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Global Constraints

Toby Walsh National ICT Australia and University of New South Wales www.cse.unsw.edu.au/~tw. Global Constraints . Course outline. Introduction All Different Lex ordering Value precedence Complexity GAC-Schema Soft Global Constraints Global Grammar Constraints Roots Constraint

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Global Constraints

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

  2. Course outline • Introduction • All Different • Lex ordering • Value precedence • Complexity • GAC-Schema • Soft Global Constraints • Global Grammar Constraints • Roots Constraint • Range Constraint • Slide Constraint • Global Constraints on Sets

  3. ROOTS and RANGE • Large catalog of global constraints • but very little taxonomy? • global grammar constraints specify a “family” of useful global constraints • ROOTS and RANGE let us define many counting and occurrence constraints

  4. Counting and Occurrence Constraints • Many (resource bounded) optimization & decision problems contain: • occurrence constraints (on values that occur)

  5. Counting and Occurrence Constraints • Many (resource bounded) optimization & decision problems contain: • occurrence constraints (on values that occur) • E.g. the value “Thur 3-5pm” should occur once in the timetable of any PhD student following the course on global constraints

  6. Counting and Occurrence Constraints • Many (resource bounded) optimization & decision problems contain: • counting constraints (on number of vals or vars satisfying a condition)

  7. Counting and Occurrence Constraints • Many (resource bounded) optimization & decision problems contain: • counting constraints (on number of vals or vars satisfying a condition) • E.g. the value “night shift” can be used by at most 3 variables in any 7!

  8. Using ROOTS and RANGE • Simple, declarative language for specifying many counting & occurrence constraints • two new primitives: • ROOTS and RANGE global constraints • Executable language • propagate primitives • need just to provide propagators within the constraint solver for ROOTS and RANGE

  9. Specification language • Based around properties of functions • consider sequence of vars X1,..,Xn each taking a value in a D • then one view of X1,..,Xn is a function: X:[1,n]->D • For example X1=a, X2=b, X3=a …

  10. Specification language • Based around properties of functions • consider sequence of vars X1,..,Xn each taking a value in a D • then one view of X1,..,Xn is a function: X:[1,n]->D • For example X1=a, X2=b, X3=a … X(1)=a, X(2)=b, X(3)=a …

  11. Specification language • Based around properties of functions • consider sequence of vars X1,..,Xn each taking a value in a D • then one view of X1,..,Xn is a function: X:[1,n]->D • This view is useful for specifying global constraints • NVALUES([X1,..,Xn],N) is equivalent to |RANGE(X)|=N

  12. RANGE constraint • Restricts range of function to a given subset • RANGE([X1,..,Xn],S,T) iff X(S)=T S T [1,n]

  13. RANGE constraint • Restricts range of function to a given subset • RANGE([X1,..,Xn],S,T) iff X(S)=T e.g. RANGE([1,3,1,2,5],{2,4,5},{2,3,5}) S T [1,n]

  14. RANGE constraint • Restricts range of function to a given subset • RANGE([X1,..,Xn],S,T) iff X(S)=T • useful to describe values used (when values are resources) • Some examples • RANGE([X1,..Xn],{1,..n},T) and |T|=N is NValues • RANGE([X1,..Xn],{1,..n},T) and |T|=n is AllDifferent • …

  15. RANGE constraint • RANGE constraint involves both finite-domain variables (X1,..,Xn) and set variables(S,T) • to describe propagation, need to define new type of local consistency • hybrid consistency is a local consistency property for global constraints involving both finite domain and set variables

  16. Hybrid Consistency • Constraint is “hybrid consistent (HC) • for each a in dom(X), there is a support • each a in ub(S) appears in one support • each a in lb(S) appears in all supports • Reduces to GAC on finite domain vars,and BC on set vars

  17. Hybrid Consistency • Constraint is “hybrid consistent (HC) • for each a in dom(X), there is a support • each a in ub(S) appears in one support • each a in lb(S) appears in all supports • Consider RANGE([X1,..X4],S,T) where • X1,X2 in {1,2,5}, X3 in {3,4} and X4 in {1} • {4} subseteq S subseteq {1,2,3,4} • {} subseteq T subseteq {1,2}

  18. RANGE constraint • Restricts range of function to a given subset • RANGE([X1,..,Xn],S,T) iff X(S)=T • Polynomial to make HC • O(nd+nt^3/2) where d=max(dom(Xi)) and t=|lb(T)|

  19. ROOTS constraint • Restricts domain to those mapping to a given subset • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) S T [1,n]

  20. ROOTS constraint • Restricts domain to those mapping to a given subset • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) • E.g. ROOTS([3,1,2,3,0],{1,4,5},{0,3}) S T [1,n]

  21. ROOTS constraint • Restricts domain to those mapping to a given subset • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) • E.g. RANGE([3,1,2,3,0],{1,5},{0,3}) S T [1,n]

  22. ROOTS constraint • Restricts domain to those mapping to a given subset • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) • Useful to describe variables using particular values • An example • ROOTS([X1,..,Xn],S,{d1,..,dm}) & |S|=N is AMONG constraint

  23. ROOTS constraint • Restricts domain to those mapping to a given subset • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) • NP-hard to make HC • but O(nd) when Xi or T are ground (all cases here!)

  24. ROOTS constraint • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) • Reduction from 3SAT in N vars and M clauses • n=N+M • Xi in {i,-i} for 1<=i<=N • Xi = i iff xi is false

  25. ROOTS constraint • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) • Reduction from 3SAT in N vars and M clauses • n=N+M • Xi in {i,-i} for 1<=i<=N • XN+j in {i,-k,l} iff jth clause is (xi or -xk or xl)

  26. ROOTS constraint • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) • Reduction from 3SAT in N vars and M clauses • n=N+M • Xi in {i,-i} for 1<=i<=N • XN+j in {i,-k,l} iff jth clause is (xi or -xk or xl) • {} subseteq T subsetqe Union_i {i, -i}

  27. ROOTS constraint • ROOTS([X1,..,Xn],S,T) iff S=X-1(T) • Reduction from 3SAT in N vars and M clauses • n=N+M • Xi in {i,-i} for 1<=i<=N • XN+j in {i,-k,l} iff jth clause is (xi or -xk or xl) • {} subseteq T subseteq Union_i {i, -i} • S={N+1, … N+M}

  28. Specification language • Equalities and inequalities • X≤ N, X=N, ... • Set constraints • S subseteq T, |S|=N, ... • Two global constraints • ROOTS and RANGE

  29. Some examples • ALLDIFFERENT([X1,..,Xn]) iff RANGE([X1,..,Xn],{1,..,n},T) & |T|=n • unfortunately enforcing HC on decomposition does not make ALLDIFFERENT constraint GAC • E.g. X1, X2 in {1,2}, X3 in {1,2,3,4} • {1,2} subseteq T subseteq {1,2,3,4} • so sometimes worth developing specialized propagators

  30. Some examples • PERMUTATION([X1,..,Xn]) iff RANGE([X1,..,Xn],{1,..,n},{1,..,n}) • special case of ALLDIFFERENT • clearly enforcing HC on “decomposition” makes constraint GAC • so sometimes this specification language is a good way to implement specific global constraints

  31. Some examples • NVALUES([X1,..,Xn,N]) • |{Xi | 1≤i≤n }|=N • RANGE([X1,..,Xn],{1,..,n},T) & |T|=N • enforcing HC on decomposition does not enforce GAC on NVALUES • but this is to be expected as it is NP-hard to do so! • one way (at least) to implement this global constraint

  32. Some examples • AMONG([X1,..,Xn],[d1,..,dm],N) • |{i | Xi=dj }|=N • ROOTS([X1,..,Xn],S,{d1,..,dm}) & |S|=N • enforcing HC on decomposition enforces GAC on AMONG • again example of where specification language is a good way to implement a specific global constraint

  33. Some examples • COMMON(N,M,[X1,..,Xn],[Y1,..,Ym]) • |{i | ∃j . Xi=Yj }|=N and |{j | ∃i . Xi=Yj }|=M RANGE([Y1,..,Ym],{1,..m},T) & ROOTS([X1,..,Xn],S,T) & |S|=N & RANGE([X1,..,Xn],{1,..n},V) & ROOTS([Y1,..,Yn],U,V) & |U|=M

  34. Some examples • COMMON(N,M,[X1,..,Xn],[Y1,..,Ym]) • |{i | ∃j . Xi=Yj }|=N and |{j | ∃i . Xi=Yj }|=M Enforcing HC on decomposition does not enforce GAC on COMMON • but to be expected as NP-hard to do so!

  35. Some examples • LINKSET2BOOLEANS(S,[B1,..Bn]) holds iff Bi=j iff j in S • ROOTS([B1,..Bn],S,{1}) • Decomposition clearly does not hinder propagation • ROOTS is polynomial to enforce HC

  36. Some examples • NOTALLEQUAL([X1,..Xn]) • Nasty large disjunction • RANGE([X1,..Xn],{1,..n},T) & |T|>1

  37. Other examples GCC SYMALLDIFFERENT ELEMENT DOMAIN CONTIGUITY • ... Whilst we can express many such global constraints, not all give effective propagators (e.g. GCC).

  38. Global grammar constraints • Filtering algorithms from finite sized automata/membership of a regular language • How do they compare? • Such approaches are complementary • these automata can specify CONTIGUITY • but not PERMUTATION

  39. Conclusions • Counting and Occurrence constraints can be specified using simple declarative language • needs two new global constraints: ROOTS and RANGE • Efficient and effective means to implement a number of such constraints: • when propagating specification achieves GAC • or constraint is NP-hard to propagate • however, we may still need to design a specialized propagator (e.g. gcc)

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