Global Constraints

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
SLIDE meta-constraint

• Even hotter off the press than the value
  PRECEDENCE constraint
• Under review for IJCAI 07!

4
SLIDE meta-constraint

• A constructor for generating many sequencing and
  related global constraints
• REGULAR
• CONTIGUITY
• LEX
• CARD PATH
• Slides a constraint down one or more sequences of
  variables
• Ensuring constraint holds at every point
• Fixed parameter tractable

5
Basic SLIDE

• SLIDE(C,X1,..Xn) holds iff
• C(Xi,..Xik) holds for every i
• AMONG SEQ constraint
• Used by ILOG for assembly line car sequencing at
  Renault
• At most 1 in 3 cars have a sun roof
• SLIDE(C,X1,..Xn) where C(X1,X2,X3) holds iff
  AMONG(X1,X2,X3,0,1,D) where D is set of cars
  ordered with sunroofs

6
GSC

• Global sequence constraint in ILOG Solver
• Combines AMONG SEQ with GCC
• Regin and Puget give partial propagator
• We can show why!

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GSC

• Global sequence constraint in ILOG Solver
• Combines AMONG SEQ with GCC
• Regin and Puget give partial propagator
• We can show why!
• It is NP-hard to enforce GAC on GSC
• Can actually prove this when GSC is AMONG SEQ
  plus an ALL DIFFERENT
• ALL DIFFERENT is a special case of GCC

8
GSC

• Reduction from 1in3 SAT on positive clauses
• The jth block of 2N clauses will ensure jth
  clause
• Even numbered CSP vars represent truth assignment
• Odd numbered CSP vars junk to ensure N odd
  values in each block
• X_2jN2i odd iff xi true
• AMONG SEQ(X1,,N,N,2N,1,3,..)
• This ensures truth assignment repeated along
  variables!

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GSC

• Reduction from 1in3 SAT on positive clauses
• Suppose jth clause is (x or y or z)
• X_2jN2x, X_2jN2y, X_2jN2z in 4NM4j,
  4NM4j1, 4NM4j2
• As ALL DIFFERENT, only one of these odd
• For i other than x, y or z, X_2jN2i in 4jN4i,
  4jN4i1 and X_2jN2i1 in 4jN4i2, 4jn4i3

10
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Introduce sequence of Boolean vars B1,..Bn1
• Play role of alpha in LEX propagator

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SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10, and Bn11 (strict lex), Bn1 in 0,1
  (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1) holds iff
  C(Xi,Yi,Bi,Bi1) holds for each i

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SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10, and Bn11 (strict lex), Bn1 in 0,1
  (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1)
• C(Xi,Yi,Bi,Bi1) holds iff Bi1 or (BiBi10 and
  XiYi) or (Bi0, Bi11 and XiltYi)

13
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10, and Bn11 (strict lex), Bn1 in 0,1
  (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1)
• C(Xi,Yi,Bi,Bi1) holds iff Bi1 or (BiBi10 and
  XiYi) or (Bi0, Bi11 and XiltYi)
• Highly efficient, incremental, ..

14
SLIDE down multiple sequences

• CONTIGUITY(X1,..Xn)
• 001100
• Two simple SLIDEs

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SLIDE down multiple sequences

• CONTIGUITY(X1,..Xn)
• 001100
• Two simple SLIDEs
• Introduce Yi in 0,1,2
• SLIDE(gt,Y1,..Yn)

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SLIDE down multiple sequences

• CONTIGUITY(X1,..Xn)
• 001100
• Two simple SLIDEs
• Introduce Yi in 0,1,2
• SLIDE(gt,Y1,..Yn)
• SLIDE(C,X1,..Xn,Y1,..Yn) where C(Xi,Yi) holds
  iff Xi1 lt-gt Yi1

17
SLIDE down multiple sequences

• REGULAR(Q,X1,..Xn)
• X1 .. Xn is a string accepted by FDA Q
• Encodes into simple SLIDE
• Introduce Yi to represent state of the automaton
  after i symbols

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SLIDE down multiple sequences

• REGULAR(Q,X1,..Xn)
• X1 .. Xn is a string accepted by FDA Q
• Introduce Yi to represent state of the automaton
  after i symbols
• SLIDE(C,X1,..Xn,Y1,..Yn1) where
• Y1 is starting state of Q
• Yn1 is limited to accepting states of Q
• C(Xi,Yi,Yi1) holds iff Q moves from state Yi to
  state Yi1 on seeing Xi

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SLIDE down multiple sequences

• REGULAR(A,X1,..Xn)
• X1 .. Xn is a string accepted by FDA A
• Introduce Qi to represent state of the automaton
  after i symbols
• SLIDE(C,X1,..Xn,Q1,..Qn1) where
• Y1 is starting state of A
• Yn1 is limited to accepting states of A
• C(Xi,Qi,Qi1) holds iff A moves from state Qi to
  state Qi1 on seeing Xi
• Gives highly efficient and effective propagator!

20
SLIDE with counters

• AMONG(X1,..Xn,v,N)
• Introduce sequence of counts, Yi
• SLIDE(C,X1,..Xn,Y1,..Yn1) where
• Y10, Yn1N
• C(Xi,Yi,Yi1) holds iff (Xi in v and Yi11Yi)
  or (Xi not in v and Yi1Yi)

21
SLIDE with counters

• CARD PATH
• SLIDE is a special case of CARD PATH
• SLIDE(C,X1,..Xn) iff CARD PATH(C,X1,..Xn,n-k1
  )

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SLIDE with counters

• CARD PATH
• SLIDE is a special case of CARD PATH
• SLIDE(C,X1,..Xn) iff CARD PATH(C,X1,..Xn,n-k1
  )
• CARD PATH is a special case of SLIDE
• SLIDE(D,X1,..Xn,Y1,..Yn1) where
• Y10, Yn1N, and
• D(Xi,..Xik,Yi,Yi1) holds iff
• (Yi11Yi and C(Xi,..Xik)) or
• (Yi1Yi and not C(Xi,..Xik))

23
SLIDE with parameters

• Slide constraints may share parameters
• LINKSET2BOOLEANS(S,X1,..Xn)
• Converts set variable into characteristic
  function
• Encodes as SLIDE(C,X1,..Xn) where
• C(S,Xi) holds iff Xi in S
• S is parameter common to each slide constraint

24
SLIDE over sets

• Value precedence for set vars
• PRECENDCE(vj,vk,S1,..Sn) holds iff
• min(i,i vj in Si and vk not in Si or in1) lt
• min(i,i vk in Si and vj not in Si or in2)

25
SLIDE over sets

• Value precedence for set vars
• PRECENDCE(vj,vk,S1,..Sn) holds iff
• min(i,i vj in Si and vk not in Si or in1) lt
• min(i,i vk in Si and vj not in Si or in2)
• Introduce sequence of Booleans to indicate
  whether vars have been distinguished apart yet or
  not

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SLIDE over sets

• Value precedence for set vars
• PRECENDCE(vj,vk,S1,..Sn) holds iff
• min(i,i vj in Si and vk not in Si or in1) lt
• min(i,i vk in Si and vj not in Si or in2)
• SLIDE(C,S1,..Sn,B1,..Bn1) where
• B10 and
• C(Si,Bi,Bi1) holds iff
• BiBi11,
• or BiBi10 and (vj, vk in Si or vj, vk not
  in Si), or Bi0, Bi11, vj in Si and vk not in Si

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SLIDE over sets

• Open stacks problem
• IJCAI 05 modelling challenge
• Three SLIDEs and one ALL DIFFERENT
• First SLIDE Si1 Si u customer(Xi)
• Second SLIDE Ti-1 Ti u customer(Xi)
• Third SLIDE Si intersect Ti lt OpenStacks

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Circular SLIDE

• STRETCH used in shift rostering
• Given sequence of vars X1,.. Xn
• Each stretch of identical values a occurs at
  least shortest(a) and at most longest(a) time
• For example, at least 0 and at most 3 night
  shifts in a row
• Each transition Xi/Xi1 is limited to given
  patterns
• For example, only Xinight, Xi1off is permitted

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Circular SLIDE

• STRETCH can be efficiently encoded using SLIDE
• SLIDE(C,X1,..Xn,Y1,..Yn1) where
• Y11
• C(Xi,Xi1,Yi,Yi1) holds iff
• XiXi1, Yi11Yi, Yi1ltlongest(Xi),
• or Xi/Xi1, Yigtshortest(Xi) and (Xi,Xi1) in
  set of permitted changes

30
Circular SLIDE

• Circular forms of STRETCH are needed for
  repeating shift patterns
• Circular form of SLIDE useful in such situations
• SLIDEo(C,X1,..Xn) holds iff
• C(Xi,..X1(ik-1)mod n) holds for 1ltiltn

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SLIDE algebra

• SLIDEOR(C,X1..Xn) holds iff
• C(Xi,..Xik) holds for some I
• Encodes as CARD PATH (and thus as SLIDE)
• Other more complex combinations
• NOT(SLIDE(C,X1,..Xn)) iff SLIDEOR(C,X1,..Xn)
• SLIDE(C1,X1,..Xn) and SLIDE(C2,X1,..Xn) iff
  SLIDE(C1 and C2,X1,..Xn)
• ..

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Propagating SLIDE

• But how do we propagate global constraints
  expressed using SLIDE?

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Propagating SLIDE

• SLIDE(C,X1,..Xn)
• Just post sequence of constraints, C(Xi,..Xik)
• If constraint graph is Berge acyclic, then we
  will achieve GAC
• Gives efficient GAC propagators for CONTIGUITY,
  DOMAIN, ELEMENT, LEX, PRECEDENCE, REGULAR,

34
Propagating SLIDE

• SLIDE(C,X1,..Xn)
• Just post sequence of constraints, C(Xi,..Xik)
• If constraint graph is Berge acyclic, then we
  will achieve GAC
• Gives efficient GAC propagators for CONTIGUITY,
  DOMAIN, ELEMENT, LEX, PRECEDENCE, REGULAR,
• But what about case constraint graph is not
  Berge-acyclic?
• Slide constraints overlap on more than one
  variable

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Propagating SLIDE

• SLIDE(C,X1,..Xn)
• Slide constraints overlap on more than one
  variable
• Enforce GAC using dynamic programming
• pass support down sequence

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Propagating SLIDE

• SLIDE(C,X1,..Xn)
• Equivalently a dual encoding
• Consider AMONG SEQ(2,2,3,X1,..X5,a) where
• X1a, X2,X3,X4,X5 in a,b
• AMONG(X1,X2,X3,2,a)
• AMONG(X2,X3,X4,2,a)
• AMONG(X3,X4,X5,2,a)
• Enforcing GAC sets X4a
• GAC can be enforced in O(ndk1) time and O(ndk)
  space where constraints overlap on k variables
• Fixed parameter tractable

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Conclusions

• SLIDE is a very useful meta-constraint
• Many global constraints for sequencing and other
  problems can be encoded as SLIDE
• SLIDE can be propagated easily
• Constraints overlap on just one variable gt
  simply post slide constraints
• Constraints overlap on more than one variable gt
  use dynamic programming or equivalently a simple
  dual encoding