1 Constraints Foreign Keys Local and Global Constraints Triggers Source: slides by Jeffrey Ullman.
Global Constraints
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Global Constraints Toby Walsh National ICT Australia and University of New South Wales www.cse.unsw.edu.au/~tw
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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 - PowerPoint PPT Presentation
Transcript of Global Constraints
Global Constraints
Toby WalshNational ICT Australia and
University of New South Waleswww.cse.unsw.edu.au/~tw
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
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
Counting and Occurrence Constraints
●Many (resource bounded) optimization & decision problems contain:–occurrence constraints (on values that occur)
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
Counting and Occurrence Constraints
●Many (resource bounded) optimization & decision problems contain:–counting constraints (on number of vals or vars satisfying a condition)
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!
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
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 …
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 …
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
RANGE constraint
●Restricts range of function to a given subset–RANGE([X1,..,Xn],S,T) iff X(S)=T
[1,n]
S T
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})
[1,n]
S T
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●…
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
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
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}
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)|
ROOTS constraint
●Restricts domain to those mapping to a given subset–ROOTS([X1,..,Xn],S,T) iff S=X-1(T)
[1,n]
S T
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})
[1,n]
S T
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})
[1,n]
S T
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
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!)
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
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)
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}
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}
Specification language
●Equalities and inequalities–X≤ N, X=N, ...●Set constraints–S subseteq T, |S|=N, ...●Two global constraints–ROOTS and RANGE
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
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
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
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
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
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!
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
Some examples
● NOTALLEQUAL([X1,..Xn])
● Nasty large disjunction
● RANGE([X1,..Xn],{1,..n},T) & |T|>1
Other examples
GCCSYMALLDIFFERENTELEMENTDOMAINCONTIGUITY●...
Whilst we can express many such global constraints, not all give effective propagators (e.g. GCC).
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
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)
