Global%20Constraints

1
Global Constraints

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

2
Quick advert

• UNSW is in Sydney
• Regularly voted in top 10 cities in World
• UNSW is one of top universities in Australia
• In top 100 universities in world
• Talk to me about our PhD programme!
• Also happy to have PhDs/PostDocs visit for
  weeks/months/years

3
Quick advert

• UNSW is in Sydney
• Regularly voted in top 10 cities in World
• UNSW is one of top universities in Australia
• In top 100 universities in world
• Talk to me about our PhD programme!
• Also happy to have PhDs/PostDocs visit for
  weeks/months/years
• Attend CP/KR/ICAPS in Sept

4
Constraint programming

• Constraint programming represents one of the
  closest approaches computer science has yet made
  to the Holy Grail of programming the user states
  the problem, the computer solves it.
• Gene Freuder, Constraints, April 1997

5
Suduko

• Puzzle
• X11,X12,X13, X19,
• X21,X22, X29,
• ..
• X91,X92, .X99
• Puzzle 1..9
• alldifferent(X11,..X19)
• alldifferent(X21,..X29)
• ..
• alldifferent(X11,..X91)
• ..
• alldifferent(X11,..X33)
• ..

6
Comparison with OR

• CP
• rich modelling language
• many different global constraints
• fast local inference on these global constraints
• good for finding feasible and tightly constrained
  solutions

• (I)LP
• everything has to be a linear inequality
• limited range of solution methods (simplex, ...)
• good at proving optimality

7
Global constraints

• Any non-binary constraint
• AllDifferent
• Nvalues
• Element
• Lex ordering
• Stretch constraint
• Sequence constraint
• ...
• Each comes with an efficient propagator ...

8
All Different
9
Golomb rulers

• Mark ticks on a rule
• Distance between any two ticks (not just
  neighbouring) is distinct
• Special type of graceful graph
• Applications
• Radio-astronomy, crystallorgraphy, ...
• Prob006 in CSPLib

10
Golomb rulers

• Simple solution
• Exponentially long ruler
• Ticks at 0, 1, 3, 7, 15, 31, ...
• Goal is to find minimal length ruler
• Sequence of optimization problems
• Is there a ruler of length m?
• Is there a ruler of length m-1?
• ...

11
Optimal Golomb rulers

• Known for up to 23 ticks
• Distributed internet project to find larger
• 0,1
• 0,1,3
• 0,1,4,6
• 0,1,4,9,11
• 0,1,4,10,12,17
• Solutions grow as approximately O(n2)

12
Golomb rulers as CSP

• Variable Xi for each tick
• Value is position
• Auxiliary variable Dij for each inter-tick
  distance
• DijXi-Xj
• Two (global) constraints
• X1ltX2lt..Xn
• AllDifferent(D11,D12,D13,...)

13
Golomb ruler as CSP

• Not yet achieved dream of declarative
  programming
• Need to break symmetry of inverting ruler
• D12lt Dn-1n
• Add implied constraints
• D12ltD13 ...
• Pure declarative specifications not quite enough!

14
AllDifferent

• AllDifferent(X1,..Xn) iff Xi/Xj for iltj
• Useful in wide range of applications
• Timetabling (exams with common student must occur
  at different times)
• Production (each product must be produced at a
  different time)
• ...
• Can propagate just using binary inequalities
• But this decomposition hurts propagation

15
AllDifferent

• AllDifferent(X1,..Xn) iff Xi/Xj for iltj
• Can propagate just using binary inequalities
• But this decomposition hurts propagation
• X1 in 1,2, X2 in 1,2, X3 in 1,2,3
• X3 cannot be 1 or 2
• How can we automate such reasoning?
• How can we do this efficiently (less than O(n2)
  time)

16
AllDifferent

• One of the oldest global constraints
• In ALICE language Lauriere 78
• Found in every constraint solver today
• GAC algorithm based on matching theory due to
  Regin AAAI 94, runs in O(dn3/2)
• BC algorithm using combinatorics due to Puget
  AAAI98, runs in O(nlogn)

17
BC on AllDifferent

• Application of Hall's Theorem
• Sometimes called the marriage theorem
• Given k sets
• Then there is an unique and distinct element in
  each set iff any union of j of the sets has at
  least j elements for 0ltjltk
• E.g. S11,2, S21,2 but not S11,2,S21,2
  and S31,2

18
Hall's theorem

• You wish to marry n men and women
• Each woman declares who they are willing to marry
  (some set of men)
• Each man will be happy with whoever is willing
  to marry them
• Given any subset of j women, the number of men
  they are willing to marry must be j or more (thus
  this condition is necessary)
• What is surprising is that it is also sufficient!

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BC on AllDifferent

• Hall Interval
• Interval of values in which as many variables as
  domain values
• E.g. X1 in 1,2,3, X2 in 1,2, X3 in 1,2,3
• 3 variables in the interval 1..3
• AllDifferent(X1,..Xn) is BC iff
• Each interval, the number of vars it covers is
  less than or equal to the width of the interval
• No variable outside a Hall Interval has a value
  within it

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BC on AllDifferent

• Consider X1 in 1,2, X2 in 1,2, X3 in 1,2,3

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BC on AllDifferent

• Consider X1 in 1,2, X2 in 1,2, X3 in 1,2,3
• Then 1..2 is a Hall Interval covered by X1 and
  X2
• X3 has values inside this Hall Interval
• We can prune these and make AllDifferent BC

22
BC on AllDifferent

• Naïve algorithm considers O(n2) intervals
• Puget orders intervals
• Ordering has O(nlogn) cost
• Then can go through them in order

23
Beyond AllDifferent

• NValues(X1,...,Xn,M) iff j
  XijM
• AllDifferent is special case when Mn
• Useful when values represent a resource
• Minimize the number of resources used

24
Beyond AllDifferent

• Global cardinality constraint
• GCC(X1,..Xn,a1,..am,b1,...bm) iff
  aj lt i Xij lt bj for
  all j
• In other words, j occurs between aj and bj times
• Again useful when values represent a resource
• You have at least one night shift but no more
  than four each week

25
Conclusions

• AllDifferent is one of the oldest (and most
  useful) global constraints
• Efficient propagators exist for achieving GAC and
  BC
• When to choose BC over GAC?
• Heuristic choice BC often best when many more
  values than variables, GAC when we are close to a
  permutation (number of varsnumber of values)

26
Lex ordering

• Widely useful
• Especially for symmetry breaking
• Breaking row and column symmetry in matrix models
• Available in most (all?) solvers
• Good example of pointer based global constraint
• Pointers save re-doing work
• Good incremental behaviour
• O(n) in general, but amortised O(n) cost down a
  branch

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Motivation

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

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Motivation

• Many problems can be modelled by matrices of
  decision variables.
• E.g. Combinatorial Problems
• Balanced Incomplete Block Design.
• Xi,j0 or 1
• 0 1 1 0 0 1 0
• 1 0 1 0 1 0 0
• 0 0 1 1 0 0 1
• 1 1 0 0 0 0 1 Parameters
  (7,7,3,3,1)
• 0 0 0 0 1 1 1
• 1 0 0 1 0 1 0
• 0 1 0 1 1 0 0

0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 1 0 0
0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1
0 0
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Motivation

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

30
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

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Motivation

• When rows and columns can be permuted
• An n by m matrix model with row and column
  symmetry has
• n! m! symmetries
• grows super-exponentially
• Too many symmetric search states
• It can be very expensive to visit all the
  symmetric branches of a search tree

32
Motivation

• Breaking symmetry is very important!
• Breaking all row and column symmetries is
  difficult
• No one has an effective way of dealing with all
  row and column symmetries.
• Symmetry breaking methods have to deal with very
  large number of symmetries.
• The effort required could easily be exponential.

33
Symmetry in CP

• Add symmetry breaking constraints
• Leave at least one solution
• Eliminate some/all symmetric solutions
• Modify search algorithm
• Ignore symmetric parts of the search space
• Adapt branching heuristic
• To explore branches which are most likely not to
  be symmetric

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Example

• Consider 2 identical bins

B
A
35
Example

• Consider 2 identical bins

B
A

• We must pack 6 items

3
4
1
6
5
2
36
Example

• Here is one solution

6
5
3
4
1
2
B
A
37
Example

• Here is another

5
6
3
4
1
2
B
A
38
Example

• Is there any fundamental difference?

5
6
a)
3
4
1
2
A
B
5
6
b)
3
4
1
2
B
A
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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
40
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
41
Example

• Notice that items 3 and 4 are identical.

1 2 3 4 5 6 A 1 0 1 0
1 0 B 0 1 0 1 0 1
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
42
Example

• Notice that items 3 and 4 are identical.

1 2 3 4 5 6 A 1 0 1 0
1 0 B 0 1 0 1 0 1
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
43
Lexicographic Ordering

• Used to order dictionaries
• A,B,C lex D,E,F
• AltD or
• (AD and BltE ) or
• (AD and BE and CltF) or
• (AD and BE and CF)

44
Breaking Row (Column) Symmetry

• Lexicographic ordering is total
• Forcing the rows to be lexicographically ordered
  breaks all row symmetry

lexicographic ordering
A B C
D E F
G H I
A B C ? lex D E F ? lex G H I
anti-lexicographic ordering
G H I
45
Breaking Row and Column Symmetries

• Breaking both row and column symmetries is
  difficult
• Rows and columns intersect
• After constraining the rows to be
  lexicographically ordered
• we distinguish the columns
• the columns are not symmetric
  anymore!

46
Good News ?

• Each symmetry class of assignments has at least
  one element where both the rows and the columns
  are lexicographically ordered
• But there may be no element with rows lex ordered
  and columns anti-lex ordered
• To break row and column symmetries, we can insist
  that the rows and columns are both
  lexicographically ordered (double-lex)
• Extends to higher dimensions

47
Bad News ?

• A symmetry class of assignments may have more
  than one element where both the rows and the
  columns are lexicographically ordered
• Double-lex does not break all row and column
  symmetries

1
0
1
0
1
0
0
1
swap the columns swap row 1 and row 3
0
1
0
1
48
GACLex

• A new family of global constraints.

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GACLex

• A new family of global constraints.

Non-binary constraint. Specialised propagator.
50
GACLex

• A new family of global constraints
• Linear time complexity

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GACLex

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

0 1 4 2
? lex
2 9 8 7
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How GACLex Works

• Consider the following example

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

• We want to enforce GAC on x ?lex y.

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A Tale of Two Pointers

• We use two pointers, alpha and beta, to avoid
  repeatedly traversing the vectors

54
A Tale of Two Pointers
x 2 1,3,4 2,3,4 1 3,4,5
y 0,1,2 1 1,2,3 0 0,1,2
Most significant index
Alpha all variables at more significant indices
are ground and equal
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A Tale of Two Pointers
x 2 1,3,4 2,3,4 1 3,4,5
y 0,1,2 1 1,2,3 0 0,1,2

• beta most significant index from which the two
  vectors tails necessarily violate the constraint.

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Pointer initialization

• alpha
• Scan through vector from start
• At most O(n) time
• beta
• Scan through vector from end
• At most O(n) time

57
Failure
Inconsistent if beta lt alpha.

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

58
How GACLex Works
We maintain alpha and beta as assignments made.

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

59
How GACLex Wroks
We maintain alpha and beta as assignments
made. When beta alpha 1 we enforce bounds
consistency on xalpha lt yalpha

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

60
How GACLex Works
We maintain alpha and beta as assignments
made. When beta alpha 1 we enforce bounds
consistency on xalpha lt yalpha When beta gt alpha
1 we enforce bounds consistency on xalpha lt
yalpha

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

61
Complexity

• Initialisation O(n)
• Propagation
• We enforce bounds consistency between at most n
  pairs of variables xalpha lt yalpha or xalpha ?
  yalpha.
• Cost b. Overall cost O(nb).
• Complexity can be amortized down branch of search
  tree
• alpha and beta move at most O(n) steps

62
Conclusions

• Global constraints
• One of the special features of constraint
  programming
• Interesting meeting point for algorithms,
  combinatorics, computational complexity, formal
  languages,