1
Chapter 3
Fall 2003 ICS 275A - Constraint Networks 2
Approximation of inference:• Arc, path and i-consistecy
Methods that transform the original network into a tighter and tighter representations
Fall 2003 ICS 275A - Constraint Networks 3
Arc-consistency
32,1,
32,1, 32,1,
1 ≤≤≤≤ X, Y, Z, T ≤≤≤≤ 3X <<<< YY = ZT <<<< ZX ≤≤≤≤ T
X Y
T Z
32,1,<<<<
=
<
∧
Fall 2003 ICS 275A - Constraint Networks 4
1 ≤≤≤≤ X, Y, Z, T ≤≤≤≤ 3X <<<< YY = ZT <<<< ZX ≤≤≤≤ T
X Y
T Z
<<<<
=
<
∧
1 3
2 3
Arc-consistency
Fall 2003 ICS 275A - Constraint Networks 5
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∏←A BABA DRR
.2,1 to of domain reduces
constriant ,3,2,1 ,3,2,1=
<==X
YX
RXYXRR
Fall 2003 ICS 275A - Constraint Networks 7
)( jijiii DRDD ⊗∩← π
Fall 2003 ICS 275A - Constraint Networks 8
!"
Fall 2003 ICS 275A - Constraint Networks 9
#
)( 3enkO Complexity (Mackworth and Freuder, 1986): e = number of arcs, n variables,k values (ek^2, each loop, nk number of loops), best-case = ek,
Arc-consistency is: )( 2ekΩ
Fall 2003 ICS 275A - Constraint Networks 10
)( 3ekO Complexity: Best case O(ek), since each arc may be processed in O(2k)
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$% &"" '
( % (
&&"
Fall 2003 ICS 275A - Constraint Networks 12
)
)( 2ekO Complexity: (Counter is the number of supports to ai in xi from xj. S_(xi,ai) is the
set of pairs that (xi,ai) supports)
Fall 2003 ICS 275A - Constraint Networks 13
*
&&
Implement AC-1 distributedly.
Node x_j sends the message to node x_i
Node x_i updates its domain:
Messages can be sent asynchronously or scheduled in a topological order
)( jijij
i DRh ⊗← π
)( jijiii DRDD ⊗∩← π
jiii
jijiii
hDD
DRDD
∩←
=⊗∩← )(π
Fall 2003 ICS 275A - Constraint Networks 14
+,
Example: a triangle graph-coloring with 2 values.• Is it arc-consistent?• Is it consistent?
It is not path, or 3-consistent.
Fall 2003 ICS 275A - Constraint Networks 15
-
Fall 2003 ICS 275A - Constraint Networks 16
-
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)( kjkikijijij RDRRR ⊗⊗∩← π
Complexity: O(k^3) Best-case: O(t) Worst-case O(tk)
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-#
Complexity: O(n^3) triplets, each take O(k^3) steps O(n^3 k^3) Max number of loops: O(n^2 k^2) .
)( 55knO
Fall 2003 ICS 275A - Constraint Networks 19
-.
)( 53knO Complexity: Optimal PC-4: (each pair deleted may add: 2n-1 triplets, number of pairs: O(n^2 k^2)
size of Q is O(n^3 k^2), processing is O(k^3))
)( 33knO
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$% &" &
PC-1 requires 2 processings of each arc while PC-2 may not Can we do path-consistency distributedly?
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+
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/""'
""
Fall 2003 ICS 275A - Constraint Networks 23
)( ikO Complexity: for binary constraints For arbitrary constraints: ))2(( ikO
Fall 2003 ICS 275A - Constraint Networks 24
)!% &"
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+
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0"(
Complexity: O(t k), t bounds number of tuples.Relational arc-consistency:
)( xSSxxx DRDD −⊗∩← π
)( xSxSxS DRR ⊗← −− π
Fall 2003 ICS 275A - Constraint Networks 27
$% &""(
x+y+z <= 15, z >= 13 x<=2, y<=2
Example of relational arc-consistency
BAGGBA ¬∨¬¬→∧ ,,
Fall 2003 ICS 275A - Constraint Networks 28
Global constraints: e.g., all-different constraints• Special semantic constraints that appears
often in practice and a specialized constraint propagation. Used in constraint programming.
Bounds-consistency: pruning the boundaries of domains
Do exercise 16
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$% &"""
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 kn^1.5 (Lopez-Ortiz, et. Al, IJCAI-03 pp 245 (A fast and simple
algorithm for bounds consistency of alldifferent constraint)
Fall 2003 ICS 275A - Constraint Networks 30
0""
Alldifferent Sum constraint Global cardinality constraint (a value can
be assigned a bounded number of times) The cummulative constraint (related to
scheduling tasks)
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1
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1 ""
Fall 2003 ICS 275A - Constraint Networks 33
1"&&
(A V ~B) and (B)• B is arc-consistent relative to A but not vice-versa
Arc-consistency achieved by resolution:res((A V ~B),B) = A
Given also (B V C), path-consistency means:res((A V ~B),(B V C)) = (A V C)
What can generalized arc-consistency do to cnfs?Relational arc-consistency rule = unit-resolution
Fall 2003 ICS 275A - Constraint Networks 34
1"&&
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?
) (or, BA BA ∨¬→) (or, ACA C ∨¬→
e?satisfiabl Is
¬∨¬∨¬= C B, A,C B,Aϕtheorynalpropositio
Fall 2003 ICS 275A - Constraint Networks 35
&&1"
2&&
Fall 2003 ICS 275A - Constraint Networks 36
" ""
"(
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 problem where each subset of
a scope is presented.• Is unit propagation equivalent to AC-4?
Fall 2003 ICS 275A - Constraint Networks 37
3,10,7
3],10,10[
5,10]9,5[],5,1[
10],15,5[],10,1[
−≥−−=+−−≥−−
≤+−∈
−≤−=+−−∈∈−
=+∈∈
zyyxaddingbyobtained
zxyconsistencpath
zy
z
yyxaddingby
yxyconsistencarc
yx
yx
Fall 2003 ICS 275A - Constraint Networks 38
3""
Fall 2003 ICS 275A - Constraint Networks 39
&"
'& )
Fall 2003 ICS 275A - Constraint Networks 40
*
&&
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:
)( jijij
i DRh ⊗← π
)( jijiii DRDD ⊗∩← π
jiii hDD ∩←
)( xSxSxS DRR ⊗← −− π
Fall 2003 ICS 275A - Constraint Networks 41
21
3
A
232C
1A
12
32
1323
32B
11A
13F
232C
1B
3121322131
23D
23
32B
11A
33G
12F
21D
1R
2R
4R
3R
5R
6R
* ""
% &"
A
B C
D F
G
The message that R2 sends to R1 is
R1 updates its relation and domains and sends messages to neighbors
Fall 2003 ICS 275A - Constraint Networks 42
*
b) Constraint network
DR-AC can be applied to the dual problem of any constraint network.
A
AB AC
ABD BCF
DFG
A
AB
A
A
AB
C
B
D F
A
Fall 2003 ICS 275A - Constraint Networks 43
2
1
3
A
232C
1A
12321323
32B
11A
13F
232C
1B
3121322131
23D
23
32B
11A
33G
12F
21D
1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
* "4&
Fall 2003 ICS 275A - Constraint Networks 44
21
3
A
232C
1A
12321323
32B
11A
13F
232C
1B
3121322131
23D
23
32B
11A
33G
12F
21D
21
3
A21
3
A
21
3
A13
A
21
3
B13
B
21
3
A12
D
13
F
21
3
D
2C
21
3
B
2C
13
F
21
3
A
21
3
B
+#
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
21h
31h
13h1
2h 14h
35h
24h2
5h
41h4
6h 42h
65h6
4h
52h 5
3h 56h
1R
2R
4R
3R
5R
6R
Fall 2003 ICS 275A - Constraint Networks 45
13
A
232C
1A12
3213
3B
1A
13F
232C
1B
1322131
2D
23
3B
1A
3G
1F
2D
1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
+#
Fall 2003 ICS 275A - Constraint Networks 46
13
A
232C
1A12
3213
3B
1A
13F
232C
1B
1322131
2D
23
3B
1A
3G
1F
2D
13
A
13
A
21
3
A13
A
13
A2D
13
F
21D
2C
13
B
2C
1F
21
3
A
13
B
21
3
B
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
21h
31h
13h1
2h 14h
35h
24h2
5h
41h4
6h 42h
65h6
4h
52h 5
3h 56h
1R
2R
4R
3R
5R
6R
+.
13
B
Fall 2003 ICS 275A - Constraint Networks 47
13
A
232C
1A13
3B
1A
1F
23CB
2132D
3B
1A
3G
1F
2D
+.1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
Fall 2003 ICS 275A - Constraint Networks 48
13
A
13
A
13
A
3B
13
A2D
1F
2D
2C
13
B
2C
1F
13
A13
A13
A
232C
1A13
3B
1A
1F
23CB
2132D
3B
1A
3G
1F
2D
13
B
13
B
21h
31h
13h1
2h 14h
35h
24h2
5h
41h4
6h 42h
65h6
4h
52h 5
3h 56h
1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
+
Fall 2003 ICS 275A - Constraint Networks 49
13
A
232C
1A
3B
1A
1F
23CB
2132D
3B
1A
3G
1F
2D
1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
+
Fall 2003 ICS 275A - Constraint Networks 50
13
A
13
A
1A
3B
13
A2D
1F
2D
2C
3B
2C
1F
13
A13
A13
A
232C
1A
3B
1A
1F
23CB
2132D
3B
1A
3G
1F
2D
13
B
13
B
21h
31h
13h1
2h 14h
35h
24h2
5h
41h4
6h 42h
65h6
4h
52h 5
3h 56h
1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
+)
Fall 2003 ICS 275A - Constraint Networks 51
1A
232C
1A
3B
1A
1F
23CB
2D
3B
1A
3G
1F
2D
1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
+)
Fall 2003 ICS 275A - Constraint Networks 52
1A
1A
1A
3B
3B
3B
1A
2D
1F
2D
2C
3B
2C
1F
1A
1A
1A
232C
1A
3B
1A
1F
23CB
2D
3B
1A
3G
1F
2D
21h
31h
13h1
2h 14h
35h
24h2
5h
41h4
6h 42h
65h6
4h
52h 5
3h 56h
1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
+5
Fall 2003 ICS 275A - Constraint Networks 53
1A
2C
1A
3B
1A
1F
23CB
2D
3B
1A
3G
1F
2D
1R
2R
4R
3R
5R
6R
A
AB AC
ABD BCF
DFG
B
4 5
3
6
2
B
D F
A
A
A
C
1
+5