Global Constraints Toby Walsh National ICT Australia and University of New South Wales

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

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..

GAC Schema 5-ary global constraint: –Compatible(motherboard345,intelCPU,2GHz,1GBRam,8 0GBdrive) –Compatible(motherboard346,intelCPU,3GHz,2GBRam,1 00GBdrive) –Compatible(motherboard346,amdCPU,2GHz,2GBRam,1 00GBdrive) –…

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

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)

GAC-Schema ● Bessiere and Regin, IJCAI’97 ● 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 X=a and this contains Y=b ● Then we automatically have a support for Y=b

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

Example ● X(C)={x1,x2,x3} D(xi)={a,b} ● T(C)={(a,a,a),(a,b,b),(b,b,a),(b,b,b)}

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.

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

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.

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...

GAC-Schema: complexity ● In worst case, GAC schema enforces GAC in – O(d^k) time and – O(k^2d) space ● Hence, k cannot be too large! – ILOG Solver limits it to 3 or so – Recall want local consistency to be O(d^2) or less – Hence all this work on specialized propagators that exploit the constraint semantics to be faster than O(d^k) for k>3

Exploiting constraint semantics ● Speed-up the search for a support

Exploiting constraint semantics ● Speed-up the search for a support ● x < y, D(x)=[ ], D(y)=[ ] – support for (x,9000) – immediate any value greater than 9000 in D(y)

Semantics of a constraint ● Design of an ad-hoc filtering algorithm: x < y : ● Two invariants (a)max(x) = max(y) -1 (b)min(y) = min(x) +1

Exploiting constraint semantics ● Design of an ad-hoc filtering algorithm: x < y : ● Two invariants (a)max(x) = max(y) -1 (b)min(y) = min(x) +1 ● Triggering of the filtering algorithm: no possible pruning of D(x) while max(y) is not modified no possible pruning of D(y) while min(x) is not modified

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

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

Conclusions ● GAC Schema is a generic propagator for global constraints – Useful when constraints lacks any special semantics we can exploit – Time complexity is O(d^k) 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