Local Search and Consistency Techniques for CSP

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Local Search and Consistency Techniques for CSP

• -Constraint propagation for CSP

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Local Search

• Local search usually refers to search methods
  that are incomplete. Stochastic search is one
  example (a later lecture), and general
  i-consistency is another.

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Contents

• I-consistency
• Path-consistency
• Singleton consistency
• Global constraints
• Bounds-consistency

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Higher level consistency

• Arc-consistency checks inconsistencies between
  two variables, which is inadequate sometimes.
• Example
• Dx Dy D z red, blue
• x ! y, y ! z, z ! x
• Although we can have assignments ltx, redgt and
• lty, bluegt, satisfying the first constraint, no
  color for z will be consistent with ltx, redgt and
  lty, bluegt, respectively.

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I-consistency

• A CSP is i-consistent iff given any consistent
  instantiation of i-1 variables, there exists an
  instantiation of any ith variable such that the i
  values taken together satisfy all of the
  constraints among the i variables.
• A CSP is strongly i-consistent iff it is
  j-consistent for all j lt i.
• Globally consistent A CSP is strongly
  i-consistent where i is the number of variables.
• Globally consistent CSPs immediate solution
  without search. But enforcing global consistency
  takes exponential time.

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• i1 Node-consistency
• e.g. x gt 100
• All values of domain of x less than 100
• can be removed.
• i2 Arc-consistency (AC)
• i3 3-consistency (coincides with
  path-consistency for binary CSPs)

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Path-consistency

• Given a CSP, a two variable pair (x,y) is
  path-consistent
• relative to z iff for any consistent assignment
  x ? a,
• y ? b, there is a value c ? Dz such that x ? a,
  z ? c is
• consistent and y ? b, z? c is consistent.
• A binary constraint c(x,y) is path consistent
  relative to z iff for every pair (a,b) ? c(x,y),
  there is a value c ? Dz, which is a neighbor of
  both x and y, such that (a,c) ? c(x,z) and (b,c)
  ? C(y,z).
• A CSP is path consistent iff for every constraint
  c(x,y) and for any z, z is a neighboring variable
  of both x and y, but different from x and y,
  c(x,y) is path consistent relative to z.

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Example

• Domains of all variables, x , y and z, are 0,1
• c(x,y) (0,1),(1,0)
• c(x,z) (0,1),(1,0)
• c(z,y) (0,1),(1,0)
• This CSP is arc-consistent but not
  path-consistent.
• Consider (0,1) ? c(x,y), there is no value v for
  z such that (0,v) ?c(x,z) and (v,1) ? c(z,y).

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Example

• All domains are 0,1,2,3
• c(x,y) (0,0),(0,1),(1,2),(1,3),(2,0),(2,1),
  (3,2),(3,3)
• c(x,z) (0,0),(0,1),(1,2),(1,3),(2,0),(2,1),
  (3,2),(3,3)
• c(z,y) (0,2),(1,3),(2,0),(3,1),(1,2),(0,3),
  (3,0),(2,1)
• This CSP is arc-consistent.
• But for any pair (a,b) ? c(x,y), there is no
  value v ? Dz that can be used to verify
  path-consistency.

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Higher level consistency is about tuples being no
good

• Unlike arc-consistency, enforcing
  path-consistency doesnt reduce domains directly.
• Rather, when a CSP is not path-consistent, it
  simply means some pair of values cannot take
  place simultaneously in any solution of the CSP.
• Therefore, enforcing path-consistency may add
  constraints or tighten constraints (delete
  tuples).

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In picture

x
y
z
Picture of x and y being path-consistent relative
to z.
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CSP in Picture DxDyDz red, blue
x ! y, y ! z, z ! x

x
r
b
b
r
y
r
z
b
Enforcing the constraint (x ! y)
path-consistent relative to z will empty it,
declaring inconsistency.
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Enforcing path-consistency may add constraints

• Example
• DxDyDz 1,2
• c(x,z) (1,1),(2,2), c(y,z)
  (1,1),(2,1)
• No constraint over x and y, meaning every
  tuple is allowed. But variable pair ltx,ygt is not
  path-consistent w. r.t. z. Enforcing
  path-consistency on ltx,ygt results in a new, but
  equivalent CSP, by adding
• c(x,y) (1,1), (1,2)
• This means, the tuples over x and y, namely
  (2,1) and (2,2) are inconsistent.

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Singleton arc-consistency (SAC)

• Idea (filtering of single values)
• Suppose we commit a value v to a variable x,
  and then
• enforce AC. If any domain becomes empty, then it
  means the value v for x cannot lead to any
  solution. So, we can safely remove v from Dx.
• Definition of SAC
• A CSP is singleton arc-consistent iff for any
  variable x with a non-empty domain, and for any a
  ? Dx, the CSP obtained by restricting the domain
  Dx to a, can be made arc-consistent with
  non-empty domains.

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Singleton i-consistency

• Similarly, the idea of SAC can be applied to any
  i-consistency
• Restricting a variable to a value, and enforce
  i-consistency If this leads to inconsistency
  (some constraint cannot be satisfied), then the
  value is removed from the domain of the variable.
• The process is done for each value and each
  variable. So, zero or more values may be removed.

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Example (earlier)

• All domains are 0,1,2,3
• c(x,y) (0,0),(0,1),(1,2),(1,3),(2,0),(2,1),
  (3,2),(3,3)
• c(x,z) (0,0),(0,1),(1,2),(1,3),(2,0),(2,1),
  (3,2),(3,3)
• c(z,y) (0,2),(1,3),(2,0),(3,1),(1,2),(0,3),
  (3,0),(2,1)
• This CSP is arc-consistent but not SAC Suppose
  we restrict Dx to 0, we get a CSP
• c(x,y) (0,0),(0,1)
• c(x,z) (0,0),(0,1)
• c(z,y) (0,2),(1,3),(2,0),(3,1),(1,2),(0,3),
  (3,0),(2,1)
• Enforcing arc-consistency removes 2 and 3 from Dy
  and all the values from Dz. The fact that a
  domain becomes empty means ltx,0gt is no good.

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Global Constraints

• Two objectives
• Easier modeling
• Efficient processing Specialized propagators
  designed for frequently used constraints
• allDifference(x1,,xn)
• values assigned to these variables be mutually
  different.

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AllDifference

• Alldifference can be expressed by a collection of
  binary non-equal constraints but applying
  arc-consistency often has no effect.
• CSP
• DxDyDz Ds 1,2
• x ! y, x ! z, x ! s
• y ! z, y ! s
• z ! s
• Though clearly this CSP has no solution, it is
  arc-consistent.

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Global Constraints

• A large collection, E.g.,
• Sum making one variable equal to the sum of k
  variables.
• Knapsack constraint A generalization of Sum
• Global cardinality constraint (generalizes
  allDifference) A set of variables
• X x1,xn
• take their values in a subset of
• V v1,vm
• such that the number of times a value vi is
  assigned to a variable in X in constrained in an
  interval 1,k.

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Global Constraints

• Cumulative constraints
• Enforce, at each point in time, that the
  cumulative
• resource consumption of a set of tasks that
• overlaps does not exceed the specified capacity.
• Each task has a start, a duration, and a resource
  
• consumption (which are all domain variables).

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Bounds-consistency

• Idea
• For large integer domains, we can narrow variable
  domains by moving the bounds inwards (increase
  lower bound and reduce the upper bound).
• E.g. DxDy1,1000
• x gt y 500
• Immediately we know the domains can be cut,
• New domain of x 501,1000
• New domain of y 1,499

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Bounds-consistency Definition

• Let c(xi,,xj) be a constraint where the variable
  domains are all well-defined orders.
• Notation min(Dx ) denotes the minimum value of
  Dx .
• A variable x ? xi,,xj is bounds-consistent
  relative to c(xi,,xj) if the value min(Dx)
  (resp. max(Dx)) can be extended to a full tuple t
  of c(xi,,xj).
• A constraint is bounds-consistent if each of its
  variables is bounds-consistent.

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Example

• Variables x1, , x6, all with domain 1,6
• c1 x4 gt x1 3 c3 x5
  gt x3 3
• c2 x4 gt x2 3 c4 x5
  gt x4 1
• c5 allDifferent(x1, x2, x3, x4, x5)
• Enforcing bounds-consistency
• Using constraints c1 through c4 (repeatedly)
• D1D2 1,2, D31,2,3,
  D44,5, D55,6
• Using C5 produces
• D3 3
• Now, C3 is no longer bounds-consistent, using it
  we get
• D5 6.
• Is the resulting CSP arc-consistent?