problem solving with constraints csce421/821, spring 2011 cse.unl/~ choueiry/f11-496-896

Problem Solving with Constraints

Arc Consistency 1

Problem Solving With ConstraintsCSCE421821 Spring 2011

wwwcseunledu~choueiryF11-496-896All questions cse496cpcseunledu

Berthe Y Choueiry (Shu-we-ri)Avery Hall Room 360Tel +1(402)472-5444

Arc Consistency


Arc Consistency 2

Lecture SourcesRequired reading1 Algorithms for Constraint Satisfaction Problems Mackworth and

Freuder AIJ85 (focus on Node consistency arc consistency)2 Sections 31 amp 32 Chapter 3 Constraint Processing Dechter

Recommended reading1 Bartak Consistency Techniques (link)2 Constraint Propagation with Interval Labels Davis AIJ87


Arc Consistency 3

Outline1 Motivation (done) amp background

2 Node consistency and its complexity

3 Arc consistency and its complexity

4 Criteria for performance comparison and CSP parameters

5 Experimentsndash Random CSPs (Model B)ndash Statistical Analysis


Arc Consistency 4

Basic Consistency Notation

Examining finite binary CSPs

Notation

Given variables ijk with values xyz the predicate

Pijk(xyz) is true iff the 3-tuple (xyz) Rijk

Node consistency checking Pi(x)

Arc consistency checking Pij(xy)


Arc Consistency 5

Worst-case complexity as a function of the input parameters

Upper bound f(n) is O(g(n)) means that f(n) cg(n)

f grows as g or slower

Lower bound f(n) is (h(n)) means that f(n) ch(n)

f grows as g or faster

Input parameters for a CSP

n = number of variables

a = (max) size of a domain

dk = degree of Vk ( n-1)

e = number of edges (or constraints) [(n-1) n(n-1)2]

Worst-case asymptotic complexitytime

space


Arc Consistency 6





5 Experiments1 Random CSPs (Model B)

2 Statistical Analysis


Arc Consistency 7

Node consistency (NC)Procedure NC(V1V2hellipVn)

For i 1 until n do DVi DVi x | Pi(x)

bull For each variable we check a valuesbull We have n variables we do na checksbull NC is O(an)bull Alert check for domain wipe-out domain annihilation

If Dvi (Di= ) empty Then BREAK


Arc Consistency 8



3 Arc consistency and its complexityndash Algorithms AC1 AC3 AC4 AC2001




Arc Consistency 9

Arc Consistency Adapted from Dechter

Definition Given a constraint graph G

bull A variable Vi is arc consistent relative to Vj iff for every value a DVi there exists b DVj | (ab) RViVj

bull The constraint CViVj is arc consistent iff ndash Vi is arc consistent relative to Vj and ndash Vj is arc consistent relative to Vi

bull A binary CSP is arc-consistent iff every constraint (or sub-graph of size 2) is arc consistent

1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency

Main Functionsbull Variable-value pair

ndash (Via) ⟨Via variable value from its domain⟩

bull CHECK(⟨Via ⟩ ⟨Vjb )⟩ndash Returns true if (ab)isinRViVj false otherwise

bull SUPPORTED(⟨Via ⟩ Vj)

ndash Verifies that ⟨Via has at least one support in ⟩ CViVj

ndash Is not standard lsquoterminologyrsquo but my lsquoconventionrsquo

bull REVISE(ViVj)

ndash Updates DVi given RViVj

ndash BREAK when domain wipe-out

10


Arc Consistency

Arc Consistency Algorithmsbull AC-1 AC-3 hellip AC2001 arc consistency algorithms

ndash Update all domains given all constraints

ndash Call REVISE

ndash Differ in how they manage updates (queue)

bull Complexityndash Nbr of variables n Domain size d Nbr of constraints e degree of

graph deg

bull CC is a global variable ndash Keeps track of the number of times that the definition of a binary relation

is accessed

ndash Is a cost measure to compare algorithmsrsquo performance in practive

bull Assumptionndash Domains are sorted alphabetically (lexicographic)

11


Arc Consistency

CHECK(⟨Via ⟩ ⟨Vjb )⟩bull Operation

ndash Verifies whether a constraint exists between ViVj

ndash If a constraint existsbull Increases CC

bull Accesses the constraint between ViVj

bull Verifies if (ab)isinRViVj

ndash Returns true if (ab) is consistent

ndash Returns false if (ab) is not consistent

ndash If no constraints existbull Returns true (universal constraint)

bull Pseudo code necessarybull Complexity cost in practice

12


Arc Consistency

SUPPORTED(⟨Via ⟩ Vj)SUPPORTED(⟨Via ⟩ Vj)

1 support nil

2 b DVj

3 If CHECK(⟨Via ⟩ ⟨Vib ) ⟩

4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support

bull Complexitybull Once you find a support stop lookingbull Cost in practice depends on implementation

13


Arc Consistency

REVISE(Vi Vj) Description

bull Function

bull Updates DVi given the constraint CViVj

bull Is directional does not update DVj

bull Calls SUPPORTED(⟨Vi ⟩ Vj)

bull If DVi is modified

ndash Returns true false otherwise

14


Arc Consistency

REVISE(Vi Vj) Pseudocode REVISE(Vi Vj)

1 revised false

2 x DVi

3 found SUPPORTED(⟨Vix ⟩ Vj)

4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16

REVISE example R Dechter

bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17

Arc Consistency (AC-1)Procedure AC-11 NC(Problem)2 Q (i j) | (ij) directed arcs in constraint network of Problem i j 3 Repeat4 change false5 Foreach (i j) Q do 6 Begin for each1 updated REVISE(i j)

2 If DVi = Then Return false Else change (updated or change)3 End for each4 Until change = false5 Return Problem

6 AC-1 does not update Q the queue of arcs7 No algorithm can have time complexity below O(ea2)8 AC1 should test for empty domains (does not in the paper)


Arc Consistency

Warningbull Most papers do not check for domain wipe-out

bull In your code have AC1

ndash Always check for domain wipe out andndash terminateinterrupt the algorithm when that occurs

bull Complexity versus performancendash Does not affect worst-case complexityndash In practice saves lots of CC and cycles

18


Arc Consistency 19

Arc Consistency (AC-1)bull If a domain is wiped out AC1 returns nilbull Otherwise returns the problem given as input (alternatively change)

Procedure AC-11 NC(Problem)2 Q (i j) | (ij) directed arcs in constraint network of Problem i j bull Repeatbull change falsebull Foreach (i j) Q do 6 Begin for each1 updated REVISE(i j)



Arc Consistency 20

Arc consistency

1 AC may discover the solution Example borrowed from Dechter

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency

2 AC may discover inconsistency Example borrowed from Dechter

1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22

NC amp AC Example courtesy of M Fromherz

[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y

AC propagates boundUnary constraint xgt3 is imposedAC propagates bounds again


Arc Consistency 23

Complexity of AC-1Procedure AC-11 begin2 for i 1 until n do NC(i)3 Q (i j) | (ij) arcs(G) i j4 repeat5 begin6 CHANGE false7 for each (i j) Q do CHANGE (REVISE(i j) or CHANGE)8 end9 until not CHANGE10 end

Note Q is not modified and |Q| = 2ebull 4 9 repeats at most na timesbull Each iteration has |Q| = 2e calls to REVISEbull Revise requires at most a2 checks of Pij

AC-1 is O(a3 n e)


Arc Consistency 24

AC versions

bull AC-1 does not update Q the queue of arcs

bull AC-2 iterates over arcs connected to at least one node whose domain has been modified Nodes are ordered

bull AC-3 same as AC-2 nodes are not ordered


Arc Consistency 25

AC-3AC-3 iterates over arcs connected to at least one

node whose domain has been modified

Procedure AC-3bull NC(Problem)bull for i 1 until n do NC(i)bull Q (i j) | (i j) directed arcs in constraint network of Problem i j bull While Q is not empty dobull Beginbull select and delete any arc (km) Qbull If REVISE(km) Then Q Q (Ik) | (Ik) arcs(G) ik im bull End(Donrsquot forget to make AC-3 check for domain wipe out)


Arc Consistency 26

Complexity of AC-3Procedure AC-3

1 begin

2 for i 1 until n do NC(i)

3 Q (i j) | (Ij) arcs(G) i j

4 While Q is not empty do

5 begin

bull select and delete any arc (km) Q

7 If Revise(k m) then Q Q (ik) | (Ik) arcs(G) ik im

8 end

9 end

bull First |Q| = 2e then it grows and shrinks 48

bull Worst case ndash 1 element is deleted from DVk per iteration

ndash none of the arcs added is in Q

bull Line 7 a(dk - 1)

bull Lines 4 - 8

bull Revise a2 checks of Pij

AC-3 is O(a2(2e + a(2e-n)))

Connected graph (e n ndash 1) AC-3 is O(a3e)

If AC-p AC-3 is O(a2e) AC-3 is (a2e) Complete graph O(a3n2)

)2(2121

neaedaen

k k


Arc Consistency 27

Example Apply AC-3Example Apply AC-3 Thanks to Xu Lin

DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28

Applying AC-3 Thanks to Xu Lin

bull Queue = CV2 V4 CV4V2 CV1V3 CV2V3 CV3 V1 CV3V2

bull Revise(V2V4) DV2 DV2 5 = 1 2 3 4

bull Queue = CV4V2 CV1V3 CV2V3 CV3 V1 CV3 V2

bull Revise(V4 V2) DV4 DV4 4 = 1 2 3 5

bull Queue = CV1V3 CV2V3 CV3 V1 CV3 V2

bull Revise(V1 V3) DV1 1 2 3 4 5

bull Queue = CV2V3 CV3 V1 CV3 V2

bull Revise(V2 V3) DV2 1 2 3 4


Arc Consistency 29

Applying AC-3 (cont) Thanks to Xu Lin

bull Queue = CV3 V1 CV3 V2



bull Revise(V2 V3) DV2 DV2

bull Queue = CV3 V2

bull Revise(V3 V2) DV3 1 3 5

bull Queue = CV1 V3

bull Revise(V1 V3) DV1 1 3 5 END


Arc Consistency 30

AC-4Mohr amp Henderson (AIJ 86) bull AC-3 O(a3e) AC-4 O(a2e)bull AC-4 is optimalbull Trade repetition of consistency-check operations with heavy

bookkeeping on which and how many values support a given value for a given variable

bull Data structures it usesndash m values that are active not filteredndash s lists all vvps that support a given vvpndash counter given a vvp provides the number of support provided by a

given variable

bull How it proceedsndash Generates data structuresndash Prepares data structuresndash Iterates over constraints while updating support in data structures


Arc Consistency 31

Step 1 Generate 3 data structures

bull m and s have as many rows as there are vvprsquos in the problembull counter has as many rows as there are tuples in the constraints

TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32

Step 2 Prepare data structuresData structures s and counter

bull Checks for every constraint CViVj all tuples Vi=ai Vj=bj_)bull When the tuple is allowed then update

ndash s(Vjbj) s(Vjbj) (Vi ai) andndash counter(Vjbj) (Vjbj) + 1

Update counter ((V2 V3) 2) to value 1Update counter ((V3 V2) 2) to value 1Update s-htable (V2 2) to value ((V3 2))Update s-htable (V3 2) to value ((V2 2))



Arc Consistency 33

Constraints CV2V3 and CV3V2 S

(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m

Note that (V3 V2)4 0 thus

we remove 4 from the domain of V3

and update (V3 4) nil in m

Updating counter

Since 4 is removed from DV3 then

for every (Vk l) | (V3 4) s[(Vk l)]

we decrement counter[(Vk V3) l] by 1


Arc Consistency

AC2001

bull When checking (Via) against Vj

ndash Keep track of the Vj value that supports (Via)

ndash Last((Via)Vi)

bull Next time when checking again (Via) against Vj we start from Last((Via)Vi) and donrsquot have to traverse again the domain of Vj again or list of tuples in CViVj

bull Big savings

34


Arc Consistency 35

Summary of arc-consistency algorithmsbull AC-4 is optimal (worst-case) [Mohr amp Henderson AIJ 86]

bull Warning worst-case complexity is pessimistic Better worst-case complexity AC-4

Better average behavior AC-3 [Wallace IJCAI 93]

bull AC-5 special constraints [Van Hentenryck Deville Teng 92]

functional anti-functional and monotonic constraints

bull AC-6 AC-7 general but rely heavily on data structures for bookkeeping[Bessiegravere amp Reacutegin]

bull Now back to AC-3 AC-2000 AC-2001equivAC-31 AC33 etc

bull Non-binary constraints ndash GAC (general) [Mohr amp Masini 1988]

ndash all-different (dedicated) [Reacutegin 94]


Arc Consistency 36




4 Performance comparisonndash Criteria and CSP parameters



Arc Consistency 37

CSP parameters ⟨na t d⟩bull n is number of variablesbull a is maximum domain sizebull t is constraint tightness

bull d is constraint density

where e is the constraints emin=(n-1) and emax = n(n-1)2

Lately we use constraint ratio p = eemax

rarr Constraints in random problems often generated uniform

rarr Use only connected graphs (throw the unconnected ones away)

tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38

Criteria for performance comparison

1 Bounding time and space complexity (theoretical)ndash worst-case alwaysndash average-case neverndash Best-case sometimes

2 Counting CC and NV (theoretical empirical)

3 Measuring CPU time (empirical)


Arc Consistency 39

Performance comparison Courtesy of Lin XU

AC-3 AC-7 AC-4 on n=10a=10td=06 displaying CC and CPU time

AC-4

AC-7

AC-3


Arc Consistency 40

Wisdom () Free adaptation from Bessiegravere

bull When a single constraint check is very expensive to make use AC-7

bull When there is a lot of propagation use AC-4bull When little propagation and constraint checks

are cheap use AC-3x


Arc Consistency

Performance comparison Courtesy of Shant

41

AC3 AC31 AC4 on on n=40a=16td=015 displaying CC and CPU time


Arc Consistency

AC-what

Instructorrsquos personal opinionbull Used to recommend using AC-3bull Now recommend using AC2001bull Do the project on AC- if you are curious [Reacutegin 03]

42


Arc Consistency 43

AC is not enough Example borrowed from Dechter

Arc-consistentSatisfiable

seek higher levels of consistency

V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44

WARNINGrarr Completeness (ie for solving the CSP) Running the Waltz

Algorithm does not solve the problem

A=2 B=3 is still not a solution

rarr Quiescence The Waltz algorithm may go into infinite loops even if problem is solvable

x [0 100] x = y

y [0 100] x = 2y

rarr Davis characterizes the completeness and quiescence of the Waltz algorithm (see Table 3) in terms ofndash constraint types

ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46

bull Various models exist (use Model B)ndash Models A B C E F etc

bull Vary parameters ltn a t pgtndash Number of variables nndash Domain size a dndash Constraint tightness t = |forbidden tuples| | all tuples |

ndash Proportion of constraints (aka constraint density constraint probability) p1 = e emax

bull Issues ndash Uniformityndash Difficulty (phase transition)ndash Solvability of instances (for incomplete search techniques)

Empirical evaluations random problems


Arc Consistency 47

1 Input n a t p12 Generate n nodes3 Generate a list of n(n-1)2 tuples of all combinations of

2 nodes4 Choose e elements from above list as constraints to

between the n nodes5 If the graph is not connected throw away go back to

step 4 else proceed6 Generate a list of a2 tuples of all combinations of 2

values7 For each constraint choose randomly a number of

tuples from the list to guarantee tightness t for the constraint

Model B


Arc Consistency 48

Cos

t of

sol

ving

Mostly solvable problems

Mostly un-solvable problems

Order parameterCritical value of order parameter

bull Significant increase of cost around critical valuebull In CSPs order parameter is constraint tightness amp ratiobull Algorithms compared around phase transition

Phase transition [Cheeseman et al lsquo91]


Arc Consistency 49

bull Fix n a p1 and ndash Vary t in 01 02 hellip09

bull Fix n a t and ndash Vary p1 in 01 02 hellip09

bull For each data point (for each value of tp1)ndash Generate (at least) 50 instances

ndash Store all instances

bull Make measurementsndash CC CPU time (in other contexts NV messages

etc)

Tests


Arc Consistency

Comparing two algorithms A1 and A2

bull Store all measurements in Excelbull Use Excel R SAS etc for statistical

measurementsbull Use the t-test paired test

bull Comparing measurementsndash A1 A2 a significantly different

bull Comparing ln measurementsndash A1is significantly better than A2

For Excel Microsoft button Excel Options Adds in Analysis ToolPak Go check the box for Analysis ToolPak Go Intallhellip

CC ln(CC)

A1 A2 A1 A2

i1 100 200 hellip hellip

i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel

bull Using ln values

ndash p ttest(array1array2tailstype)bull tails=1 or 2 bull type1 (paired)

ndash t tinv(pdf)bull degree of freedom = instances ndash 2

51


Arc Consistency

t-test with 95 confidencebull One-tailed test

ndash Interested in direction of change

ndash When t gt 1645 A1 is larger than A2

ndash When t -1645 A2 is larger than A1

ndash When -1645 t 1645 A1 and A2 do not differ significantly

ndash |t|=1645 corresponds to p=005 for a one-tailed test

bull Two-tailed testndash Although it tells direction not as accurate as the one-tailed test




ndash |t|=196 corresponds to p=005 for a two-tailed test

bull p=005 is a US Supreme Court ruling any statistical analysis needs to be significant at the 005 level to be admitted in court

52


Arc Consistency

Computing the 95 confidence interval

bull The t test can be used to test the equality of the means of two normal populations with unknown but equal variance

bull We usually use the t-testbull Assumptions

Normal distribution of data

Sampling distributions of the mean approaches a uniform distribution (holds when instances 30)

Equality of variances

Sampling distribution distribution calculated from all possible samples of a given size drawn from a given population

53


Arc Consistency

Alternatives to the t testbull To relax the normality assumption a non-parametric

alternative to the t test can be used and the usual choices are ndash for independent samples the Mann-Whitney U testndash for related samples either the binomial test or the

Wilcoxon signed-rank test

bull To test the equality of the means of more than two normal populations an Analysis of Variance can be performed

bull To test the equality of the means of two normal populations with known variance a Z-test can be performed

54


Arc Consistency

Alertsbull For choosing the value of t in general check

httpwwwsocruclaeduAppletsdirT-tablehtml bull For a sound statistical analysis

ndash consult the Help Desk of the Department of Statistics at UNL

ndash held at least twice a week at Avery Hall

bull Acknowledgments Dr Makram Geha Department of Statistics UNL All errors are mine

55


Arc Consistency 56

Summary1 Alert

ndash Do not confuse a consistency property with the algorithms for reinforcing it

ndash For each property many algorithms may exist

2 Local consistency methodsndash Remove inconsistent values (node arc consistency)ndash Remove Inconsistent tuples (path consistency)ndash Get us closer to the solutionndash Reduce the lsquosizersquo of the problem amp thrashing during search ndash Are lsquocheaprsquo (ie polynomial time)


Arc Consistency 2

Lecture SourcesRequired reading1 Algorithms for Constraint Satisfaction Problems Mackworth and

Freuder AIJ85 (focus on Node consistency arc consistency)2 Sections 31 amp 32 Chapter 3 Constraint Processing Dechter

Recommended reading1 Bartak Consistency Techniques (link)2 Constraint Propagation with Interval Labels Davis AIJ87


Arc Consistency 3







Arc Consistency 4



Notation






Arc Consistency 5












space


Arc Consistency 6








Arc Consistency 7






Arc Consistency 8







Arc Consistency 9






1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 3







Arc Consistency 4



Notation






Arc Consistency 5












space


Arc Consistency 6








Arc Consistency 7






Arc Consistency 8







Arc Consistency 9






1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 4



Notation






Arc Consistency 5












space


Arc Consistency 6








Arc Consistency 7






Arc Consistency 8







Arc Consistency 9






1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 5












space


Arc Consistency 6








Arc Consistency 7






Arc Consistency 8







Arc Consistency 9






1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 6








Arc Consistency 7






Arc Consistency 8







Arc Consistency 9






1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 7






Arc Consistency 8







Arc Consistency 9






1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 8







Arc Consistency 9






1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 9






1

22

3

1

2

3

1

2

3

Vi VjVjVi


Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency







bull REVISE(ViVj)



10


Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency



ndash Call REVISE



graph deg


is accessed



11


Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency










12


Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency


1 support nil

2 b DVj


4 Then Begin

5 support true

6 RETURN support

7 End

8 RETURN support


13


Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency


bull Function






14


Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency


1 revised false

2 x DVi


4 If found = false

5 Then Begin

6 revised true

7 DVi DVi x

8 End

9 RETURN revised

1 Complexity

15


Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 16


bull REVISE(ViVj)

bull REVISE(ViVj)

1

2 2

3

1

2

3

1

2

3

Vi Vj Vi Vj


Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 17





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency





18


Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 19





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 20

Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 21

Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 22


[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y



Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 23



AC-1 is O(a3 n e)


Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 24

AC versions





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 25





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 26


1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 27


DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 28











Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 29






bull Queue = CV3 V2


bull Queue = CV1 V3



Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 30




given variable



Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 31



TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter

MM


Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 32







Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 33


(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency

AC2001



ndash Last((Via)Vi)


bull Big savings

34


Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 35











Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 36







Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 37







tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 38






Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 39



AC-4

AC-7

AC-3


Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 40




are cheap use AC-3x


Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency


41



Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency

AC-what


42


Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 43




V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 44





x [0 100] x = y

y [0 100] x = 2y


ndash domain types

1 2 3 2 3 4

BA BA

2 3 2 3


Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 45







Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 46







Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 47







Model B


Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 48

Cos

t of

sol

ving







Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 49






etc)

Tests


Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency







CC ln(CC)

A1 A2 A1 A2


i2 hellip

i3

hellip

i50

50


Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency

t-test in Excel




51


Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency













52


Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency








53


Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency






54


Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency






55


Arc Consistency 56

Summary1 Alert





Arc Consistency 56

Summary1 Alert




problem solving with constraints csce421/821, spring 2011 cse.unl/~ choueiry/f11-496-896

Documents