1 Directional consistency Chapter 4 ICS-275 Fall 2010 Fall 2010 ICS 275 - Constraint Networks
1
Directional consistency
Chapter 4
ICS-275
Fall 2010
Fall 2010 ICS 275 - Constraint Networks
Fall 2010 2
Tractable classes
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Backtrack-free search: or
What level of consistency will guarantee global-
consistency
Backtrack free and queries:
Consistency,
All solutions
Counting
optimization
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Directional arc-consistency:
another restriction on propagation
D4={white,blue,black}
D3={red,white,blue}
D2={green,white,black}
D1={red,white,black}
X1=x2, x1=x3,x3=x4
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Directional arc-consistency:
another restriction on propagation
D4={white,blue,black}
D3={red,white,blue}
D2={green,white,black}
D1={red,white,black}
X1=x2,
x1=x3,
x3=x4
After DAC:
D1= {white},
D2={green,white,black},
D3={white,blue},
D4={white,blue,black}
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Algorithm for directional arc-
consistency (DAC)
)( 2ekO Complexity:
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Directional arc-consistency may not be enough
Directional path-consistency
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Algorithm directional path consistency (DPC)
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Example of DPC
E
D
A
C
B
}2,1{
}2,1{}2,1{
}2,1{ }3,2,1{
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Directional i-consistency
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Algorithm directional i-consistency
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The induced-width
DPC recursively connects parents in the ordered graph,
yielding:
Width along ordering d, w(d):
• max # of previous parents
Induced width w*(d):
• The width in the ordered
induced graph
Induced-width w*:
• Smallest induced-width
over all orderings
Finding w*
• NP-complete (Arnborg,
1985) but greedy heuristics
(min-fill).
E
D
A
C
B
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Induced-width
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Induced-width and DPC
The induced graph of (G,d) is denoted
(G*,d)
The induced graph (G*,d) contains the
graph generated by DPC along d, and
the graph generated by directional i-
consistency along d.
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Refined complexity using induced-width
Consequently we wish to have ordering with minimal
induced-width
Induced-width is equal to tree-width to be defined later.
Finding min induced-width ordering is NP-complete
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Greedy algorithms for induced-width
• Min-width ordering
• Max-cardinality ordering
• Min-fill ordering
• Chordal graphs
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Min-width ordering
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Min-induced-width
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Min-fill algorithm
Prefers a node who adds the least
number of fill-in arcs.
Empirically, fill-in is the best among the
greedy algorithms (MW,MIW,MF,MC)
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Cordal graphs and max-
cardinality ordering
A graph is cordal if every cycle of length at
least 4 has a chord
Finding w* over chordal graph is easy using
the max-cardinality ordering
If G* is an induced graph it is chordal
K-trees are special chordal graphs.
Finding the max-clique in chordal graphs is
easy (just enumerate all cliques in a max-
cardinality ordering
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Example
We see again that G in Figure 4.1(a) is not chordal
since the parents of A are not connected in the max-
cardinality ordering in Figure 4.1(d). If we connect B
and C, the resulting induced graph is chordal.
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Max-cardinality ordering
Figure 4.5 The max-cardinality (MC) ordering procedure.
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Width vs local consistency:
solving trees
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Tree-solving
)(: 2nkOcomplexity
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Width-2 and DPC
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Width vs directional consistency
(Freuder 82)
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Width vs i-consistency
DAC and width-1
DPC and width-2
DIC_i and with-(i-1)
backtrack-free representation
If a problem has width 2, will DPC make it
backtrack-free?
Adaptive-consistency: applies i-consistency
when i is adapted to the number of parents
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Adaptive-consistency
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Bucket E: E D, E C
Bucket D: D A
Bucket C: C B
Bucket B: B A
Bucket A:
A C
widthinduced -*
*
w
))exp(w O(n :Complexity
contradiction
=
D = C
B = A
Bucket Elimination
Adaptive Consistency (Dechter & Pearl, 1987)
=
Fall 2010 ICS 275 - Constraint Networks 30
dordering along widthinduced -(d)
,
*
*
w
(d)))exp(w O(n :space and Time
E
D
A
C
B
}2,1{
}2,1{}2,1{
}2,1{ }3,2,1{
:)(
AB :)(
BC :)(
AD :)(
BE C,E D,E :)(
ABucket
BBucket
CBucket
DBucket
EBucket
A
E
D
C
B
:)(
EB :)(
EC , BC :)(
ED :)(
BA D,A :)(
EBucket
BBucket
CBucket
DBucket
ABucket
E
A
D
C
B
|| RD
BE ,
|| RE
|| RDB
|| RDCB
|| RACB
|| RAB
RA
RC
BE
Bucket Elimination
Adaptive Consistency (Dechter & Pearl, 1987)
Fall 2010 ICS 275 - Constraint Networks 31
The Idea of Elimination
project and join E variableEliminate
ECDBC EBEDDBC RRRR
3
value assignment
D
B
C
RDBC
eliminating E
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Variable Elimination
Eliminate variablesone by one:“constraintpropagation”
Solution generation after elimination is backtrack-free
3
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Adaptive-consistency, bucket-elimination
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Properties of bucket-elimination
(adaptive consistency)
Adaptive consistency generates a constraint network that is backtrack-free (can be solved without dead-ends).
The time and space complexity of adaptive consistency along ordering d is respectively, or O(r k^(w*+1)) when r is the number of constraints.
Therefore, problems having bounded induced width are tractable (solved in polynomial time)
Special cases: trees ( w*=1 ), series-parallel networks(w*=2 ), and in general k-trees ( w*=k ).
1*w1*w (k ) O (n),(2 k ) O (n
1*w
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Back to Induced width
Finding minimum-w* ordering is NP-complete
(Arnborg, 1985)
Greedy ordering heuristics: min-width, min-degree,
max-cardinality (Bertele and Briochi, 1972; Freuder
1982), Min-fill.
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Solving Trees
(Mackworth and Freuder, 1985)
Adaptive consistency is linear for trees andequivalent to enforcing directional arc-consistency (recording only unary constraints)
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Summary: directional i-consistency
D CBR
A
E
CD
B
D
CB
E
D
CB
E
D
CB
E
:A
B A:B
BC :C
AD C,D :D
BE C,E D,E :E
Adaptive d-arcd-path
D BD C RR ,
CBRDR
CR
DR
Fall 2010 ICS 275 - Constraint Networks 38
Relational consistency
(Chapter 8)
Relational arc-consistency
Relational path-consistency
Relational m-consistency
Relational consistency for Boolean and linear constraints:• Unit-resolution is relational-arc-consistency
• Pair-wise resolution is relational path-consistency
Fall 2010 ICS 275 - Constraint Networks 39
Sudoku’s propagation
http://www.websudoku.com/
What kind of propagation we do?
Sudoku
Each row, column and major block must be
alldifferent
“Well posed” if it has unique solution: 27 constraints
2 34 62
Constraint propagation
•Variables: 81 slots
•Domains = {1,2,3,4,5,6,7,8,9}
•Constraints: • 27 not-equal
Sudoku
Each row, column and major block must be alldifferent
“Well posed” if it has unique solution