Non-Binary Constraint Satisfaction

1
Non-Binary Constraint Satisfaction

• Toby Walsh
• Cork Constraint Computation Center

2
Overview

• Introduction to CSPs
• Local Consistencies, search algorithms
• Modelling (4 case studies)
• Auxiliary variables, implied constraints
• Redundant models and channelling constraints
• Eliminating non-binary constraints
• Encodings, decomposition
• Theoretical properties
• Tightness, relational consistencies

3
Resources

• Course links
• www.cs.york.ac.uk/tw/Links/csps/
• Benchmark problems
• www.csplib.org
• Constraints solvers
• LP based like ECLIPSE, Java based liked Jsolver,
  open source like Choco,

4
Constraint programming

• Ongoing dream of declarative programming
• State the constraints
• Solver finds a solution
• Paradigm of choice for many hard combinatorial
  problems
• Scheduling, assignment, routing,

5
Constraints are everywhere!

• No meetings before 10am
• Network traffic lt 100 Gbytes/sec
• PCB width lt 21cm
• Salary gt 45k Euros

6
Constraint satisfaction

• Constraint satisfaction problem (CSP) is a triple
  ltV,D,Cgt where
• V is set of variables
• Each X in V has set of values, D_X
• Usually assume finite domain
• true,false, red,blue,green, 0,10,
• C is set of constraints
• Goal find assignment of values to variables to
  satisfy all the constraints

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Example CSP

• Course timetabling
• Variable for each course
• X1, X2 ..
• Domain are possible times for course
• Wed9am, Fri10am, ..
• Constraints
• X1 \ Wed9am
• Capacity constraints atmost(3,X1,X2..,Wed9am)
• Lecturer constraints alldifferent(X1,X5,)

8
Constraint optimization

• CSP objective function
• E.g. objective is Profit Income - Costs
• Find assignment of vals to vars that
• Satisfies constraints
• Maximizes (minimizes) objective
• Often solved as sequence of satisfaction problems
• Profit gt 0, Profit gt Ans1, Profit gt Ans2,

9
Constraint programming v. Constraint logic
programming

• Constraints declaratively specify problem
• Logic programming natural approach
• Assert constraints, call labelling strategy
  (backtracking search predicate)
• But can also define constraint satisfaction or
  optimization within an imperative of functional
  language
• Popular toolkits in C, Java, CAML,

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Constraints

• Constraints are tuples ltS,Rgt where
• S is the scope, X1,X2, Xm
• list of variables to which constraint applies
• R is relation specifying allowed values (goods)
• Subset of D_X1 x D_X2 x x D_Xm
• May be specified intensionally or extensionally

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Constraints

• Extensional specification
• List of goods (or for tight constraints, nogoods)
• Intensional specification
• X1 / X2
• 5X1 6X2 lt X3
• alldifferent(X1,X2,X3,X4),

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Constraint tightness

• Informally, tight constraints admit few tuples
• E.g. on 0/1 vars
• X1 / X2 is loose as half tuples satisfy
• X1X2X3X4X5 lt 1 is tight as only 5 out of 32
  tuples satisfy
• More formal definition later

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Binary v non-binary

• Binary constraint
• Scope covers 2 variables
• E.g. not-equals constraint X1 / X2.
• E.g. ordering constraint X1 lt X2
• Non-binary constraint
• Scope covers 3 or more variables
• E.g. alldifferent(X1,X2,X3).
• E.g. tour(X1,X2,X3,X4).
• Non-binary constraints usually do not
  include unary constraints!

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Constraint graph

• Nodes variables
• Edge between 2 nodes iff constraint between 2
  associated variables
• Few constraints, sparse constraint graph
• Lots of constraints, dense constraint graph

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Some non-binary examples

• Timetabling
• Variables Lecture1, Lecture2,
• Values time1, time2,
• Constraint that lectures taught by same lecturer
  do not conflict
• alldifferent(Lecture1,Lecture5,).

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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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Modelling with non-binary constraints

• Benefits include
• Compact, declarative specifications
• (discussed next)
• Efficient constraint propagation
• (discussed second)

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Modelling with non-binary constraints

• Consider writing your own alldifferent
  constraint
• alldifferent().
• alldifferent(HeadTail)-
• onediff(Head,Tail),
• alldifferent(Tail).

• onediff(El,).
• onediff(El,HeadTail)-
• El \ Head,
• onediff(El,Tail).

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Modelling 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?
• And there exist very efficient algorithms
  for reasoning efficiently with many specialized
  non-binary constraints

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Constraint solvers

• Two main approaches
• Systematic, tree search algorithms
• Local search or repair based procedures
• Other more exotic possibilities
• Hybrid algorithms
• Quantum algorithms

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Systematic solvers

• Tree search
• Assign value to variable
• Deduce values that must be removed from
  future/unassigned variables
• Propagation to ensure some level of consistency
• If future variable has no values, backtrack else
  repeat
• Number of choices
• Variable to assign next, value to assign
• Some important refinements like nogood learning,
  non-chronological backtracking,

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Local search

• Repair based methods
• Generate complete assignment
• Change value to some variable in a violated
  constraint
• Number of choices
• Violated constraint, variable within it,
• Unable to exploit powerful constraint propagation
  techniques

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Constraint propagation

• Arc-consistency (AC)
• 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
• A problem is AC iff every constraint is AC

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Enforcing arc-consistency

• Remove all values that are not AC
• (i.e. have no support)
• May remove support from other values
• (often queue based algorithm)
• Best AC algorithms (AC7, AC-2000) run in O(ed2)
• Optimal if we know nothing else about the
  constraints

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Enforcing arc-consistency

• 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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Properties of AC

• Unique maximal AC subproblem
• Or problem is unsatisfiable
• Enforcing AC can process constraints in any order
• But order does affect (average-case) efficiency

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Non-binary constraint propagation

• Most popular is generalized arc-consistency (GAC)
• 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 again prune values that are not supported
• GAC AC on binary constraints

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GAC on alldifferent

• alldifferent(X1,X2,X3)
• Constraint is not GAC
• X12 cannot be extended
• X2 would have to be 3
• No value left then for X3
• X11 is GAC

X1

• 1,2

2,3
2,3
X2
X3
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Enforcing GAC

• Enforcing GAC is expensive in general
• GAC schema is O(dk)
• On k-ary constraint on vars with domains of size
  d
• Trick is to exploit semantics of constraints
• Regins all-different algorithm
• Achieves GAC in just O(k2 d2)
• On k-ary all different constraint with domains of
  size d
• Based on finding matching in value graph

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Other types of constraint propagation

• (i,j)-consistency due to Freuder, JACM 85
• Non-empty domains
• Any consistent instantiation for i variables can
  be extended to j others
• Describes many different consistency techniques

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(i,j)-consistency

• Generalization of arc-consistency
• AC (1,1)-consistency
• Path-consistency (2,1)-consistency
• Strong path-consistency AC PC
• Path inverse consistency (1,2)-consistency

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Enforcing (i,j)-consistency

• problem is (1,1)-consistent (AC)
• BUT is not (2,1)-consistent (PC)
• X12, X23 cannot be extended to X3
• Need to add constraints
• not(X12 X23)
• not(X12 X33)
• Nor is it (1,2)-consistent (PIC)
• X12 cannot be extended to X2 X3 (so needs to
  be deleted)

1,2
X1
\
\
2,3
2,3
\
X3
X2
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Other types of constraint propagation

• Singleton arc-consistency (SAC)
• Problem resulting from instantiating any variable
  can be made AC
• Restricted path-consistency (RPC)
• AC if a value has just one support then any
  third variable has a consistent value

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Other types of consistency

• problem is (1,1)-consistent (AC)
• BUT is not singleton AC (SAC)
• Problem with X12 cannot be made AC (so value
  should be deleted)
• Nor is it restricted PC (RPC)
• X12 has only one support in X2 (the value 3) but
  X3 then has no consistent values
• X12 can therefore be deleted

1,2
X1
\
\
2,3
2,3
\
X3
X2
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Other types of constraint propagation

• Neighbourhood inverse consistency (NIC)
• For all vals for a var, there are consistent vals
  for all vars in the immediate neighbourhood
• Bounds consistency (BC)
• With ordered domains
• Enforce AC just on max/min elements

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Comparing local consistencies

• Formal definition of tightness introduced by
  Debruyne Bessiere IJCAI-97
• A-consistency is tighter than B-consistency iff
• If a problem is A-consistent -gt it is
  B-consistent
• We write A gt B

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Properties

• Partial ordering
• reflexive A ? A
• transitive A ? B B ? C implies A ? C
• Defined relations
• tighter A gt B iff A ? B not B ?
  A
• incomparable A _at_ B iff neither A ? B
• 
  nor B ? A

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Comparison of consistency techniques

• Exercise for the reader, prove the following
  identities!
• Strong PC gt SAC gt PIC gt RPC gt AC gt BC
• NIC gt PIC
• NIC _at_ SAC
• NIC _at_ Strong PC
• NB gaps can reduce search exponentially!

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Which to choose?

• For binary constraints, AC is often chosen
• Space efficient
• Just prune domains (cf PC)
• Time efficient
• For non-binary constraints GAC is often chosen
• If we can exploit the constraint semantics to
  keep it cheap!

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Why consider these other consistencies?

• Promising experimental results
• Useful pruning for their additional cost
• Theoretical value
• E.g. GAC on non-binary constraints may exceed SAC
  on equivalent binary model

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Maintaining a local consistency property

• Tree search
• Assign value to variable
• Enforce some level of local consistency
• Remove values/add new constraints
• If any future variable has no values, backtrack
  else repeat
• Two popular algorithms
• Maintaining arc-consistency (MAC)
• Forward checking (very restricted form of AC
  maintained)

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Forward checking

• Binary constraints (FC)
• Make constraints involving current variable and
  one future variable arc-consistent
• No need to look at any other constraints!
• Non-binary constraints
• Several choices as to how to do forward checking

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Forward checking with non-binary constraints

• nFC0 makes AC only those k-ary constraints with
  k-1 variables set
• nFC1 applies one pass of AC on constraints and
  projections involving current var and one future
  var
• nFC2 applies one pass of GAC on constraints
  involving current var and at least one future var
• nFC3 enforces GAC on this set
• nFC4 applies one pass of GAC on constraints
  involving at least one past and one future var
• nFC5 enforces GAC on this set

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Summary

• Constraint solving
• Constraint propagation central part of many
  algorithms
• Binary v non-binary constraints
• Compact, more efficient representation