Constraint( slides courtesy o( y

based on slides court

t Networksof Natalia Flerova,,tesy of Rina Dechter)

Read

Russell and NorvigCh t 6 (3rd d ) ChChapter 6 (3rd ed.); Cha

For next week:Ch t 7 (f T d )Chapter 7 (for Tuesday)Chapter 8 (for Thursday

ding

t 5 (2nd d )apter 5 (2nd ed.)

))y)

Good sourceGood sourceinform

Rina Dechter,

Constraint Processing,g,Morgan Kaufmann

e of advancede of advanced mation

3

Out

Constraint Satisfaction ProDefinition and simple mod Definition, and simple mod

Representing constraintsC i l i h ( Consistency algorithms (ar

General search strategy Look-ahead meth

tline

oblems (CSPs):deling examplesdeling examples

h d i i )rc-, path- and i-consistency)

hods

4

Constra

ExampleVariables - countries (AValues - colors (e.g., rConstraints: A≠ B,

A B Ared greenred blackgreen redgreen blackbl k

B

D

black greenblack red

C

aint Satisfaction

: map coloringA,B,C,etc.)red, green, black)

A≠ D, D≠ E , etc .

ED

FF

G

Constraint Netw A constraint network is: R=(X,D,C X variables X = {X D domains

C constraints

X {X

D= {D C constraints

R expresses allowed tuples over scopesC= {C

A solution is an assignment to all variall relations).

T k i t ? ll l ti Tasks: consistency?, one or all solutio

work; DefinitionC)

X1 X }X1 , .. . ,X n}D1 ,. . . , Dn} , Di= {v1 , . .. vk }

s1 ,. .. Ct },,, Ci= �Si ,Ri �

ables that satisfies all constraints (join of

ti ti i tions, counting, optimization

6

Example: The N-queens

The network has four variables, all w(a) The labeled chess board. (b) The c

Sp

problem

with domains Di = {1, 2, 3, 4}. constraints between variables.

ring 2009 7

Example The 4pPlace 4 Queens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal.

QQ Q

Q

QQ

Q QQ

QD {1 2 3 4}• Domains: Di= {1,2,3,4} .

• Constraints: There are = 6 constr42( )2

R12= {�1,3��1,4��2,4 ��3,1��4,1��4,2�}R13= {�1,2 ��1,4 ��2,1��2,3��3,2��3,4��4,1��4R = {�1 2��1 3��2 1��2 3��2 4��3 1��3 2��3R14= {�1,2��1,3��2,1��2,3��2,4��3,1��3,2��3R23= {�1,3��1,4��2,4 ��3,1��4,1��4,2 �}R24= {�1,2 ��1,4��2,1��2,3��3,2��3,4��4,1��4R {�1 3��1 4 ��2 4��3 1��4 1��4 2�}R34= {�1,3��1,4 ��2,4��3,1��4,1��4,2�}

4-queen problemq pStandard CSP formulation of the problem:• Variables: each row is a variable.

1 2 3 4

QQX1

X 2

1 2 3 4

QQ

QX 4

X 3

2

raints involved:

4,3�}�3 4��4 2 ��4 3�}

• Constraint Graph :X1 X 3

�3,4��4,2 ��4,3�}

�4,3�}X 2 X 4

A solution and a pA solution and a ptup

Not all consistent instantiations are part o(a)A consistent instantiation that is not pa(b) Th l t f th(b) The placement of the queens corresponsolution (2, 4, 1,3). c) The placement of the queens correspon(3, 1, 4, 2).

Sp

partial consistentpartial consistent pleof a solution: art of a solution.

di t thnding to the

nding to the solution

ring 2009 9

Example: configu

Sp

uration and design

ring 2009 10

Configuratio Want to build: recreation area, apartm

dump

Recreation area near lake Steep slopes avoided except for recreation a

P il id d f d l t Poor soil avoided for developments Highway far from apartments, houses and r Dump not visible from apartments, houses Lots 3 and 4 have poor soil Lots 3 and 4 have poor soil Lots 3, 4, 7, 8 are on steep slopes Lots 2, 3, 4 are near lake Lots 1, 2 are near highwayg y

Sp

on and designments, houses, cemetery,

area

recreationand lake

ring 2009 11

Out

CSP: Definition, and simpRepresenting constraints Representing constraints

Consistency algorithms (ar General search strategy

Look-ahead meth

tline

ple modeling examples

rc-, path- and i-consistency)

hods

12

Constraint’s re

Relation: allowed tuplesp

Algebraic expression:X

Propositional formula:X

Sp

epresentationsX Y Z1 3 2

X Y 2 10 X≠ Y

2 1 3

X �Y 2≤ 10 , X≠ Y

� �b���a�b�� ¬c

ring 2009 13

Operations w Intersection

U i Union Difference Selection Projectionj Join

Composition Composition

with relations

14

Figure 1.8: Example of set operationapplied to

Sp

ns intersection, union, and difference o relations.

ring 2009 15

selection, projection, and jo

Sp

oin operations on relations.

ring 2009 16

ConstrainPrimal, Dual an

A (primal) constraint graph: a node per variablet t i d i bl arcs connect constrained variables.

A dual constraint graph: a node per constraint’s scarc connect nodes sharing variables =hypergraph

1 2 4

1,2,3,4,5

536

9 7 11 5

12 8 105,7,11

Sp

13

(a)

nt Graphs: nd Hypergraphs

cope, an

12,133,6,9,12

3 12

139

8,9,10,11 10,131011

ring 2009 17(b)

Out

CSP: Definition, and simpRepresenting constraints Representing constraints

Consistency algorithms (ar General search strategy

Look-ahead meth

tline

ple modeling examples

rc-, path- and i-consistency)

hods

18

Consistenc Constraint propagation

constraintsconstraints Can get such an explic

ill fi d th l tiwill find the solution w Approximation of infer

Arc, path and i-consiste Methods that transform

into a tighter and tighte

cy methodsn – inferring new

it network that the search ith t d d dwithout dead-ends.

rence:ency

m the original network

19

er representations

Arc-consArc-cons- infer constraints based on p

1 X Y Z T 3

X

32,1,1 X, Y, Z, T 3X YY = ZT Z

32,1,

T ZX T

,,

T

Insures that every legal value in the doma legal match In the domain of any othe

sistencysistencypairs of variables

32,1,

Y

=

32,1, ,,

Z

20

main of a single variable haser selected variable

AArc-con

1 X Y Z T 3

X

11 X, Y, Z, T 3

X YY = ZT Z

T ZX T

2

T

i tnsistencyY

3

=

3

Z

21

AArc-con i tnsistency

22

Revise for arc

D � D ∩ �RDi � Di∩ π i�R

c-consistency

23R � D �Rij � D j �

ACAC

Complexity (Mackworth and Freuder, 1986): e = number of arcs, n variables, k values (ek^2, each loop, nk number of loops), best Arc-consistency is:

� �ek 2�

C-1C-1

O�enk 3�s

24

t-case = ek,

ACAC

O�ek 3� Complexity: Best case O(ek) since each arc may Best case O(ek), since each arc may e = number of arcs, n variables, k valu

C-3C-3

25be processed in O(2k)be processed in O(2k)ues

Distributed arDistributed ar(Constraint p

Implement AC-1 distributedly.

Node x j sends the message to node _j gx_i

Node x_i updates its domain:

Messages can be sent asynchronously or scheduled in a topological order

rc-consistencyrc-consistencypropagation)

Di � Di∩ π i�Rij � D j �

hij � π i�Rij � D j �

Di � Di∩ π i�Rij � D j �=D D ∩ h jDi � Di∩ hi

j

26

Distributed ArA i b Arc-consistency can be algorithm:

rc-Consistencyf l d di ib dformulated as a distributed

A

B C

D F

G

27a Constraint networka Constraint network

Relational Ar

The message that R2 sends to R1 is

R1 updates its relation and domains and sendmessages to neighbors

rc-consistencyA123

A B1 21 3

R1R2 R3A 3 A C

1 23 2

2 1

2 3

3 13 2

A

B C

3 2

B C F

R5D F

ds

B C F1 2 33 2 1A B D

1 2 31 3 2

R4D F

G

28

1 3 22 1 32 3 13 1 2

D F G1 2 32 1 3

R6

3 2 1 2 1 3

A1

R1DRAC on

23

A B1 2

R2

1 21 32 12 3 2 A

3 13 2

R4

AB

B

4

A

A B D1 2 31 3 22 1 3

4

ABD

4

D

2 1 32 3 13 1 23 2 1 D

1

R6

12

A1the dual join-graph

R1

A C1 23 2

R3A

3A

1

R5

AC

5

B C

B C F1 2 33 2 1

BCF

DFG

5

6F

29F G2 3

DFG

2 31 3

SudConstraint

•Constraint P ti•Propagation

•Inference

Each row, column and major blo

“Well posed” if it has unique solu

doku –t Satisfactionf

•Variables: empty slots

•Domains = {1,2,3,4,5,6,7,8,9}

•Constraints: 27 all-different

2 34 62

30ock must be alldifferent

ution: 27 constraints

Path-connsistency

31

Path-connsistency

32

ReviRevi

Complexity: O(k^3) Best-case: O(t) Worst-case O(tk) e = number of arcs, n variables, k value number of arcs, n variables, k valu

constraint relation

ise-3ise-3

33ues, t = tightness: number of tuples in theues, t tightness: number of tuples in the

PC

Complexity: e = number of arcs, n variables, k values

constraint relation

O�n5 k5�

constraint relation

C-1

34s, t = tightness: number of tuples in the

PC-PC-

O�n3 k5� Complexity: Optimal PC-4: O�n3 k3� (each pair deleted may add: 2n-1 triplets, numb

O(n^3 k^2), processing is O(k^3))b f i bl k l t

O n k

e = number of arcs, n variables, k values, tconstraint relation

-2-2

35

ber of pairs: O(n^2 k^2) size of Q is

t ti ht b f t l i tht = tightness: number of tuples in the

Example: beforeExample: beforeconsis

PC-1 requires 2 processings of each ar Can we do path-consistency distribute

e and after path-e and after path-stency

36

rc while PC-2 may notedly?

I-consiistency

37

Out

CSP: Definition, and simpRepresenting constraints Representing constraints

Consistency algorithms (ar General search strategy

Look-ahead meth

tline

ple modeling examples

rc-, path- and i-consistency)

hods

38

Backtrackin

No matter how much we reaare still left with choices → possible choices.

Backtracking search – assiki h iturn, making sure each assig

values assigned thus far.E t i bl ith d Encounter variable with no dprevious assignment – dead-the value of previous variabl

ng algorithm

ason about the problem – we need to search the space of

ign values to each variable in d l i i i hgned value is consistent with

d i l i t t ithdomain value consistent with -end → backtrack – change les.

39

Backtrracking

The sear

A tree of all partial soluA i l l i ( 1 A partial solution: (a1,relevant constraints

The size of the underlyon: Variable ordering Level of consistency po

rch space

utionsj) i f i ll…,aj) satisfying all

ying search space depends

ossessed by the problem

S h d hSearch space and thh ff f d ihe effect of ordering

Z evenly divides x,y,l

20 legal states1 dead-end leaf

48 legal states18 dead-end leaves

Dependency on cDependency on c

After arc-consistency z=5 and l=5 are removed

After path-consistency R’_zx R’_zy R’_zl R’_xy R’_xl R’_yl

consistency levelconsistency level

Backtrack-ffree network

BacktrBacktrrackingracking

Complexity of extending a partial solution: Complexity of consistent O(e log

t), t bounds tuples, e constraints Complexity of selectvalue O(e k

log t)log t)

Improving b

Before search: (reducingArc consistenc path consist Arc-consistency, path-consist

Variable ordering (fixed)

During search:g Look-ahead schemes:

value ordering, variable ordering (if not fixe variable ordering (if not fixe

Look-back schemes: Backjump

Constraint recording Constraint recording Dependency-directed backtr

backtracking

g the search space)tenctency

ed)ed)

racking

Out

CSP: Definition, and simpRepresenting constraints Representing constraints

Consistency algorithms (ar General search strategy

Look-ahead meth

tline

ple modeling examples

rc-, path- and i-consistency)

hods

51

Improving b

Before search: (reducinA i t th Arc-consistency, path-c

Variable ordering (fixeDuring search: During search:

Look-ahead schemes: Look-ahead schemes: value order

maximizesfuture assig

variable ori t ti tinstantiate variable

backtracking

ng the search space)i tconsistency

ed)

ring - first assign value that

s the number of options for pgnmentsrdering (if not fixed)- first

t hi hl t i dmost highly constrained

Look-ahead: v

Intuition: Choose value least likely to yield a y y Approach: apply propagation at eac

Forward-checking (check each unassigned variable sep(check each unassigned variable sep

Maintaining arc-consistency (MAC) (apply full arc-consistency)

Full look-ahead Full look ahead One pass of arc-consistency (AC-1)

Partial look-ahead directional-arc-consistency directional-arc-consistency

value orderings

dead-endch node in the search tree

parately)parately)

)

Generalized loook-ahead

Forward-

The most limited form propagation during seapropagation during sea

Propagates the effect oselection to each futureselection to each future

If domain of one of thebecomes empty – valuenext value.

-checking

of constraint archarchf a tentative value

e variable separatelye variable separatelye future variables e is not selected → try

Forward-checcking example

Forward-

Complexity of selectValue-forward-chep y

-checking

O�ek 2�cking at each node: O�ek �g

Arc-consisten(Gashni

Applies full arc-consistencvariables following each vvariables following each vcurrent variable.

Complexity:p y If optimal arc-consisten

SelectValue-Arc-Consisteessentially AC-1 with som

Maintaining-Arc-Consistearc-consistency after each

cy look-aheadig, 1977)

cy on all un-instantiated value assignment to thevalue assignment to the

O�ek 3�ncy is used:

O�ek �

ency: The repeat...until loop –me variables instantiated. ency (MAC): Performs full h domain value is rejected.

Arc-consisten(maintaining arc(maintaining arc-

cy look-ahead:consistency MAC)-consistency MAC)

Full and parti

Full looking ahead:M k f ll th Make one full pass thro(remove repeat-until froconsistency)consistency)

Partial look-ahead:Applies (similar to) dir Applies (similar-to) dirfuture variables. - futurcompared with those vaO�e

Complexity: also More efficient than MA

O�e

ial look-ahead

h f t i blough future variables om selectValue-arc-

rectional arc consistency torectional arc-consistency to re variables are only ariables following them. ek 3�

AC

ek �

Example of parrtial look-ahead

Out

CSP: Definition, and simpRepresenting constraints Representing constraints

Consistency algorithms (a General search strategy

Look-ahead meth Branching-ahead

tline

ple modeling examples

cr-, path- and i-consistency)

hodsd

62

Branching-ahead: Dy

Rank order the promise in non Rank functionsRank functions

MC (min conflict) – chooseof values from domains of f

MD (min domain) – choose MD (min domain) – choosedomain size in future variab

SC (expected solution countof solutions by multiplying to so ut o s by u t p y g tafter incompatible values ar

ynamic Value Ordering

n-rejected values

s value that removes the smallest number future variabless value that creates the largest minimums value that creates the largest minimum

blests) – computes upper bound on number the domains sizes of each future variable t e do a s s es o eac utu e va ab e

re removed.

BranchinDynamic Variable

Following constraint propaconstrained variableconstrained variable

Intuition: early discovery Highly effective: the singleg y e ec ve: t e s g e

cut down search space Most popular with FC Dynamic search rearrangem

(Purdon,1983)

ng-ahead: e Ordering (DVO)

agation, choose the most

of dead-endse most important heuristic to e ost po ta t eu st c to

ment (Bitner and Reingold, 1975)