplify store by constraint propagation
x�{3,4,5} y�{4,5}
xty y>3
ID2204, L01, Christian Schulte, EECS, KTH 30
2018-03-19
Propagators
� Amplify store by constraint propagation
x�{3,4,5} y�{4,5}
xty y>3
ID2204, L01, Christian Schulte, EECS, KTH 31
2018-03-19
Propagators
� Amplify store by constraint propagation
x�{4,5} y�{4,5}
xty y>3
ID2204, L01, Christian Schulte, EECS, KTH 32
2018-03-19
Propagators
� Amplify store by constraint propagation� Disappear when done (subsumed, entailed)
� no more propagation possible
x�{4,5} y�{4,5}
xty y>3
ID2204, L01, Christian Schulte, EECS, KTH 33
2018-03-19
Propagators
� Amplify store by constraint propagation� Disappear when done (subsumed, entailed)
� no more propagation possible
x�{4,5} y�{4,5}
xty
ID2204, L01, Christian Schulte, EECS, KTH 34
2018-03-19
Propagation for SMM
� Results in storeS�{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
ID2204, L01, Christian Schulte, EECS, KTH 35
2018-03-19
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
ID2204, L01, Christian Schulte, EECS, KTH 36
Search
2018-03-19
Search: Branching
� Create subproblems with additional information� enable further constraint propagation
x�{4,5} y�{4,5}
xty
x�{4} y�{4}
xty
x�{5} y�{4,5}
xty
x=4 xz4
ID2204, L01, Christian Schulte, EECS, KTH 39
2018-03-19
Example Branching Strategy
� Pick variable x with at least two values� Pick value n from domain of x� Branch with
x=n and xzn
� Part of model
ID2204, L01, Christian Schulte, EECS, KTH 40
2018-03-19
Search: Exploration
� Iterate propagation and branching� Orthogonal: branching ' exploration� Nodes:
x Unsolved x Failed x Succeeded
ID2204, L01, Christian Schulte, EECS, KTH 41
2018-03-19
SMM: Unique Solution
SEND+ MORE= MONEY
9567+ 1085= 10652
ID2204, L01, Christian Schulte, EECS, KTH 42
2018-03-19
Heuristics for Branching
� Which variable� least possible values (first-fail)� application dependent heuristic
� Which value� minimum, median, maximum
x=m or xzm� split with median m
x<m or xtm
� Problem specific
ID2204, L01, Christian Schulte, EECS, KTH 43
2018-03-19
SMM: Solution With First-fail
SEND+ MORE= MONEY
9567+ 1085= 10652
ID2204, L01, Christian Schulte, EECS, KTH 44
2018-03-19
Send Most Money (SMM++)
� Find distinct digits for letters, such that
and MONEY maximal
SEND
+ MOST
= MONEY
ID2204, L01, Christian Schulte, EECS, KTH 45
2018-03-19
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
ID2204, L01, Christian Schulte, EECS, KTH 46
2018-03-19
Branch-and-bound Search
� Find first solutionID2204, L01, Christian Schulte, EECS, KTH 47
2018-03-19
Branch-and-bound Search
� Explore with additional constraintID2204, L01, Christian Schulte, EECS, KTH 49
2018-03-19
Branch-and-bound Search
� Guarantees better solutionsID2204, L01, Christian Schulte, EECS, KTH 51
2018-03-19
Branch-and-bound Search
� Last solution bestID2204, L01, Christian Schulte, EECS, KTH 52
2018-03-19
Branch-and-bound Search
� Proof of optimalityID2204, L01, Christian Schulte, EECS, KTH 53
2018-03-19
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
ID2204, L01, Christian Schulte, EECS, KTH 54
2018-03-19
SMM++: Branch-and-bound
SEND+ MOST= MONEY
9782+ 1094= 10876
ID2204, L01, Christian Schulte, EECS, KTH 55
2018-03-19
SMM: Strong Propagation
SEND+ MORE= MONEY
9567+ 1085= 10652
ID2204, L01, Christian Schulte, EECS, KTH 63
2018-03-19
Acknowledgments
� I am grateful to Pierre Flener for helpful comments and bugreports on these slides
ID2204, L01, Christian Schulte, EECS, KTH 97
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