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
GAC Schema

• Not all global constraints have nice semantics we
  can exploit to devise an efficient propagator
• Consider product configuration
• Compatibility constraints on hardware components
• Only certain combinations of components work
  together
• Compatibility may not be a simple pairwise
  relationship
• Video cards supported function of motherboard,
  CPU, clock speed, O/S ..

4
GAC Schema

• 5-ary global constraint
• Compatible(motherboard345,intelCPU,2GHz,1GBRam,80G
  Bdrive)
• Compatible(motherboard346,intelCPU,3GHz,2GBRam,100
  GBdrive)
• Compatible(motherboard346,amdCPU,2GHz,2GBRam,100GB
  drive)

5
Crossword puzzle

• Word(X1,X2,X3,X4)
• Word(X2,X15,X17)
• No simple way to
• decide acceptable
• words other than to
• put them in a table

6
GAC schema

• Generic propagator
• Enforces GAC on global constraint given by
• Set of allowed tuples OR
• Set of disallowed tuples OR
• Predicate answering if a constraint is satisfied
  or not
• ..
• Sometimes called the table constraint (e.g.
  user supplies table of acceptable values)

7
GAC-Schema

• Bessiere and Regin, IJCAI97
• You just have to say how to compute a solution.
• Works incrementally (notion of support)
• Keeps supports found to save re-finding them
• Exploits multi-directionality
• If we find support for Xa and this contains Yb
• Then we automatically have a support for Yb

8
GAC-Schema

• Idea tuple solution of the
  constraint support valid tuple - while the
  tuple remains do nothing - if the tuple
  is no longer possible, then search for a new
  support for the values it contains
• a solution (support) can be computed by any
  algorithm

9
Example

• X(C)x1,x2,x3 D(xi)a,b
• T(C)(a,a,a),(a,b,b),(b,b,a),(b,b,b)

10
Example

• X(C)x1,x2,x3 D(xi)a,b
• T(C)(a,a,a),(a,b,b),(b,b,a),(b,b,b)
• Support for (x1,a) (a,a,a) is computed and
  (a,a,a) is added to S(x2,a) and S(x3,a), (x1,a)
  in (a,a,a) is marked as supported.

11
Example

• X(C)x1,x2,x3 D(xi)a,b
• T(C)(a,a,a),(a,b,b),(b,b,a),(b,b,b)
• Support for (x1,a) (a,a,a) is computed and
  (a,a,a) is added to S(x2,a) and S(x3,a), (x1,a)
  in (a,a,a) is marked as supported.
• Support for (x2,a) (a,a,a) is in S(x2,a) it is
  valid, therefore it is a support.
  (Multidirectionnality). No need to compute a
  solution

12
Example

• X(C)x1,x2,x3 D(xi)a,b
• T(C)(a,a,a),(a,b,b),(b,b,a),(b,b,b)
• Support for (x1,a) (a,a,a) is computed and
  (a,a,a) is added to S(x2,a) and S(x3,a), (x1,a)
  in (a,a,a) is marked as supported.
• Value a is removed from x1, then all the tuples
  in S(x1,a) are no longer valid (a,a,a) for
  instance. The validity of the values supported by
  this tuple must be reconsidered.

13
Example

• X(C)x1,x2,x3 D(xi)a,b
• T(C)(a,a,a),(a,b,b),(b,b,a),(b,b,b)
• Support for (x1,a) (a,a,a) is computed and
  (a,a,a) is added to S(x2,a) and S(x3,a), (x1,a)
  in (a,a,a) is marked as supported.
• Support for (x1,b) (b,b,a) is computed, and
  updated ...

14
GAC-Schema complexity

• In worst case, GAC schema enforces GAC in
• O(dk) time and
• O(k2d) space
• Hence, k cannot be too large!
• ILOG Solver limits it to 3 or so
• Recall want local consistency to be O(d2) or
  less
• Hence all this work on specialized propagators
  that exploit the constraint semantics to be
  faster than O(dk) for k3

15
Exploiting constraint semantics

• Speed-up the search for a support

16
Exploiting constraint semantics

• Speed-up the search for a support
• x
• support for (x,9000)
• immediate any value greater than 9000 in D(y)

17
Semantics of a constraint

• Design of an ad-hoc filtering algorithmx
• Two invariants
• (a) max(x) max(y) -1(b) min(y) min(x) 1

18
Exploiting constraint semantics

• Design of an ad-hoc filtering algorithmx
• Two invariants
• (a) max(x) max(y) -1(b) min(y) min(x)
  1
• Triggering of the filtering algorithmno
  possible pruning of D(x) while max(y) is not
  modifiedno possible pruning of D(y) while min(x)
  is not modified

19
Building constraint propagators

• When to wake constraint?
• Only want this to happen when it is likely to
  prune
• When any domain changes?
• When upper bound changes?
• When is a constraint no longer useful?
• If a constraint is logically entailed, it can no
  longer prune
• Never want it to wake up
• Set flag and ignore till backtrack out of this
  point

20
Building constraint propagators

• How to avoid re-doing work?
• When constraint re-awakes, how do we re-build all
  the data structures it needs
• Remember the network flow for the GCC constraint
  or the Hall intervals in the AllDifferent
  constraint
• Remember the pointers used in the LEX constraint
  to avoid re-traversing the vectors

21
Conclusions

• GAC Schema is a generic propagator for global
  constraints
• Useful when constraints lacks any special
  semantics we can exploit
• Time complexity is O(dk) in general where k is
  the constraint arity
• Only useful than for relatively small k
• Useful nevertheless for product configuration and
  other real world domans