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Non-binary Constraints Toby Walsh tw@cs.york.ac.uk

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Outline Definition of non-binary constraints Modeling with non-binary constraints Constraint propagation with non-binary constraints Practical benefits: case study – Golomb rulers

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Definitions Binary constraint – Relation on 2 variables identifying those pairs of values disallowed (nogoods) – E.g. not-equals constraint: X1 \= X2. Non-binary constraint – Relation on 3 or more variables identifying tuples of values disallowed – E.g. alldifferent(X1,X2,X3).

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Some non-binary examples Timetabling – Variables: Lecture1. Lecture2, … – Values: time1, time2, … – Constraint that lectures do not conflict: alldifferent(Lecture1,Lecture2,…).

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Some non-binary examples Scheduling – Variables: Job1. Job2, … – Values: machine1, machine2, … – Constraint on number of jobs on each machine: atmost(2,[Job1,Job2,…],machine1), atmost(1,[Job1,Job2,…],machine2).

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Why use non-binary constraints? Binary constraints are NP-complete – Any non-binary constraint can be represented using binary constraints – E.g. alldifferent(X1,X2,X3) is equivalent to X1 \= X2, X1 \= X3, X2 \= X3 In theory therefore theyre not needed – But in practice, they are!

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Modeling with non-binary constraints Benefits include: – Compact, declarative specifications (discussed next) – Efficient constraint propagation (discussed after next section)

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Modeling with non-binary constraints Consider writing your own alldifferent constraint: alldifferent([]). alldifferent([Head|Tail]):- onedifferent(Head,Tail), alldifferent(Tail). onedifferent(El,[]). onedifferent(El,[Head|Tail]):- El \= Head, onedifferent(El,Tail).

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Modeling with non-binary constraints Its possible but its not very pleasant! Nor is it very compact – alldifferent([X1,…Xn]) expands into n(n-1)/2 binary not-equals constraints, Xi \= Xj – one non-binary constraint or O(n^2) binary constraints?

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Theoretical comparison Constraint algorithms: – Tree search (labeling) – Constraint propagation at each node Binary constraint propagation – Arc-consistency Non-binary constraint propagation – Generalized arc-consistency

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Binary constraint propagation Arc-consistency (AC) is very popular – A binary constraint r(X1,X2) is AC iff for every value for X1, there is a consistent value (often called support) for X2 and vice versa – We can prune values that are not supported – A problem is AC iff every constraint is AC AC offers good tradeoff between amount of pruning and computational effort

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Binary constraint propagation X2 \= X3 is AC X1 \= X2 is not AC – X2=1 has no support so can this value can be pruned X2 \= X3 is now not AC – No support for X3=2 – This value can also be pruned Problem is now AC {1} {1,2}{2,3} \= X1 X3X2

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Non-binary constraint propagation generalized arc-consistency (GAC) for non- binary constraints – A non-binary constraint is GAC iff for every value for a variable there are consistent values for all other variables in the constraint – We can prune values that are not supported GAC = AC on binary constraints

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GAC is stronger than AC Pigeonhole problem – 3 pigeons in 2 holes Non-binary model – alldifferent(X1,X2,X3) is not GAC Binary model – X1 \= X2, X1 \= X3, X2 \= X3 are all AC {2,3} X1 X3X2 \=

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Using GAC within search Tree search – Instantiate chosen variable with value (label) – Maintain (incrementally enfoce) some level of consistency Maintaining GAC can be exponentially better than maintaining AC – Construct generalized pigeonhole example on which well explore exponentially fewer nodes

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Achieving GAC By exploiting semantics of constraints, we can often enforce GAC efficiently – Consider alldifferent([X1,…Xn]) with each Xi having domain of size m – Generic GAC algorithm runs in O(m^n) – Specialized GAC algorithm for alldifferent runs in O(m^2 n^2) based on network flow

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Practical benefits How do these (theoretical) differences affect you practically? Case study – Golomb rulers

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Golomb rulers Mark ticks on a ruler – Distance between any two (not necessarily consecutive) ticks is distinct Applications in radio-astronomy – http://csplib.cs.strath.ac.uk/prob006

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Golomb rulers Simple solution – Exponentially long ruler – Ticks at 0,1,3,7,15,31,63,… Goal is to find miminal length rulers – turn optimization problem into sequence of satisfaction problems Is there a ruler of length m? Is there a ruler of length m-1? ….

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Optimal Golomb rulers Known for up to 23 ticks Distributed internet project to find large rulers 0,1 0,1,3 0,1,4,6 0,1,4,9,11 0,1,4,10,12,17 0,1,4,10,18,23,25

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Modeling the Golomb ruler Variable, Xi for each tick Value is position on ruler Naïve model with quaternary constraints – For all i,j,k,l |Xi-Xj| \= |Xk-Xl|

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Problems with naïve model Large number of quaternary constraints – O(n^4) constraints Looseness of quaternary constraints – Many values satisfy |Xi-Xj| \= |Xk-Xl| – Limited pruning

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A better non-binary model Introduce auxiliary variables for inter-tick distances – Dij = |Xi-Xj| – O(n^2) ternary constraints Post single large non-binary constraint – alldifferent([D11,D12,…]). – Relatively tight!

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Other modeling issues Symmetry – A ruler can always be reversed! – Break this symmetry by adding constraint: D12 < Dn-1,n – Also break symmetry on Xi X1 < X2 < … Xn – Such tricks important in many problems

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Other modeling issues Additional (implied) constraints – Dont change set of solutions – But may reduce search significantly E.g. D12 < D13, D23 < D24, … Pure declarative specifications are not enough!

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Solving issues Labeling strategies often very important – Smallest domain often good idea – Focuses on hardest part of problem Best strategy for Golomb ruler is instantiate variables in strict

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Experimental results Runtime/secNaïve modelAlldifferent model 8-Find2.00.1 8-Prove12.010.2 9-Find31.71.6 9-Prove1689.7 10-Find65724.3 10-Prove> 10^568.3

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Something to try at home? Circular (or modular) Golomb rulers 2-d Golomb rulers

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Conclusions Benefits of non-binary constraints – Compact, declarative models – Efficient and effective constraint propagation Supported by many constraint toolkits – alldifferent, atmost, cardinality, …

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Conclusions Modeling decisions: – Auxiliary variables – Implied constraints – Symmetry breaking constraints More to constraints than just declarative problem specifications!

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