Global%20Constraints

1
Global Constraints

• Toby Walsh
• National ICT Australia and
• University of New South Wales
• www.cse.unsw.edu.au/tw

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
  X1,n-gtD
• For example
• X1a, X2b, X3a

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
  X1,n-gtD
• For example
• X1a, X2b, X3a
• 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
  X1,n-gtD
• 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 TN is NValues
• RANGE(X1,..Xn,1,..n,T) and Tn 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

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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

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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

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RANGE constraint

• Restricts range of function to a given subset
• RANGE(X1,..,Xn,S,T) iff X(S)T
• Polynomial to make HC
• O(ndnt3/2) where dmax(dom(Xi)) and tlb(T)

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ROOTS constraint

• Restricts domain to those mapping to a given
  subset
• ROOTS(X1,..,Xn,S,T) iff SX-1(T)

S
T
1,n
20
ROOTS constraint

• Restricts domain to those mapping to a given
  subset
• ROOTS(X1,..,Xn,S,T) iff SX-1(T)
• E.g. ROOTS(3,1,2,3,0,1,4,5,0,3)

S
T
1,n
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ROOTS constraint

• Restricts domain to those mapping to a given
  subset
• ROOTS(X1,..,Xn,S,T) iff SX-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 SX-1(T)
• Useful to describe variables using particular
  values
• An example
• ROOTS(X1,..,Xn,S,d1,..,dm) SN is AMONG
  constraint

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ROOTS constraint

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

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ROOTS constraint

• ROOTS(X1,..,Xn,S,T) iff SX-1(T)
• Reduction from 3SAT in N vars and M clauses
• nNM
• Xi in i,-i for 1ltiltN
• Xi i iff xi is false

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ROOTS constraint

• ROOTS(X1,..,Xn,S,T) iff SX-1(T)
• Reduction from 3SAT in N vars and M clauses
• nNM
• Xi in i,-i for 1ltiltN
• XNj in i,-k,l iff jth clause is (xi or -xk or
  xl)

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ROOTS constraint

• ROOTS(X1,..,Xn,S,T) iff SX-1(T)
• Reduction from 3SAT in N vars and M clauses
• nNM
• Xi in i,-i for 1ltiltN
• XNj in i,-k,l iff jth clause is (xi or -xk or
  xl)
• subseteq T subsetqe Union_i i, -i

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ROOTS constraint

• ROOTS(X1,..,Xn,S,T) iff SX-1(T)
• Reduction from 3SAT in N vars and M clauses
• nNM
• Xi in i,-i for 1ltiltN
• XNj in i,-k,l iff jth clause is (xi or -xk or
  xl)
• subseteq T subseteq Union_i i, -i
• SN1, NM

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Specification language

• Equalities and inequalities
• X N, XN, ...
• Set constraints
• S subseteq T, SN, ...
• Two global constraints
• ROOTS and RANGE

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Some examples

• ALLDIFFERENT(X1,..,Xn) iff RANGE(X1,..,Xn,1,.
  .,n,T) Tn
• 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

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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

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Some examples

• NVALUES(X1,..,Xn,N)
• Xi 1in N
• RANGE(X1,..,Xn,1,..,n,T) TN
• 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 Xidj N
• ROOTS(X1,..,Xn,S,d1,..,dm) SN
• enforcing HC on decomposition enforces GAC on
  AMONG
• again example of where specification language is
  a good way to implement a specific global
  constraint

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Some examples

• COMMON(N,M,X1,..,Xn,Y1,..,Ym)
• i ?j . XiYj N and j ?i . XiYj M
• RANGE(Y1,..,Ym,1,..m,T)
• ROOTS(X1,..,Xn,S,T) SN
  RANGE(X1,..,Xn,1,..n,V)
• ROOTS(Y1,..,Yn,U,V) UM

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Some examples

• COMMON(N,M,X1,..,Xn,Y1,..,Ym)
• i ?j . XiYj N and j ?i . XiYj 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
• Bij iff j in S
• ROOTS(B1,..Bn,S,1)
• Decomposition clearly does not hinder propagation
• ROOTS is polynomial to enforce HC

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Some examples

• NOTALLEQUAL(X1,..Xn)
• Nasty large disjunction
• RANGE(X1,..Xn,1,..n,T) Tgt1

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Other examples

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

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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

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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)