forbidden patterns in constraint satisfaction problems · introduction flat patterns existential...

IntroductionFlat patterns

Existential patternsConclusion

Forbidden Patternsin Constraint Satisfaction Problems

Guillaume Escamocher,under the supervision of Martin Cooper and Pierre Régnier

Institut de Recherche en Informatique de Toulouse

12/11/2014

Forbidden Patterns in Constraint Satisfaction Problems 1/ 65



1 IntroductionConstraint Satisfaction ProblemClassical methods to look for tractable classesA new approach

2 Flat patternsDefinitionClassical elimination operationsThe reductionPatterns on two constraints

3 Existential patternsDefinitionsVariable eliminationTractable classes

4 Conclusion




Constraint Satisfaction ProblemClassical methods to look for tractable classesA new approach

Constraint Satisfaction Problem

Data :A set V of n variables {v1, . . . , vn}.A set of n domains {D1, . . . ,Dn}. Each domain Di contains the setof possible values for vi . 〈vi , a〉 represents the assignment of thevalue a ∈ Di to the variable vi . (assignment ≡ point)A set C of constraints {C1, . . . ,Cm}. Each constraint Cj isassociated to a kj -uple of variables

⟨vj1 , . . . , vjkj

⟩and describes the

kj -uples of assignments allowed for this kj -uple of variables. kj isthe arity of Cj .

Question :Is there an assignment of the values to the n variables such that allconstraints in C are satisfied?





binary CSP

Data :A set V of n variables {v1, . . . , vn}.A set of n domains {D1, . . . ,Dn}. Each domain Di contains the setof possible values for vi . 〈vi , a〉 represents the assignment of thevalue a ∈ Di to the variable vi . (assignment ≡ point)A set C of n(n + 1)/2 binary constraints {C1, . . . ,Cn(n+1)/2}. Eachconstraint Cj is associated to a couple of variables

⟨vj1 , vj2

⟩and

describes the couples of assignments allowed for these 2variables.

Question :Is there an assignment of the values to the n variables such that allconstraints in C are satisfied?





A CSP instance

j i

ec

a

bb

dd

gg

hh

f

v1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}

A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?Two compatible points.

b

h

Two incompatible points.

h

e

A trivial constraint.The constraint graph.

v1

v2

v3

v4

Answer: Yes! (b,d,g,h)





A CSP instance

j i

ec

a

bb

dd

gg

hh

fv1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}

A set of 4 domains {D1,D2,D3,D4}

A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

b

b

dd

gg

hh

fv1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}

D1 = {a,b}

D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

b

b

d

d

g

g

h

h

f

v1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}

D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}

C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

b

b

d

d

g

g

h

h

f

v1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}

A set of 6 constraints {C1,C2,C3,C4,C5,C6}

C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

b

b

d

d

g

g

h

h

f

v1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}

C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}

D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

b

b

d

d

g

g

h

h

f

v1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}

C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}

Question : Is there a solution ?Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

b

b

d

d

g

g

h

h

f

v1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}

Question : Is there a solution ?

Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

b

b

d

d

g

g

h

h

f

v1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

bb

d

d

g

g

hh

f

v1

v2

v3

v4

D1

D2

D3

D4

A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?

Two compatible points.

b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

e

c

a

b

b

d

d

g

g

hh

f

v1

v2

v3

v4

D1

D2

D3

D4


b

h


h

e


v1

v2

v3

v4






A CSP instance

j i

ec

a

b

b

d

d

g

g

h

h

f

v1

v2

v3

v4

D1

D2

D3

D4


b

h


h

e

A trivial constraint.

The constraint graph.

v1

v2

v3

v4






A CSP instance

j i

ec

a

bb

dd

gg

hh

fv1

v2

v3

v4

D1

D2

D3

D4A set of 4 variables {v1, v2, v3, v4}A set of 4 domains {D1,D2,D3,D4}A set of 6 constraints {C1,C2,C3,C4,C5,C6}C1 = {〈〈v1,a〉 , 〈v3, f 〉〉 , 〈〈v1,b〉 , 〈v3,g〉〉}D1 = {a,b}D2 = {c,d ,e}, D3 = {f ,g}, D4 = {h, i , j}C2 = {〈〈v2,d〉 , 〈v4,h〉〉 , 〈〈v2,d〉 , 〈v4, i〉〉 , 〈〈v2,d〉 , 〈v4, j〉〉}Question : Is there a solution ?Two compatible points.

b

h


h

e

A trivial constraint.

The constraint graph.

v1

v2

v3

v4






Why the CSP ?

The CSP is NP-Complete : it can represent many problems.It is an intuitive problem; reducing other NP-Complete problemsto it can be done quickly.

Reduction from 3-SAT can be done in linear time.

It is used in:Planning.Bioinformatics.Computer vision.and many other fields. . .





The search for tractable classes

Tractable classThe CSP is NP-Complete. A polynomial subset of the CSP is atractable class.

Examples1 The set of CSP instances where the size of the domains is

smaller or equal than 2.2 The set of CSP instances where the constraint graph is a tree.





Methods

Two main approaches are used :

Restrictions on the constraints.Restrictions on the constraint graph.





Restrictions on the constraints

Some constraints are forbidden.All constraint graphs are allowed.

Example : ZOA

∀v , v ′ ∀a ∈ Av , a can be compatible with :ZeroOneor All

points in Av ′ .





ZOA

a a′





Restrictions on the constraint graphs

Some constraint graphs are forbidden.All constraints are allowed.

Example : Bounded Tree Width

The set of instances where the constraint graph has a tree width ≤ k .





Forbidden pattern

Pattern=Part of an instance

The pattern T4.





Forbidden pattern

We associate to a pattern the set of CSP instances in which thepattern does NOT appear.





The assets of forbidden patterns

1 Patterns can define classes we could not get with restrictions ononly the constraints or only the graphs.





More refined classes

A lot of patterns (for instance T4) define classes of instances where :

All possible constraints are represented.All possible constraint graphs are represented.







2 Patterns can also define many existing classes based onconstraints or on graphs.





Binary Boolean CSP





ZOA





Forbidden patterns are not perfect...

There is no forbidden pattern equivalent to the set of CSP instanceswhose constraint graph is a tree.

...but we can improve them.

For example, BTP is a tractable class generalizing the class of CSPinstances whose constraint graph is a tree.





Broken Triangle Property

Avi

Avj

Avk

i < j < k

There exists an ordering of the variables such that this pattern doesnot appear.







2 Patterns can also define many existing classes based onconstraints or on graphs.

3 Patterns are very malleable, their potential can be easilyincreased.

4 Patterns can be detected in polynomial time.5 A pattern can still appear a limited number of times; one can just

isolate the part of the instance where it does.




DefinitionClassical elimination operationsThe reductionPatterns on two constraints




4 Conclusion





A pattern 〈V ,A, var ,E , cpt〉 is defined by:

A set of variables V .A set of points A.A function var which associates a variable with each point.A set of edges E .A function cpt which assigns a compatibility to each edge (eitherT , or F ).





Example

Av1 Av2

Av0

a b

c

d

e

V = {v0, v1, v2}

A = {a,b, c,d ,e}var(a) = v1, var(b) = v2, var(c) = var(d) = var(e) = v0

E = {(a, c), (a,d), (a,e), (b, c), (b,d)}cpt(a, c) = T , cpt(a,d) = T , cpt(a,e) = F , cpt(b, c) = F , cpt(b,d) = T





Example

Av1 Av2

Av0

a b

c

d

e

V = {v0, v1, v2}A = {a,b, c,d ,e}var(a) = v1, var(b) = v2, var(c) = var(d) = var(e) = v0

E = {(a, c), (a,d), (a,e), (b, c), (b,d)}cpt(a, c) = T , cpt(a,d) = T , cpt(a,e) = F , cpt(b, c) = F , cpt(b,d) = T





CSP instance

A CSP instance is a pattern 〈V ,A, var ,E , cpt〉 where E is the set ofcouples (a,b) such that a and b belong to different domains.





Extension

→

P P ′





Merging

→

P ′P





Occurence in a pattern

Definition

We say that a pattern P occurs in a pattern P ′ if P ′ can be obtainedfrom P with extension and/or merging.

DefinitionWe say that a pattern P occurs in an instance I if I can be obtainedfrom P with extension and/or merging.





Pattern→ Class

Notation

We note CSP(P) the set of CSP instances in which the pattern Pdoes NOT occur.

CSP(P) is polynomial → P is tractable.CSP(P) is NP-Complete → P is NP-Complete.





Arc consistency

j i h

dc e

c e

a

b

g

fAv1

Av2

Av3

Av4





Arc consistency

j i h

d

c e

c e

a

b

g

fAv1

Av2

Av3

Av4





Arc consistency

j i h

d

c ec e

a

b

g

fAv1

Av2

Av3

Av4





Single valued variable elimination

j i h

d

a

b

g

f

Av2

Av1

Av1

Av3

Av4





Neighborhood substitution

j i

h

h

b

g

fAv1

Av3

Av4

Av4





Neighborhood substitution

j

i h

h

b

g

fAv1

Av3

Av4

Av4





Classical operations properties

These operations can be applied in polynomial time.The order in which we do them does not matter.

Combination with forbidden patterns

Removing a point or a variable cannot introduce a pattern⇒We cancombine classical operations and forbidden patterns.





Dangling point

a





DP elimination

→

P P ′





Let P be a pattern and P ′ the result of a DP elimination on P.

P ′ occurs in an instance I⇒ (arc consistency)

P occurs in I

RemarkThe same can be said of merging.





Reduction

DefinitionWe say that a pattern P can be reduced to a pattern Q, and that Q isa reduction of P, if one of the following conditions is fulfilled:

1 Q is the result of an extension applied to P.2 Q is the result of a merging applied to P.3 Q is the result of a DP elimination applied to P.4 There is a recursive combination of the above conditions.





P → P ′

P can be reduced to P ′

P occurs in I ⇐ P ′ occurs in IP does not occur in I ⇒ P ′ does not occur in I

I ∈ CSP(P) ⇒ I ∈ CSP(P ′)

CSP(P) ⊆ CSP(P ′)P is tractable ⇐ P ′ is tractable

P is NP-Complete ⇒ P ′ is NP-Complete

Usefulness of the reduction

Small number of results→ Large number of complexity classes

To establish a dichotomy, we just need to find the patterns located atthe “border” between tractability and NP-Completeness.





Two constraints

TheoremLet P be a non mergeable flat pattern on two constraints. P istractable if and only if P can be extended to one of the patterns of T .

T1 T2 T3

T4 T5 2I





Proof

⇐ 1 We give 6 tractability proofs, one for each pattern in T .2 From the properties of the reduction, we have the result.

⇒ 1 We give a set G of gadgets.2 We show that all gadgets in G are NP-Complete.3 We show that for every flat pattern on two constraints P, either P

contains a gadget from G, or P can be extended to one of thepatterns in T .

4 From the properties of the reduction, we have the result.





Examples of NP-Complete gadgets

ab

a 6= b




DefinitionsVariable eliminationTractable classes

Variable eligible for elimination

Definition

Let I = 〈V ,A, var ,E , cpt〉 be a CSP instance satisfying arcconsistency. If the satisfiability of I is the same as the satisfiability ofI \ v , we say that v can be eliminated.

QuestionHow to detect such variables?








4 Conclusion





Quantified patterns

A quantified pattern 〈V ,A, var ,E , cpt ,v〉 is defined by:

A set of variables V .A set of points A.A function var which associates a variable with each point.A set of edges E .A function cpt which assigns a compatibility to each edge (eitherT , or F ).A distinguished variable v ∈ V .





Example

v

∃v

A quantified pattern 〈V ,A, var ,E , cpt , v〉.





Occurence on a variable

∃v

v

BTP

Av0

Av1 Av2

An instance I

BTP occurs on v0.BTP does not occur on v1.BTP does not occur on v2.





Existential patterns

An existential pattern 〈V ,A, var ,E , cpt ,a〉 is defined by:

A set of variables V .A set of points A.A function var which associates a variable with each point.A set of edges E .A function cpt which assigns a compatibility to each edge (eitherT , or F ).A distinguished point a ∈ A.





Example

a

∃a

An existential pattern 〈V ,A, var ,E , cpt ,a〉.






∃a

a

X1

a0 b0 c0Av0

Av0

Av1 Av2

An instance I

a0 b0 c0a0 b0 c0

X1 occurs on a0.X1 occurs on b0.X1 does not occur on c0.

⇒ X1 does not occur on v0.






∃a

a

X1

a0

b0 c0Av0

Av0

Av1 Av2

An instance I

a0

b0 c0a0 b0 c0

X1 occurs on a0.

X1 occurs on b0.X1 does not occur on c0.







∃a

a

X1

a0

b0

c0Av0

Av0

Av1 Av2

An instance I

a0

b0

c0a0 b0 c0

X1 occurs on a0.X1 occurs on b0.

X1 does not occur on c0.







∃a

a

X1

a0 b0

c0

Av0

Av0

Av1 Av2

An instance I

a0 b0

c0

a0 b0 c0








∃a

a

X1

a0 b0 c0Av0

Av0

Av1 Av2

An instance I

a0 b0 c0

a0 b0 c0







VE pattern

DefinitionWe say that a pattern P is a Variable Elimination pattern (or VEpattern) if for every instance I = 〈V ,A, var ,E , cpt〉 satisfying arcconsistency and for every variable v ∈ V we have:P does not occur on v ⇒ v can be eliminated





The reduction for VE patterns

P → P ′

P can be reduced to P ′

P occurs on v ⇐ P ′ occurs on vP does not occur on v ⇒ P ′ does not occur on v

P is a VE pattern ⇐ P ′ is a VE patternP is not a VE pattern ⇒ P ′ is not a VE pattern





Why the arc consistency requirement?

1 Because we can: arc consistency can be established inquadratic time.

2 Because otherwise VE patterns would not be interesting: anypattern containing 1C would not be a VE pattern.

1C





In what order should we eliminate?

It does not matter:

Eliminating a variable cannot create an occurence of a pattern.Actually, eliminating a variable can remove occurences of apattern, and make other variables eligible for elimination!





TheoremLet P be a non mergeable pattern. P is a VE pattern if and only if Pcan be extended to one of the patterns in B.

Av

∃v

BTP

a

∃a

∃subBTP

a ∃a

∃invsubBTP

a ∃a

∃snake(2)Forbidden Patterns in Constraint Satisfaction Problems 54/ 65




Proof

⇐ 1 We give 4 proofs to show that the 4 patterns in B allow variableelimination.

2 From the properties of the reduction, we have the result.

⇒ 1 We give a set G of gadgets.2 We show that none of the gadgets from G allows variable

elimination.3 We show that for any non mergeable pattern P, either P contains a

gadget from G, or P can be extended to one of the patterns in B.4 From the properties of the reduction, we have the result.





Examples of gadgets which are not VE

Av

∃v

a

∃a





Tractable classes

VE patterns define tractable classes.

P is a VE pattern⇒ P is a tractable pattern

One can just eliminate all variables from the instance.





Occurence in an instance

∃a

a

X1

a2

b2

Av0

Av2

Av1 Av2

An instance I

a2

b2


⇒ X1 occurs on v2.⇒ X1 occurs in I.






∃a

a

X1

a2

b2

Av0

Av2

Av1 Av2

An instance I

a2

b2

X1 occurs on a2.

X1 occurs on b2.⇒ X1 occurs on v2.⇒ X1 occurs in I.






∃a

a

X1

a2

b2

Av0

Av2

Av1 Av2

An instance I

a2

b2








∃a

a

X1

a2

b2

Av0

Av2Av1

Av2

An instance I

a2

b2


⇒ X1 occurs on v2.

⇒ X1 occurs in I.






∃a

a

X1

a2

b2

Av0

Av2

Av1 Av2

An instance I

a2

b2







Existential pattern→ Class

Notation

We note CSP(P) the set of CSP instances in which the existentialpattern P does NOT occur.

CSP(P) is polynomial → P is tractable.CSP(P) is NP-Complete → P is NP-Complete.





Two constraints

TheoremLet P be a non mergeable existential pattern on two constraints. P istractable if and only if P can be extended to one of the patterns in X .

∃a

a

X1

∃aa

X2

∃a

a

X3





Dichotomy on two constraints

TheoremLet P be a non mergeable pattern on two constraints. P is tractable ifand only if P is a subpattern of T1, T2, T3, T4, X1, X2 or X3.

RemarkT5 and 2I are not mentioned because they are subpatterns of X2 andX3 respectively.







4 Conclusion




Some results from my PhD

Formal definition of forbidden patterns.Flat patterns.Quantified patterns.Existential patterns.

New tool: the reduction.New tractable classes.

Tractable forbidden patterns on two constraints.Several tractable forbidden patterns on three variables.Tractable Max-CSP subproblems.

Instance simplification operations.Forbidden patterns allowing variable elimination.Subdomain fusion.




Recent forbidden patterns related work

Value elimination.Fusion of values.Topological minors: related to the notion of minor graph.




Question(s) time

Thank you very much for your attention!


forbidden patterns in constraint satisfaction problems · introduction flat patterns existential...

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