Global Constraints for Lexicographic Orderings

1
Global Constraints for Lexicographic Orderings

• Alan Frisch, Ian Miguel (University of York)
• Brahim Hnich, Toby Walsh (4C)
• Zeynep Kiziltan (Uppsala University)

2
GACLex

• A new family of global constraints.

3
GACLex

• A new family of global constraints.

Non-binary constraint. Specialised propagator.
4
GACLex

• A new family of global constraints.

Non-binary constraint. Specialised propagator.
GACLex maintains Generalised Arc
Consistency Given an assignment to any
constrained variable, we can find consistent
assignments for the remainder.
5
GACLex

• A new family of global constraints.
• Linear time complexity.

6
GACLex

• A new family of global constraints.
• Linear time complexity.
• Ensures that a pair of vectors of variables are
  lexicographically ordered.

7
GACLex

• A new family of global constraint.
• Linear time complexity.
• Ensures that a pair of vectors of variables are
  lexicographically ordered.

0 1 4 2
? lex
2 9 8 7
8
Motivation

• Many problems can be modelled by matrices of
  decision variables.

9
Motivation

• Many problems can be modelled by matrices of
  decision variables.
• E.g. Combinatorial Problems
• Balanced Incomplete Block Design.

10
Motivation

• Many problems can be modelled by matrices of
  decision variables.
• E.g. Configuration Problems
• Rack Configuration.

11
Motivation

• Many problems can be modelled by matrices of
  decision variables.
• E.g. Scheduling Problems
• Social Golfers.

12
Motivation

• Many problems can be modelled by matrices of
  decision variables.
• E.g. Design Problems
• Steel Mill Slab Design.

13
Motivation

• Many problems can be modelled by matrices of
  decision variables.
• Frequently these matrices exhibit row and/or
  column symmetry.

14
Motivation

• Many problems can be modelled by matrices of
  decision variables.
• Frequently these matrices exhibit row and/or
  column symmetry.
• We can permute the rows/columns in any
  (non)solution to obtain another (non)solution.

15
Motivation

• Many problems can be modelled by matrices of
  decision variables.
• Frequently these matrices exhibit row and/or
  column symmetry.
• We can permute the rows/columns in any
  (non)solution to obtain another (non)solution.
• It can be very expensive to search fruitless
  symmetric branches of the search tree.

16
Motivation

• Many problems can be modelled by matrices of
  decision variables.
• Frequently these matrices exhibit row and/or
  column symmetry.
• Lexicographic ordering is an effective way to
  break a lot of this symmetry.

17
Example

• Consider 2 identical bins

B
A
18
Example

• Consider 2 identical bins

B
A

• We must pack 6 items

3
4
1
6
5
2
19
Example

• Here is one solution

6
5
3
4
1
2
B
A
20
Example

• Here is another

5
6
3
4
1
2
B
A
21
Example

• Is there any fundamental difference?

5
6
a)
3
4
1
2
A
B
5
6
b)
3
4
1
2
B
A
22
Example

• Consider a matrix model

1 2 3 4 5 6
A 1 0 1 0 1 0
B 0 1 0 1 0 1
5
6
a)
3
4
1
2
A
B
1 2 3 4 5 6
A 0 1 0 1 0 1
B 1 0 1 0 1 0
5
6
b)
3
4
1
2
B
A
23
Example
NB 1 means place this item in this bin

• Consider a matrix model

1 2 3 4 5 6
A 1 0 1 0 1 0
B 0 1 0 1 0 1
5
6
a)
3
4
1
2
A
B
1 2 3 4 5 6
A 0 1 0 1 0 1
B 1 0 1 0 1 0
5
6
b)
3
4
1
2
B
A
24
Example

• Consider a matrix model

1 2 3 4 5 6
A 1 0 1 0 1 0
B 0 1 0 1 0 1
1 2 3 4 5 6
A 0 1 0 1 0 1
B 1 0 1 0 1 0
5
6
b)
3
4
1
2
B
A
25
Example

• Notice that items 3 and 4 are identical.

1 2 3 4 5 6
A 0 1 0 1 0 1
B 1 0 1 0 1 0
5
6
b)
3
4
1
2
B
A
1 2 3 4 5 6
A 0 1 1 0 0 1
B 1 0 0 1 1 0
5
6
c)
4
3
1
2
B
A
26
Example

• Notice that items 3 and 4 are identical.

1 2 3 4 5 6
A 0 1 0 1 0 1
B 1 0 1 0 1 0
5
6
b)
3
4
1
2
B
A
1 2 3 4 5 6
A 0 1 1 0 0 1
B 1 0 0 1 1 0
27
How GACLex Works

• Consider the following example.
• We have two vectors of decision variables

28
How GACLex Works

• Consider the following example.
• We have two vectors of decision variables

x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
29
How GACLex Works

• Consider the following example.
• We have two vectors of decision variables

x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2

• We want to enforce GAC on x ?lex y.

30
A Tale of Two Pointers

• We use two pointers, a and ß, to avoid repeatedly
  traversing the vectors.

31
A Tale of Two Pointers

• We use two pointers, a and ß, to avoid repeatedly
  traversing the vectors.
• We index the vectors as follows

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
32
A Tale of Two Pointers

• We use two pointers, a and ß, to avoid repeatedly
  traversing the vectors.
• We index the vectors as follows

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
33
A Tale of Two Pointers

• We use two pointers, a and ß, to avoid repeatedly
  traversing the vectors.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2

• a index such that all variables at more
  significant indices are ground and equal.

34
A Tale of Two Pointers

• We use two pointers, a and ß, to avoid repeatedly
  traversing the vectors.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

35
A Tale of Two Pointers

• We use two pointers, a and ß, to avoid repeatedly
  traversing the vectors.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2

• a index such that all variables at more
  significant indices are ground and equal.
• ß If tails never violate the constraint ?

36
Pointer Initialisation

• Needs one traversal of the vectors (linear).
• Details in the paper.

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

37
Pointer Initialisation

• Needs one traversal of the vectors (linear).

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

38
Pointer Initialisation

• Needs one traversal of the vectors (linear).

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

39
Failure

• Inconsistent if ß ? a.

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

40
How GACLex Works

• We maintain a and ß as assignments made.

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

41
How GACLex Works

• We maintain a and ß as assignments made.
• When ß a 1 we enforce bounds consistency on
  xa lt ya

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

42
How GACLex Works

• We maintain a and ß as assignments made.
• When ß a 1 we enforce bounds consistency on
  xa lt ya

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

43
How GACLex Works

• We maintain a and ß as assignments made.
• When ß a 1 we enforce bounds consistency on
  xa lt ya
• When ß gt a 1 we enforce bounds consistency on
  xa ? ya

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

44
How GACLex Works

• We maintain a and ß as assignments made.
• Key we reduce GAC on vectors to BC on binary
  constraints.

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

45
How GACLex Works

• 0, 1 removed from ya.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

46
How GACLex Works

• 0, 1 removed from ya.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

47
How GACLex Works

• Update a.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

48
How GACLex Works

• Update a.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

49
How GACLex Works

• 3, 4 removed from xa.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

50
How GACLex Works

• 3, 4 removed from xa.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

51
How GACLex Works

• Update a.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

52
How GACLex Works

• Update a.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

53
How GACLex Works

• 4, 5 removed from xa, 0, 1 removed from ya.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

54
How GACLex Works

• 4, 5 removed from xa, 0, 1 removed from ya.

0 1 2 3 4
x 2 1,3,4 1,2,3,4,5 1,2 3,4,5
y 0,1,2 1 0,1,2,3,4 0,1 0,1,2
a
ß

• a index such that all variables at more
  significant indices are ground and equal.
• ß most significant index from which the two
  vectors tails necessarily violate the constraint.

55
Complexity

• Initialisation O(n)

56
Complexity

• Initialisation O(n)
• Propagation
• We enforce bounds consistency between at most n
  pairs of variables xa lt ya or xa ? ya.

57
Complexity

• Initialisation O(n)
• Propagation
• We enforce bounds consistency between at most n
  pairs of variables xa lt ya or xa ? ya.
• Cost b.

58
Complexity

• Initialisation O(n)
• Propagation
• We enforce bounds consistency between at most n
  pairs of variables xa lt ya or xa ? ya.
• Cost b.
• Overall cost O(nb).

59
Alternative 1 Decomposition

• x ?lex y can be decomposed into n non-binary
  constraints
• x0 ? y0

60
Alternative 1 Decomposition

• x ?lex y can be decomposed into n non-binary
  constraints
• x0 ? y0
• x0 y0 ? x1 ? y1

61
Alternative 1 Decomposition

• x ?lex y can be decomposed into n non-binary
  constraints
• x0 ? y0
• x0 y0 ? x1 ? y1
• x0 y0 ? x1 y1 ? x2 ? y2

62
Alternative 1 Decomposition

• x ?lex y can be decomposed into n non-binary
  constraints
• x0 ? y0
• x0 y0 ? x1 ? y1
• x0 y0 ? x1 y1 ? x2 ? y2
• x0 y0 ? x1 y1 ? xn-2 yn-2 ? xn-1 ? yn-1

63
Alternative 1 Decomposition

• x ?lex y can be decomposed into n non-binary
  constraints
• Usually, only forward checking is performed on
  such non-binary constraints.

64
Alternative 1 Decomposition

• x ?lex y can be decomposed into n non-binary
  constraints
• Usually, only forward checking is performed on
  such non-binary constraints.
• They are also cumbersome.

65
Alternative 2 Arithmetic Constraints

• To ensure that x ?lex y with domain size d
• dn-1 ? x0 dn-2 ? x1 d0 ? xn-1 ?
• dn-1 ? y0 dn-2 ? y1 d0 ? yn-1

66
Alternative 2 Arithmetic Constraints

• To ensure that x ?lex y with domain size d
• dn-1 ? x0 dn-2 ? x1 d0 ? xn-1 ?
• dn-1 ? y0 dn-2 ? y1 d0 ? yn-1
• Bounds consistency on this expression is
  equivalent to GAC on x ?lex y.

67
Alternative 2 Arithmetic Constraints

• To ensure that x ?lex y with domain size d
• dn-1 ? x0 dn-2 ? x1 d0 ? xn-1 ?
• dn-1 ? y0 dn-2 ? y1 d0 ? yn-1
• Bounds consistency on this expression is
  equivalent to GAC on x ?lex y.
• When n and d are large, dn-1 is greater than the
  word size of the computer.

68
Alternative 2 Arithmetic Constraints

• To ensure that x ?lex y with domain size d
• dn-1 ? x0 dn-2 ? x1 d0 ? xn-1 ?
• dn-1 ? y0 dn-2 ? y1 d0 ? yn-1
• Bounds consistency on this expression is
  equivalent to GAC on x ?lex y.
• When n and d are large, dn-1 is greater than the
  word size of the computer.
• This method is only feasible for small
  vectors/domain size.

69
Multiple Lex Constraints

• All rows/columns of a matrix might be lex ordered.

70
Multiple Lex Constraints

• All rows/columns of a matrix might be lex
  ordered.
• Can treat as a single global constraint.

71
Multiple Lex Constraints

• All rows/columns of a matrix might be lex
  ordered.
• Can treat as a single global constraint.
• GAC on such a constraint is future work.
• Can approximate with GACLex between all pairs, or
  adjacent pairs.
• GACLex on all pairs GAC on whole set for 0/1
  vectors.

72
Experimental Results BIBD

• Standard combinatorial problem.

• v rows.
• b columns.
• r 1s per row.
• k 1s per column.
• ? is scalar product between every pair of rows.

0 0 0 0 1 1 1
0 0 1 1 0 0 1
0 1 0 1 0 1 0
0 1 1 0 1 0 0
1 0 0 1 1 0 0
1 0 1 0 0 1 0
1 1 0 0 0 0 1
73
Results BIBD
Decomposition takes About 9 times longer on each
of these instances.
74
Results BIBD

• Fails, Choice-points very similar
• GACLex ? Decomposition.
• Why?
• We are using a good static variable ordering.
• So substantially decreased run-times due to
  efficiency gains in this case.

75
Results Social Golfers

• Schedule a golf tournament of w weeks.
• In each week, divide golfers into g groups, size
  s.
• Every golfer plays once per week.
• Every pair of golfers can meet at most once.

76
Results Social Golfers

• Schedule a golf tournament of w weeks.
• In each week, divide golfers into g groups, size
  s.
• Every golfer plays once per week.
• Every pair of golfers can meet at most once.
• 3d 0/1 model

Players
Groups
Weeks
77
Results Social Golfers

• Schedule a golf tournament of w weeks.
• In each week, divide golfers into g groups, size
  s.
• Every golfer plays once per week.
• Every pair of golfers can meet at most once.
• 3d 0/1 model

Players
Players, groups and weeks all symmetrical
Groups
Weeks
78
Results Social Golfers
Decomposition takes between 10 and 56 times
longer (at least) on these instances.
79
Results Social Golfers
More complex interactions in 3d model result in
greater pruning with GACLex.
This explains the even greater run-time
improvements.
80
Results Tournament Scheduling

• n teams play over n-1 weeks.
• Each week divided into n/2 periods.
• Each period divided into 2 slots (home/away).
• Every team
• Plays once a week.
• Plays at most twice in the same period.
• Plays every other team.

Periods
Slots(2)
Channel
Periods
Weeks
Weeks
81
Results Tournament Scheduling

• Search tree similar.
• Variable/value ordering compatible with lex.
• Speed increase around 3x.
• Vectors much smaller.

82
Results Adjacent v All-pairs

• In theory, enforcing lex between all pairs of
  vectors gives more propagation.
• We have seen no evidence of this.
• Perhaps conditions appear infrequently.

83
Conclusion

• Global constraints for lexicographic orderings.
• Linear time complexity.
• Decompositions carry a penalty in the amount or
  cost of propagation.

84
Future Work

• GACLex for multiple vectors.
• Other types of ordering, e.g. multisets.
• Compare with other decompositions
• x0 ? y0 ? (x0 y0 ? (x1 ? y1 ? (x1 y1 ?

85
Resources

• Solver code coming soon at
• http//www-users.cs.york.ac.uk/ianm/