Directional Consistency

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Directional Consistency

• Chapter 4

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Section 4.1

• Problem simplification as a justification for
  humans ability to handle difficult problems
• Abstraction, reformulation, approximation
• Approximate solutions or heuristic guidance
• Easy problems ? solvable in polynomial time
• Search easy problems ? backtrack-free
• Bactrack-free partial solutions can always be
  extended consistently to one more variable
• Tractability
• Graph topology ? focus of this chapter
• Constraint semantic

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Section 4.2

• Tractability by restricting graph topology
• Graph parameter (induced) width

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Section 4.2.1 Width

• Given an undirected graph
• Given an ordering of the nodes
• Node width in the ordering is the number of
  parents of the node in the ordering
• Width of ordering is the max of width of nodes
• Graph width is the minimal width across all
  possible orderings

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Section 4.2.1 Induced width

• Induced graph of an ordered graph (elimination,
  moralization)
• Induced width of ordered graph is the width of
  the moralized graph
• Induced width of the graph is the minimal induced
  width over all orderings

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Section 4.2.1 Width of trees

• Trees have no cycles
• G is a tree ? (induced) width of G is 1
• Width of a tree polynomial
• Induced width of a tree NP-complete
• Deciding whether there is an ordering with
  induced width k is O(nk)
• Greedy approximation eliminate node of min
  degree, connect neighbors

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Section 4.2.1 Chordal graphs

• Chordal ? triangulated, subclass of Perfect
  Graphs
• Maximal cliques is easy on chordal graphs
• Use max-cardinality ordering
• Max-cardinality ordering identifies chordal
  graphs
• Iff each vertex and its parents form a clique
• Ordered graph ? induced graph
• Width ? induced width
• Proposition 4.2.5 induced graphs are chordal

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Section 4.2.1 k-trees

• K-trees are a subclass of chordal graphs
• Max cliques are (exactly) of size (k1)
• A graph G can be embedded in a k-tree ? induced
  width of G w is ? k

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Section 4.3

• Goal amount of inference to guarantee
  backtrack-free search
• Amount of inference
• Constraint propagation level (e.g., full arc,
  full path, full i-consistency)
• Limit inference to a given ordering of the
  variables
• Example arc-consistency is required only in the
  direction to be exploited by search sic
• FAC/FPC versus DAC/DPC

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Section 4.3 directional AC/PC

• Section 4.3 directional consistency
• Directional arc, path, i-consistency
• Section 4.5 adaptive consistency
• Chapter 8 relational consistency
• DAC/DPC
• Simplest forms binary constraints
• Generalized to arbitrary constraints generalized
  arc-consistency and relational consistency

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Section 4.3.1 Directional AC

• In DAC, each constraint is processed exactly
  once, O(ek2) k is domain size
• Computational advantage of DAC vs FAC
• FAC ? DAC, DAC is weaker than FAC
• a fortiriori, DAC is not sufficient in general
  and higher levels of consistency are neccessary
• but is sometimes sufficient to yield a
  backtrack-free search (see example page 101).

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Section 4.3.2 Directional PC

• DPC vs FPC see Example 4.3.6
• DPC in Fig 8, relative to ordering d
• realizes DAC (updates domains)
• realizes DPC (updates binary constraints)
• manages addition of new arcs

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Section 4.3.3 Directional iC

• To generalize to i-consistency we need to look a
  larger scopes (than 2 or 3)