chapter 3dechter/books/chapter03.pdf · chapter 3. fall i2003 cs 275a - on st rai ne w k 2...

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


Arc-consistency

32,1,

32,1, 32,1,

1 ≤≤≤≤ X, Y, Z, T ≤≤≤≤ 3X <<<< YY = ZT <<<< ZX ≤≤≤≤ T

X Y

T Z

32,1,<<<<

=

<

∧


1 ≤≤≤≤ X, Y, Z, T ≤≤≤≤ 3X <<<< YY = ZT <<<< ZX ≤≤≤≤ T

X Y

T Z

<<<<

=

<

∧

1 3

2 3

Arc-consistency


∏←A BABA DRR

.2,1 to of domain reduces

constriant ,3,2,1 ,3,2,1=

<==X

YX

RXYXRR


)( jijiii DRDD ⊗∩← π


!"


#

)( 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


)( 3ekO Complexity: Best case O(ek), since each arc may be processed in O(2k)


$% &"" '

( % (

&&"


)

)( 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)


*

&&

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


jiii

jijiii

hDD

DRDD

∩←

=⊗∩← )(π


+,

Example: a triangle graph-coloring with 2 values.• Is it arc-consistent?• Is it consistent?

It is not path, or 3-consistent.


-


)( kjkikijijij RDRRR ⊗⊗∩← π

Complexity: O(k^3) Best-case: O(t) Worst-case O(tk)


-#

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


-.

)( 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


$% &" &

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


+


/""'

""


)( ikO Complexity: for binary constraints For arbitrary constraints: ))2(( ikO


)!% &"


+


0"(

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

)( xSSxxx DRDD −⊗∩← π

)( xSxSxS DRR ⊗← −− π


$% &""(

x+y+z <= 15, z >= 13 x<=2, y<=2

Example of relational arc-consistency

BAGGBA ¬∨¬¬→∧ ,,


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


$% &"""

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)


0""

Alldifferent Sum constraint Global cardinality constraint (a value can

be assigned a bounded number of times) The cummulative constraint (related to

scheduling tasks)


1


1 ""


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


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


&&1"

2&&


" ""

"(

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?


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


3""


&"

'& )


*

&&

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


jiii hDD ∩←

)( xSxSxS DRR ⊗← −− π


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


*

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


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&


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


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

+#


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


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


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

+


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

+


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

+)


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

+)


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


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

chapter 3dechter/books/chapter03.pdf · chapter 3. fall i2003 cs 275a - on st rai ne w k 2...

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