Sliding Constraints

1
Sliding Constraints

• Toby Walsh
• National ICT Australia and
• University of New South Wales
• www.cse.unsw.edu.au/tw
• Joint work with Christian Bessiere, Emmanuel
  Hebrard, Brahim Hnich, Zeynep Kiziltan

2
SLIDE meta-constraint

• Hot off the press
• Some of this is to appear at ECAI 06
• Some is even under review for IJCAI 07!

3
SLIDE meta-constraint

• A constructor for generating many sequencing and
  related global constraints
• REGULAR Pesant 04
• AMONG SEQ Beldiceanu and Contegjean
  94
• CARD PATH Beldiceanu and Carlsson
  01
• VALUE PRECEDENCE Lee and Law 04
• Slides a constraint down one or more sequences of
  variables
• Ensuring constraint holds at every point
• Fixed parameter tractable

4
Basic SLIDE

• SLIDE(C,X1,..Xn) holds iff
• C(Xi,..Xik) holds for every I
• INCREASING(X1,..Xn)
• SLIDE(lt,X1,..Xn)
• Unfolds into
• X1 lt X2, X2 lt X3, Xn-1 lt Xn

5
Basic SLIDE

• SLIDE(C,X1,..Xn) holds iff
• C(Xi,..Xik) holds for every I
• ALWAYS_CHANGE(X1,..Xn)
• SLIDE(/,X1,..Xn)
• Unfolds into
• X1 / X2, X2 / X3, Xn-1 / Xn

6
Basic SLIDE

• AMONG SEQ constraint Beldiceanu and Contegjean
  94
• Car sequencing, staff rostering,
• E.g. at most 1 in 3 cars have a sun roof,
• at most 3 in 7 night shifts,
• SLIDE(C,X1,..Xn) where C(X1..,Xk) holds iff
  AMONG(X1,..,Xk,l,u,v)

7
Basic SLIDE

• AMONG SEQ constraint Beldiceanu and Contegjean
  94
• Car sequencing, staff rostering,
• E.g. at most 1 in 3 cars have a sun roof,
• at most 3 in 7 night shifts,
• SLIDE(C,X1,..Xn) where C(X1..,Xk) holds iff
  AMONG(X1,..,Xk,l,u,v)
• E.g. lu2, k3 and va
• SLIDE(C,X1,..X5) where X1a, X2, .. X5 in a,b

8
Basic SLIDE

• AMONG SEQ constraint Beldiceanu and Contegjean
  94
• Car sequencing, staff rostering,
• E.g. at most 1 in 3 cars have a sun roof,
• at most 3 in 7 night shifts,
• SLIDE(C,X1,..Xn) where C(X1..,Xk) holds iff
  AMONG(X1,..,Xk,l,u,v)
• E.g. lu2, k3 and va
• SLIDE(C,X1,..X5) where X1a, X2, .. X5 in a,b
• Enforcing GAC sets X4a since only satisfying
  tuples are a,a,b,a,a and a,b,a,a,b. Enforcing GAC
  on decomposition does nothing!

9
GSC

• Global sequence constraint in ILOG Solver
• Combines AMONG SEQ with GCC
• Regin and Puget have given partial propagator

10
GSC

• Global sequence constraint in ILOG Solver
• Combines AMONG SEQ with GCC
• Regin and Puget have given 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

11
GSC

• Reduction from 1in3 SAT on positive clauses
• The jth block of 2N clauses will ensure jth
  clause satisfied
• 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!

12
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

13
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn) Frisch et
  al CP 02
• Introduce sequence of Boolean vars B1,..Bn1
• Play role of alpha in LEX propagator

14
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10
• 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

15
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10
• Bn11 (strict lex), Bn1 in 0,1 (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1)
• C(Xi,Yi,Bi,Bi1) holds iff BiBi11 or
  (BiBi10 and XiYi) or (Bi0, Bi11 and
  XiltYi)

16
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10
• Bn11 (strict lex), Bn1 in 0,1 (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1)
• C(Xi,Yi,Bi,Bi1) holds iff BiBi11 or
  (BiBi10 and XiYi) or (Bi0, Bi11 and
  XiltYi)
• 2 1,3,4 2,3,4 1 3,4,5
• 0,1,2 1 1,2,3 0 0,1,2

17
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10
• Bn11 (strict lex), Bn1 in 0,1 (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1)
• C(Xi,Yi,Bi,Bi1) holds iff BiBi11 or
  (BiBi10 and XiYi) or (Bi0, Bi11 and
  XiltYi)
• 2 1,3,42,3,4 1 3,4,5
• 2 1 1,2,3 0 0,1,2

18
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10
• Bn11 (strict lex), Bn1 in 0,1 (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1)
• C(Xi,Yi,Bi,Bi1) holds iff BiBi11 or
  (BiBi10 and XiYi) or (Bi0, Bi11 and
  XiltYi)
• 2 1 2,3,4 1 3,4,5
• 2 1 1,2,3 0 0,1,2

19
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10
• Bn11 (strict lex), Bn1 in 0,1 (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1)
• C(Xi,Yi,Bi,Bi1) holds iff BiBi11 or
  (BiBi10 and XiYi) or (Bi0, Bi11 and
  XiltYi)
• 2 1 2 1 3,4,5
• 2 1 3 0 0,1,2

20
SLIDE down multiple sequences

• Can SLIDE down more than one sequence at a time
• LEX(X1,..Xn,Y1,..Yn)
• Set B10
• Bn11 (strict lex), Bn1 in 0,1 (lex)
• SLIDE(C,X1..Xn,Y1,..Yn,B1,..Bn1)
• C(Xi,Yi,Bi,Bi1) holds iff BiBi11 or
  (BiBi10 and XiYi) or (Bi0, Bi11 and
  XiltYi)
• Highly efficient, incremental, ..

21
SLIDE down multiple sequences

• CONTIGUITY(X1,..Xn) Maher 02
• 001100
• Two simple SLIDEs

22
SLIDE down multiple sequences

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

23
SLIDE down multiple sequences

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

24
SLIDE down multiple sequences

• REGULAR(A,X1,..Xn) Pesant
  04
• X1 .. Xn is a string accepted by FDA A
• Can encode many useful constraints including LEX,
  AMONG, STRETCH, CONTIGUITY

25
SLIDE down multiple sequences

• REGULAR(A,X1,..Xn) Pesant
  04
• X1 .. Xn is a string accepted by FDA A
• Encodes into simple SLIDE
• Introduce Qi to represent state of the automaton
  after i symbols

26
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
• Q1 is starting state of A
• Qn1 is limited to accepting states of A
• C(Xi,Qi,Qi1) holds iff A moves from state Qi to
  state Qi1 on seeing Xi

27
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
• Q1 is starting state of A
• Qn1 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!
• Introducing Qi also gives modelling access to
  state variables

28
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)

29
SLIDE with counters

• CARD PATH Beldiceanu and Carlsson
  01
• SLIDE is a special case of CARD PATH
• SLIDE(C,X1,..Xn) iff CARD PATH(C,X1,..Xn,n-k1
  )

30
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))

31
Global constraints for symmetry breaking

• Decision variables
• ColItaly, ColFrance, ColAustria ...
• Domain of values
• red, yellow, green, ...
• Constraints
• binary relations like
• ColItaly/ColFrance
• ColItaly/ColAustria

32
Value symmetry

• Solution
• ColItalygreen ColFrancered
• ColSpaingreen
• Values (colours) are interchangeable
• Swap red with green everywhere will still give us
  a solution

33
Value symmetry

• Solution
• ColItalygreen ColFrancered
• ColSpaingreen
• Values (colours) are interchangeable
• ColItalyred
• ColFrancegreen
• ColSpainred

34
Value precedence

• Old idea
• Used in bin-packing and graph colouring
  algorithms
• Only open the next new bin
• Only use one new colour
• Applied now to constraint satisfaction Law and
  Lee 04

35
Value precedence

• Suppose all values from 1 to m are
  interchangeable
• Might as well let X11

36
Value precedence

• Suppose all values from 1 to m are
  interchangeable
• Might as well let X11
• For X2, we need only consider two choices
• X21 or X22

37
Value precedence

• Suppose all values from 1 to m are
  interchangeable
• Might as well let X11
• For X2, we need only consider two choices
• Suppose we try X22

38
Value precedence

• Suppose all values from 1 to m are
  interchangeable
• Might as well let X11
• For X2, we need only consider two choices
• Suppose we try X22
• For X3, we need only consider three choices
• X31, X32, X33

39
Value precedence

• Suppose all values from 1 to m are
  interchangeable
• Might as well let X11
• For X2, we need only consider two choices
• Suppose we try X22
• For X3, we need only consider three choices
• Suppose we try X32

40
Value precedence

• Suppose all values from 1 to m are
  interchangeable
• Might as well let X11
• For X2, we need only consider two choices
• Suppose we try X22
• For X3, we need only consider three choices
• Suppose we try X32
• For X4, we need only consider three choices
• X41, X42, X43

41
Value precedence

• Global constraint
• Precedence(X1,..Xn) iff
• min(i Xij or in1) lt
• min(i Xik or in2) for
  all jltk
• In other words
• The first time we use j is before the first time
  we use k

42
Value precedence

• Global constraint
• Precedence(X1,..Xn) iff
• min(i Xij or in1) lt
• min(i Xik or in2) for
  all jltk
• In other words
• The first time we use j is before the first time
  we use k
• E.g
• Precedence(1,1,2,1,3,2,4,2,3)
• But not Precedence(1,1,2,1,4)

43
Value precedence

• Global constraint
• Precedence(X1,..Xn) iff
• min(i Xij or in1) lt
• min(i Xik or in2) for
  all jltk
• In other words
• The first time we use j is before the first time
  we use k
• E.g
• Precedence(1,1,2,1,3,2,4,2,3)
• But not Precedence(1,1,2,1,4)
• Proposed by Law and Lee 2004
• Pointer based propagator (alpha, beta, gamma) but
  only for two interchangeable values at a time
  

44
Value precedence

• Precedence(i,j,X1,..Xn) iff
• min(i Xij or
  in1) lt
• min(i Xik or
  in2)
• Of course
• Precedence(X1,..Xn) iff Precedence(i,j,X1,..X
  n) for all iltj

45
Value precedence

• Precedence(i,j,X1,..Xn) iff
• min(i Xij or
  in1) lt
• min(i Xik or
  in2)
• Of course
• Precedence(X1,..Xn) iff Precedence(i,j,X1,..X
  n) for all iltj
• Precedence(X1,..Xn) iff Precedence(i,i1,X1,.
  .Xn) for all i

46
Value precedence

• Precedence(i,j,X1,..Xn) iff
• min(i Xij or
  in1) lt
• min(i Xik or
  in2)
• Of course
• Precedence(X1,..Xn) iff Precedence(i,j,X1,..X
  n) for all iltj
• But this hinders propagation
• GAC(Precedence(X1,..Xn)) does strictly more
  pruning than GAC(Precedence(i,j,X1,..Xn)) for
  all iltj
• Consider
• X11, X2 in 1,2, X3 in 1,3 and X4 in 3,4

47
Pugets method

• Introduce Zj to record first time we use j
• Add constraints
• Xij implies Zj lt i
• Zji implies Xij
• Zi lt Zi1

48
Pugets method

• Introduce Zj to record first time we use j
• Add constraints
• Xij implies Zj lt I
• Zji implies Xij
• Zi lt Zi1
• Binary constraints
• easy to implement

49
Pugets method

• Introduce Zj to record first time we use j
• Add constraints
• Xij implies Zj lt I
• Zji implies Xij
• Zi lt Zi1
• Unfortunately hinders propagation
• AC on encoding may not give GAC on
  Precedence(X1,..Xn)
• Consider X11, X2 in 1,2, X3 in 1,3, X4 in
  3,4, X52, X63, X74

50
Propagating Precedence

• Simple SLIDE encoding
• Introduce sequence of variables, Yi
• Record largest value used so far
• Y10

51
Propagating Precedence

• Simple SLIDE encoding
• SLIDE(C,X1,..Xn,Y1,..Yn1) where C(Xi,Yi,Yi1)
  holds iff
• Xilt1Yi and Yi1max(Yi,Xi)

52
Propagating Precedence

• Simple SLIDE encoding
• SLIDE(C,X1,..Xn,Y1,..Yn1) where C(Xi,Yi,Yi1)
  holds iff
• Xilt1Yi and Yi1max(Yi,Xi)
• Consider Y10, X1 in 1,2,3, X2 in 1,2,3 and
  X33

53
Precedence and matrix symmetry

• Alternatively, could map into 2d matrix
• Xij1 iff Xij
• Value precedence now becomes column symmetry
• Can lex order columns to break all such symmetry
• Alternatively view value precedence as ordering
  the columns of a matrix model

54
Precedence and matrix symmetry

• Alternatively, could map into 2d matrix
• Xij1 iff Xij
• Value precedence now becomes column symmetry
• However, we get less pruning this way
• Additional constraint that rows have sum of 1
• Consider, X11, X2 in 1,2,3 and X31

55
Precedence for set variables

• Social golfers problem
• Each foursome can be represented by a set of
  cardinality 4
• Values (golfers) within this set are
  interchangeable
• Value precedence can be applied to such set
  variables
• But idea is now conceptually more complex!

56
Precedence for set variables

• We might as well start with X11,2,3

57
Precedence for set variables

• We might as well start with X11,2,3,4
• Now 1, 2, 3 and 4 are still symmetric

58
Precedence for set variables

• We might as well start with X11,2,3,4
• Lets try X22,5,6,7

59
Precedence for set variables

• We might as well start with X11,2,3,4
• Lets try X22,5,6,7
• But if we permute 1 with 2 we get
• X11,2,3,4 and X21,5,6,7
• And X2 is lower than X2 in the multiset ordering

60
Precedence for set variables

• We might as well start with X11,2,3,4
• Lets try X21,5,6,7

61
Precedence for set variables

• We might as well start with X11,2,3,4
• Lets try X21,5,6,7
• Lets try X31,2,4,5

62
Precedence for set variables

• We might as well start with X11,2,3,4
• Lets try X21,5,6,7
• Lets try X31,2,4,5
• But if we permute 3 with 4 we get
• X11,2,3,4, X21,5,6,7 and X31,2,3,5
• This is again lower in the multiset ordering

63
Precedence for set variables

• We might as well start with X11,2,3,4
• Lets try X21,5,6,7
• Lets try X31,2,3,5

64
Precedence for set variables

• Precedence(S1,..Sn) iff
• min(i, i j in Si not(k in Si) or in1)
  lt
• min(i, i k in Si not(j in Si) or in2)
  
• for all jltk
• In other words
• The first time we distinguish apart j and k (by
  one appearing on its own), j appears and k does
  not

65
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

66
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

67
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

68
Circular SLIDE

• STRETCH used in shift rostering Hellsten et al
  04
• 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

69
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

70
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

71
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(NOT(C),X1,..X
  n)
• SLIDE(C1,X1,..Xn) and SLIDE(C2,X1,..Xn) iff
  SLIDE(C1 and C2,X1,..Xn)
• ..

72
Propagating SLIDE

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

73
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,

74
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

75
Propagating SLIDE

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

76
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

77
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