Nonbinary Constraints

1
Non-binary Constraints

• Toby Walsh
• tw_at_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).

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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)

8
Modeling with non-binary constraints

• Consider writing your own alldifferent
  constraint
• alldifferent().
• alldifferent(HeadTail)-
• onedifferent(Head,Tail),
• alldifferent(Tail).
• onedifferent(El,).
• onedifferent(El,HeadTail)-
• 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(n2) 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
• X21 has no support so can this value can be
  pruned
• X2 \ X3 is now not AC
• No support for X32
• This value can also be pruned
• Problem is now AC

1
X1
\
1,2
2,3
\
X3
X2
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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
\
\
2,3
2,3
\
X3
X2
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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(mn)
• Specialized GAC algorithm for alldifferent runs
  in O(m2 n2) 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(n4) 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(n2) 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 lt Dn-1,n
• Also break symmetry on Xi
• X1 lt X2 lt 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 lt D13, D23 lt 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
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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!