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Global Constraints

Global Constraints . Toby Walsh NICTA and University of New South Wales www.cse.unsw.edu.au/~tw. Quick advert. UNSW is in Sydney Regularly voted in top 10 cities in World UNSW is one of top universities in Australia In top 100 universities in world Talk to me about our PhD programme!

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Global Constraints

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  1. Global Constraints Toby Walsh NICTA and University of New South Wales www.cse.unsw.edu.au/~tw

  2. Quick advert • UNSW is in Sydney • Regularly voted in top 10 cities in World • UNSW is one of top universities in Australia • In top 100 universities in world • Talk to me about our PhD programme! • Also happy to have PhDs/PostDocs visit for weeks/months/years …

  3. Quick advert • UNSW is in Sydney • Regularly voted in top 10 cities in World • UNSW is one of top universities in Australia • In top 100 universities in world • Talk to me about our PhD programme! • Also happy to have PhDs/PostDocs visit for weeks/months/years … • Attend CP/KR/ICAPS in Sept

  4. Constraint programming • “Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it”. • Gene Freuder, Constraints, April 1997

  5. Suduko Puzzle = [X11,X12,X13, …X19, X21,X22, ………X29, .. X91,X92, ……….X99] Puzzle :: 1..9 alldifferent([X11,..X19]) alldifferent([X21,..X29]) .. alldifferent([X11,..X91]) .. alldifferent([X11,..X33]) ..

  6. (I)LP everything has to be a linear inequality limited range of solution methods (simplex, ...) good at proving optimality Comparison with OR • CP • rich modelling language • many different global constraints • fast local inference on these global constraints • good for finding feasible and tightly constrained solutions

  7. Global constraints • Any non-binary constraint • AllDifferent • Nvalues • Element • Lex ordering • Stretch constraint • Sequence constraint • ... • Each comes with an efficient propagator ...

  8. All Different

  9. Golomb rulers • Mark ticks on a rule • Distance between any two ticks (not just neighbouring) is distinct • Special type of graceful graph • Applications • Radio-astronomy, crystallorgraphy, ... • Prob006 in CSPLib

  10. Golomb rulers • Simple solution • Exponentially long ruler • Ticks at 0, 1, 3, 7, 15, 31, ... • Goal is to find minimal length ruler • Sequence of optimization problems • Is there a ruler of length m? • Is there a ruler of length m-1? • ...

  11. Optimal Golomb rulers • Known for up to 23 ticks • Distributed internet project to find larger • 0,1 • 0,1,3 • 0,1,4,6 • 0,1,4,9,11 • 0,1,4,10,12,17 • Solutions grow as approximately O(n^2)

  12. Golomb rulers as CSP • Variable Xi for each tick • Value is position • Auxiliary variable Dij for each inter-tick distance • Dij=|Xi-Xj| • Two (global) constraints • X1<X2<..Xn • AllDifferent(D11,D12,D13,...)

  13. Golomb ruler as CSP • Not yet achieved “dream” of declarative programming • Need to break symmetry of inverting ruler • D12< Dn-1n • Add implied constraints • D12<D13 ... • Pure declarative specifications not quite enough!

  14. AllDifferent • AllDifferent([X1,..Xn]) iff Xi=/=Xj for i<j • Useful in wide range of applications • Timetabling (exams with common student must occur at different times) • Production (each product must be produced at a different time) • ... • Can propagate just using binary inequalities • But this decomposition hurts propagation

  15. AllDifferent • AllDifferent([X1,..Xn]) iff Xi=/=Xj for i<j • Can propagate just using binary inequalities • But this decomposition hurts propagation • X1 in {1,2}, X2 in {1,2}, X3 in {1,2,3} • X3 cannot be 1 or 2 • How can we automate such reasoning? • How can we do this efficiently (less than O(n^2) time)

  16. AllDifferent • One of the oldest global constraints • In ALICE language [Lauriere 78] • Found in every constraint solver today • GAC algorithm based on matching theory due to Regin [AAAI 94], runs in O(dn^3/2) • BC algorithm using combinatorics due to Puget [AAAI98], runs in O(nlogn)

  17. BC on AllDifferent • Application of Hall's Theorem • Sometimes called the “marriage” theorem • Given k sets • Then there is an unique and distinct element in each set iff any union of j of the sets has at least j elements for 0<j<=k • E.g. S1={1,2}, S2={1,2} but not S1={1,2},S2={1,2} and S3={1,2}

  18. Hall's theorem • You wish to marry n men and women • Each woman declares who they are willing to marry (some set of men) • Each man will be “happy” with whoever is willing to marry them • Given any subset of j women, the number of men they are willing to marry must be j or more (thus this condition is necessary) • What is surprising is that it is also sufficient!

  19. BC on AllDifferent • Hall Interval • Interval of values in which as many variables as domain values • E.g. X1 in {1,2,3}, X2 in {1,2}, X3 in {1,2,3} • 3 variables in the interval [1..3] • AllDifferent([X1,..Xn]) is BC iff • Each interval, the number of vars it covers is less than or equal to the width of the interval • No variable outside a Hall Interval has a value within it

  20. BC on AllDifferent • Consider X1 in {1,2}, X2 in {1,2}, X3 in {1,2,3}

  21. BC on AllDifferent • Consider X1 in {1,2}, X2 in {1,2}, X3 in {1,2,3} • Then [1..2] is a Hall Interval covered by X1 and X2 • X3 has values inside this Hall Interval • We can prune these and make AllDifferent BC

  22. BC on AllDifferent • Naïve algorithm considers O(n^2) intervals • Puget orders intervals • Ordering has O(nlogn) cost • Then can go through them in order

  23. Beyond AllDifferent • NValues([X1,...,Xn],M) iff |{j | Xi=j}|=M • AllDifferent is special case when M=n • Useful when values represent a resource • Minimize the number of resources used

  24. Beyond AllDifferent • Global cardinality constraint • GCC([X1,..Xn],[a1,..am],[b1,...bm]) iff aj <= |{i | Xi=j }| <= bj for all j • In other words, j occurs between aj and bj times • Again useful when values represent a resource • You have at least one night shift but no more than four each week

  25. Conclusions • AllDifferent is one of the oldest (and most useful) global constraints • Efficient propagators exist for achieving GAC and BC • When to choose BC over GAC? • Heuristic choice: BC often best when many more values than variables, GAC when we are close to a permutation (number of vars=number of values)

  26. Lex ordering • Widely useful • Especially for symmetry breaking • Breaking row and column symmetry in matrix models • Available in most (all?) solvers • Good example of pointer based global constraint • Pointers save re-doing work • Good incremental behaviour • O(n) in general, but amortised O(n) cost down a branch

  27. Motivation • Many problems can be modelled by matrices of decision variables.

  28. Motivation • Many problems can be modelled by matrices of decision variables. • E.g. Combinatorial Problems • Balanced Incomplete Block Design. • X[i,j]=0 or 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 Parameters = (7,7,3,3,1) 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0

  29. Motivation • Many problems can be modelled by matrices of decision variables • Frequently these matrices exhibit row and/or column symmetry

  30. Motivation • Many problems can be modelled by matrices of decision variables • Frequently these matrices exhibit row and/or column symmetry • We can permute the rows/columns in any (non)solution to obtain another (non)solution

  31. Motivation • When rows and columns can be permuted • An n by m matrix model with row and column symmetry has • n! m! symmetries • grows super-exponentially • Too many symmetric search states • It can be very expensive to visit all the symmetric branches of a search tree

  32. Motivation • Breaking symmetry is very important! • Breaking all row and column symmetries is difficult • No one has an effective way of dealing with all row and column symmetries. • Symmetry breaking methods have to deal with very large number of symmetries. • The effort required could easily be exponential.

  33. Symmetry in CP • Add symmetry breaking constraints • Leave at least one solution • Eliminate some/all symmetric solutions • Modify search algorithm • Ignore symmetric parts of the search space • Adapt branching heuristic • To explore branches which are most likely not to be symmetric

  34. Example • Consider 2 identical bins: A B

  35. Example • Consider 2 identical bins: A B • We must pack 6 items: 1 2 3 4 5 6

  36. Example • Here is one solution: 5 6 3 4 1 2 A B

  37. Example • Here is another: 6 5 4 3 2 1 A B

  38. Example • Is there any fundamental difference? 5 6 a) 3 4 1 2 A B 6 5 b) 4 3 2 1 A B

  39. Example • Consider a matrix model: 1 2 3 4 5 6 A 1 0 1 0 1 0 B 0 1 0 1 0 1 5 6 a) 3 4 1 2 A B 1 2 3 4 5 6 A 0 1 0 1 0 1 B 1 0 1 0 1 0 6 5 b) 4 3 2 1 A B

  40. If we insist that row A lex row B, we remove a) from the solution set. Example • Consider a matrix model: 1 2 3 4 5 6 A 1 0 1 0 1 0 B 0 1 0 1 0 1 1 2 3 4 5 6 A 0 1 0 1 0 1 B 1 0 1 0 1 0 6 5 b) 4 3 2 1 A B

  41. Example • Notice that items 3 and 4 are identical. 1 2 3 4 5 6 A 1 0 1 0 1 0 B 0 1 0 1 0 1 6 5 b) 4 3 2 1 A B 1 2 3 4 5 6 A 0 1 1 0 0 1 B 1 0 0 1 1 0 6 5 c) 3 4 2 1

  42. If we insist that col 3 lex col 4, we remove c) from the solution set. Example • Notice that items 3 and 4 are identical. 1 2 3 4 5 6 A 1 0 1 0 1 0 B 0 1 0 1 0 1 6 5 b) 4 3 2 1 A B 1 2 3 4 5 6 A 0 1 1 0 0 1 B 1 0 0 1 1 0

  43. Lexicographic Ordering • Used to order dictionaries [A,B,C] ≤ lex [D,E,F] • A<D or • (A=D and B<E ) or • (A=D and B=E and C<F) or • (A=D and B=E and C=F)

  44. Breaking Row (Column) Symmetry • Lexicographic ordering is total • Forcing the rows to be lexicographically ordered breaks all row symmetry • lexicographic ordering • [A B C] lex [D E F] lex [G H I] • anti-lexicographic ordering • [G H I]

  45. Breaking Row and Column Symmetries • Breaking both row and column symmetries is difficult • Rows and columns intersect • After constraining the rows to be lexicographically ordered • we distinguish the columns • the columns are not symmetric anymore!

  46. Good News  • Each symmetry class of assignments has at least one element where both the rows and the columns are lexicographically ordered • But there may be no element with rows lex ordered and columns anti-lex ordered • To break row and column symmetries, we can insist that the rows and columns are both lexicographically ordered (double-lex) • Extends to higher dimensions

  47. Bad News  • A symmetry class of assignments may have more than one element where both the rows and the columns are lexicographically ordered • Double-lex does not break all row and column symmetries 0 1 0 1 0 1 1 0 swap the columns swap row 1 and row 3 1 0 1 0

  48. GACLex • A new family of global constraints.

  49. GACLex • A new family of global constraints. Non-binary constraint. Specialised propagator.

  50. GACLex • A new family of global constraints • Linear time complexity

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