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Non-binary Constraints

Non-binary Constraints. Toby Walsh tw@cs.york.ac.uk. Outline. Definition of non-binary constraints Modeling with non-binary constraints Constraint propagation with non-binary constraints Practical benefits: case study Golomb rulers. Definitions. Binary constraint

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Non-binary Constraints

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

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

  3. 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).

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

  5. 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).

  6. 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 they’re not needed • But in practice, they are!

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

  8. 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).

  9. Modeling with non-binary constraints • It’s possible but it’s 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?

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

  11. 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

  12. 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} X1 \= {1,2} {2,3} \= X2 X3

  13. 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

  14. 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 \= \= {2,3} {2,3} \= X2 X3

  15. 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 we’ll explore exponentially fewer nodes

  16. 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

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

  18. 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

  19. 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? ….

  20. 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

  21. 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|

  22. 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

  23. 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!

  24. 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

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

  26. 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

  27. Experimental results

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

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

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

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