Consistency algorithms

1
Consistency algorithms

• Chapter 3

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Consistency methods

• Approximation of inference
• Arc, path and i-consistecy
• Methods that transform the original network into
  a tighter and tighter representations

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Arc-consistency
X
Y
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2,
1,
3
2,
1,
1 ? X, Y, Z, T ? 3 X ? Y Y Z T ? Z X ? T

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2,
1,
3
2,
1,
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T
Z
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Arc-consistency
X
Y
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1 ? X, Y, Z, T ? 3 X ? Y Y Z T ? Z X ? T

?
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T
Z
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Arc-consistency
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Arc-consistency
Only domain constraints are recorded
Example
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Revise for arc-consistency
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Figure 3.3 (a) Matching diagram describing a
network of constraints that is not arc-consistent
(b) An arc-consistent equivalent network.
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AC-1

• Complexity (Mackworth and Freuder, 1986)
• e number of arcs, n variables,k values
• (ek2, each loop, nk number of loops), best-case
  ek,
• Arc-consistency is

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AC-3

• Complexity
• Best case O(ek), since each arc may be processed
  in O(2k)

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Example A three variable network, with two
constraints z divides x and z divides y (a)
before and (b) after AC-3 is applied.
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AC-4

• Complexity
• (Counter is the number of supports to ai in xi
  from xj. S_(xi,ai) is the set of pairs that
  (xi,ai) supports)

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Example applying AC-4
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Distributed arc-consistency(Constraint
propagation)

• Implement AC-1 distributedly.
• Node x_j sends the message to node x_i
• Node x_i updates its domain
• Messages can be sent asynchronously or scheduled
  in a topological order

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Exercise make the following network
arc-consistent

• Draw the networks primal and dual constraint
  graph
• Network
• Domains 1,2,3,4
• Constraints y lt x, z lt y, t lt z, fltt, xltt1,
  Yltf2

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Arc-consistency Algorithms

• AC-1 brute-force, distributed
• AC-3, queue-based
• AC-4, context-based, optimal
• AC-5,6,7,. Good in special cases
• Important applied at every node of search
• (n number of variables, econstraints, kdomain
  size)
• Mackworth and Freuder (1977,1983), Mohr and
  Anderson, (1985)

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Is arc-consistency enough?

• Example a triangle graph-coloring with 2 values.
• Is it arc-consistent?
• Is it consistent?
• It is not path, or 3-consistent.

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Path-consistency
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Path-consistency
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Revise-3

• Complexity O(k3)
• Best-case O(t)
• Worst-case O(tk)

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PC-1

• Complexity
• O(n3) triplets, each take O(k3) steps ? O(n3
  k3)
• Max number of loops O(n2 k2) .

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PC-2

• Complexity
• Optimal PC-4
• (each pair deleted may add 2n-1 triplets, number
  of pairs O(n2 k2) ? size of Q is O(n3 k2),
  processing is O(k3))

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Example before and after path-consistency

• PC-1 requires 2 processings of each arc while
  PC-2 may not
• Can we do path-consistency distributedly?

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Path-definition of path-consistency
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Path-consistency Algorithms

• Apply Revise-3 (O(k3)) until no change
• Path-consistency (3-consistency) adds binary
  constraints.
• PC-1
• PC-2
• PC-4 optimal

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I-consistency
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Higher levels of consistency, global-consistency
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Revise-i

• Complexity for binary constraints
• For arbitrary constraints

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4-queen example
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I-consistency
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Path-consistency vs 3-consistency
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Arc-consistency for non-binary
constraintsGeneralized arc-consistency

• Complexity O(t k), t bounds number of tuples.
• Relational arc-consistency

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Examples of generalized arc-consistency

• xyz lt 15 and z gt 13 implies
• xlt2, ylt2
• Example of relational arc-consistency

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More arc-based consistency

• Global constraints e.g., all-different
  constraints
• Special semantic constraints that appears often
  in practice and a specialized constraint
  propagation. Used in constraint programming.
• Bounds-consistency pruning the boundaries of
  domains

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Sudoku Constraint Satisfaction

• Variables empty slots
• Domains 1,2,3,4,5,6,7,8,9
• Constraints
• 27 all-different

• Constraint
• Propagation
• Inference

2 34 6
2
Each row, column and major block must be
alldifferent Well posed if it has unique
solution 27 constraints
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Global constraints
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Example for alldiff

• A 3,4,5,6
• B 3,4
• C 2,3,4,5
• D 2,3,4
• E 3,4
• Alldiff (A,B,C,D,E
• Arc-consistency does nothing
• Apply GAC to sol(A,B,C,D,E)?
• ? A 6, F 1.
• Alg bipartite matching kn1.5
• (Lopez-Ortiz, et. Al, IJCAI-03 pp 245 (A fast and
  simple algorithm for bounds consistency of
  alldifferent constraint)

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Global constraints

• Alldifferent
• Sum constraint (variable equal the sum of others)
• Global cardinality constraint (a value can be
  assigned a bounded number of times to a set of
  variables)
• The cummulative constraint (related to scheduling
  tasks)

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Bounds consistency
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Bounds consistency for Alldifferent constraints
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Boolean constraint propagation

• (A V B) and (B)
• B is arc-consistent relative to A but not
  vice-versa
• Arc-consistency by resolution
• res((A V B),B) A
• Given also (B V C), path-consistency
• Res((A V B),(B V C) (A V C)
• What will generalized arc-consistency can do to
  cnfs?
• Relational arc-consistency rule unit-resolution

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Boolean constraint propagation
Example party problem

• If Alex goes, then Becky goes
• If Chris goes, then Alex goes
• Query
• Is it possible that Chris goes to the party
  but Becky does not?

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Gausian and Boolean propagation

• Linear inequalities
• Boolean constraint propagation

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Constraint propagation for Boolean constraints
Unit propagation
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Arc-consistency for non-binary constraints
Generalized arc-consistency (reminder)

• Complexity O(t k), t bounds number of tuples.
• Relational arc-consistency

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Algorithms for relational and generalized
arc-cnsistency

• Think about the following
• GAC-i apply AC-i to the dual problem when
  singleton variables are explicit the bi-partite
  representation.
• What is the complexity?
• Relational arc-consistency imitate unit
  propagation.
• Apply AC-1 on the dual problemwhere each subset
  of a scope is presented.
• Is unit propagation equivalent to AC-4?

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Consistency for numeric constraints
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Tractable classes
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Changes in the network graph as a result of
arc-consistency, path-consistency and
4-consistency.
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Distributed arc-consistency(Constraint
propagation)

• Implement AC-1 distributedly.
• Node x_j sends the message to node x_i
• Node x_i updates its domain
• Generalized arc-consistency can be implemented
  distributedly sending messages between
  constraints over the dual graph

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Distributed Arc-Consistency

• Arc-consistency can be formulated as a
  distributed algorithm

A
B
C
D
F
G
a Constraint network
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Relational Arc-consistency
A
The message that R2 sends to R1 is R1 updates
its relation and domains and sends messages to
neighbors
B
C
D
F
G
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DRAC on the dual join-graph
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Distributed Relational Arc-Consistency

• DRAC can be applied to the dual problem of any
  constraint network

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DR-AC on the dual graph
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AB
AC
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AB
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ABD
BCF
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DFG
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Iteration 1
Node 6 sends messages
Node 5 sends messages
Node 4 sends messages
Node 3 sends messages
Node 2 sends messages
Node 1 sends messages

1
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AB
AC
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DFG
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Iteration 1
B
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Iteration 1
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Iteration 2
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Iteration 2
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Iteration 3
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Iteration 3
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Iteration 4
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Iteration 4
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Iteration 5
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Iteration 5