Conjunctive Queries on Probabilistic Graphs: Combined Complexity

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Conjunctive Queries on Probabilistic Graphs:

Combined Complexity

Antoine Amarilli¹,Mikaël Monet^1,2, Pierre Senellart^2,3 May 16th, 2017

1LTCI, Télécom ParisTech, Université Paris-Saclay; Paris, France

2Inria Paris; Paris, France

3École normale supérieure, PSL Research University; Paris, France

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Tuple-independent databases (TID)

•

Probabilistic databases:modeluncertaintyabout data

•

Simplest model:tuple-independent databases(TID)

• Arelational databaseI

• Aprobability valuationπmapping each fact ofIto[0,1]

•

^Semantics^{of a TID}^{(I, π): a}probability distributiononI⁰⊆I:

• Each factF∈Iis eitherpresentorabsentwith probabilityπ(F)

• Assumeindependenceacross facts

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Example: TID

S a b .5 a c .2

This TID(I, π)represents the followingprobability distribution:

.5×.2 S a b a c

.5×(1−.2) S

a b

(1−.5)×.2 S

a c

(1−.5)×(1−.2) S

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Example: TID

S a b .5 a c .2

.5×.2 S a b a c

.5×(1−.2) S

a b

(1−.5)×.2 S

a c

(1−.5)×(1−.2) S

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Example: TID

S a b .5 a c .2

.5×.2 S a b a c

.5×(1−.2) S

a b

(1−.5)×.2 S

a c

(1−.5)×(1−.2) S

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Example: TID

S a b .5 a c .2

.5×.2 S a b a c

.5×(1−.2) S

a b

(1−.5)×.2 S

a c

(1−.5)×(1−.2) S

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Example: TID

S a b .5 a c .2

.5×.2 S a b a c

.5×(1−.2) S

a b

(1−.5)×.2 S

a c

(1−.5)×(1−.2) S

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Example: TID

S a b .5 a c .2

.5×.2 S a b a c

.5×(1−.2) S

a b

(1−.5)×.2 S

a c

(1−.5)×(1−.2) S

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Probabilistic query evaluation (PQE)

Let us fix:

•

Relational signatureσ

•

^Class^I ^ofrelational instancesonσ (e.g., acyclic, treelike)

•

^Class^Q^ofBoolean queries(e.g., paths, trees)

Probabilistic query evaluation(PQE) problem forQandI:

•

^{Given a}^query^q^{∈ Q}

•

^{Given an}^instance^I^{∈ I} ^{and a}probability valuationπ

•

Compute theprobabilitythat(I, π)satisfiesq

→ Pr((I, π)|=q) =P

J⊆I,J|=qPr(J)

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Let us fix:

•

→ Pr((I, π)|=q) =P

J⊆I,J|=qPr(J)

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Let us fix:

•

→ Pr((I, π)|=q) =P

J⊆I,J|=qPr(J)

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Complexity of probabilistic query evaluation (PQE)

Question:what is the (data, combined)complexityof PQE depending on the classQofqueriesand classI ofinstances?

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Data complexity results

•

^Existingdata dichotomy resulton queries [Dalvi & Suciu, 2012]

• Q=UCQs

• I is all instances

• There is a classS ⊆ Qofsafe queries

→ PQE isPTIMEfor anyq∈ S

→ PQE is#P-hardfor anyq∈ Q\S

•

^Existingdata dichotomy resulton instances

→ PQE forMSOonbounded-treewidthinstances haslineardata complexity [Amarilli, Bourhis, & Senellart, 2015]

→ There is an FO query for which PQE is#P-hardonany

unbounded-treewidth graph familyI (under some assumptions) [Amarilli, Bourhis, & Senellart, 2016]

What aboutcombinedcomplexity?

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•

• Q=UCQs

•

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•

• Q=UCQs

•

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•

• Q=UCQs

•

(17)

•

• Q=UCQs

•

(18)

•

• Q=UCQs

•

unbounded-treewidth graph familyI(under some assumptions) [Amarilli, Bourhis, & Senellart, 2016]

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•

• Q=UCQs

•

unbounded-treewidth graph familyI(under some assumptions) [Amarilli, Bourhis, & Senellart, 2016]

What about complexity?

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Restrict to CQs on graph signatures

∃x y z t R(x,y)∧S(y,z)∧S(t,z)

→ x R y S z S t

R a b .1 b c .1 c d .05 d a 1.

d b .8 S b d .7

→

d

c b

a R 1.

R .1 R

.1

R .05 S

a .7 R a .8

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∃x y z t R(x,y)∧S(y,z)∧S(t,z) → x R y S z S t R

a b .1 b c .1 c d .05 d a 1.

d b .8 S b d .7

→

d

c b

a R 1.

R .1 R

.1

R .05 S

a .7 R a .8

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∃x y z t R(x,y)∧S(y,z)∧S(t,z) → x R y S z S t R

a b .1 b c .1 c d .05 d a 1.

d b .8 S b d .7

→

d

c b

a R 1.

R .1 R

.1

R .05 S

a .7 R a .8

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Restrict instances to trees

Q=one-way paths(1WP),I =polytrees(PT)

Q: T S S S T

I:

+ prob. for each edge

T T

S S

S

T S

T

Proposition

PQE of1WPonPTis#P-hard

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Q: T S S S T

I:

T T

S S

S

T S

T

Proposition

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Q: T S S S T

I:

T T

S S

S

T S

T

Proposition

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Q: T S S S T

I:

T T

S S

S

T S

T

Proposition

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Q=one-way paths,I =polytrees, without labels

•

^{What if we}do not have labels?

•

Probability that the instance graph has a path of length|Q|

•

^PTIME: Bottom-up, e.g., tree automaton

•

^Labelshave an impact!

Q: T S S S T

I:

T T

S S

S

S T

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•

Q:

I:

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•

Q:

I:

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•

Q:

I:

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•

Q:

I:

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Q=two-way paths,I =polytrees, without labels

•

^Q⁼one-way paths(1WP),I=polytrees(PT)

•

^#P-hard

•

Global orientationof thequeryhas an impact

Q:

I:

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•

^Q⁼two-way paths(2WP),I=polytrees(PT)

•

^#P-hard

•

Q:

I:

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•

^#P-hard

•

Q:

I:

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•

^#P-hard

•

Q:

I:

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Q=one-way paths,I =downwards trees

•

^Q⁼one-way paths(1WP),I=polytrees(PT)

•

^PTIME^also:^β^-acyclicity of the lineage

•

Global orientationof theinstancealso has an impact!

Q: T S S S T

I:

T T

S S

S

S T

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•

^Q⁼one-way paths(1WP),I=downwards trees(DWT)

•

Q: T S S S T

I:

T T

S S

S

S T

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•

Q: T S S S T

I:

T T

S S

S

S T

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•

Q: T S S S T

I:

T T

S S

S

S T

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Q=downwards trees,I =downwards trees, with labels

•

^Q⁼one-way paths(1WP),I=downwards trees

•

^#P-hard

•

^Branchinghas an impact!

Q: T S S S T

I:

T T

S S

S

S T

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•

^Q⁼downwards trees(DWT),I =downwards trees

•

^#P-hard

•

Q:

T

S S

S

I:

T T

S S

S

S T

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•

^#P-hard

•

Q:

T

S S

S

I:

T T

S S

S

S T

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•

^#P-hard

•

Q:

T

S S

S

I:

T T

S S

S

S T

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Our graph classes

1WP

2WP

R S S T

R S S T R

DWT PT

1WP

2WP DWT

PT Connected All

⊆ ⊆

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Results

↓Q I→ 1WP 2WP DWT PT Connected 1WP

• •

2WP

•

DWT PTIME

•

PT #P-hard

Connected

•

>2labels

•

2WP

•

DWT PTIME

•

PT #P-hard

Connected

• •

No labels

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Results

• •

2WP

•

DWT PTIME

•

PT #P-hard

Connected

•

>2labels

•

2WP

•

DWT PTIME

•

PT #P-hard

Connected

• •

No labels

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Results

↓Q I→ 1WP 2WP DWT PT Connected

1WP • •

2WP •

DWT PTIME •

PT #P-hard

Connected •

>2labels

↓Q I→ 1WP 2WP DWT PT Connected

1WP •

2WP •

DWT PTIME •

PT #P-hard

Connected • •

No labels

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Results

•

• 2WP

•

DWT PTIME

•

PT #P-hard

Connected

•

>2labels

•

2WP

•

DWT PTIME

•

PT #P-hard

Connected

• •

No labels

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Reduction forQ=one-way paths,I =polytrees

Q: T S S S T

I:

T T

S S

S

S T

Reduction from#P-hardproblem#PP2DNF:

•

^INPUT:Boolean formulaϕ=W

j=1...m(Xx_j∧Yy_j)on variables {X1, . . . ,Xn₁} t {Y1, . . . ,Yn₂}

•

^OUTPUT:number of satisfying assignments ofϕ

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Q: T S S S T

I:

T T

S S

S

S T

•

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Q: T S S S T

I:

T T

S S

S

S T

•

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Reduction forQ=one-way paths,I =polytrees ϕ=X1Y2∨X1Y1∨X2Y2

#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X2 Y1 YY22

Q:

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X2 Y1 YY22

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X2 Y1 YY22

Q:

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2

Y1 YY22

S S

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

Q:

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

Q:

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

Q:

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

T T

Q:

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

T Q:

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

T Q: −−→^T −−→^S −−→^S −−→^S −−→^S −−→^S −−→^S −−→^T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

T Q: −−→^T −−→^S −−→^S −−→^S −−→^S −−→^S −−→^S −−→^T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

T Q: −−→^T −−→^S −−→^S −−→^S −−→^S −−→^S −−→^S −−→^T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S S S T

T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S

S S T

T

T Q: −−→^T −−→^S −−→^S −−→^S −−→^S −−→^S −−→^S −−→^T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1 Y2

Y2

S S S S

S S S

S

S S T

T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1

Y2

S S S S

S S S

S

S S T

T

T Q: −−→^T −−→^S −−→^S −−→^S −−→^S −−→^S −−→^S −−→^T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1

Y2

S S S S

S S S

S

S S T

T

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#ϕ=Pr((I, π)|=Q)×2^|vars(ϕ)|

I:

•

X1 X2 Y1

Y2

S S S S

S S S

S

S S T

T

T Q: −−→^T −−→^S −−→^S −−→^S −−→^S −−→^S −−→^S −−→^T

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Disconnected graphs

We also introduce the classesF

1WP(resp.,F

2WP,F

DWT,F PT) of graphs that aredisjoint unions of1WP(resp.,2WP,DWT,PT)

↓G H→ 1WP 2WP DWT PT Connected F1WP

F2WP FDWT FPT

All

No labels

With labels, PQE ofF

1WPon1WPis already#P-hard!

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1WP(resp.,F

2WP,F

F2WP FDWT FPT

All

No labels

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1WP(resp.,F

2WP,F

F2WP FDWT FPT

All

No labels

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Conclusion

Contributions:

•

Detailed study of thecombinedcomplexity of PQE

•

Focus on CQs on arity-two signatures

•

Showed the importance of various features on the problem: labels, global orientation, branching, connectedness

•

Established the complexity for all combinations of the graph classes we considered

Drawbacks and future work:

•

Our graph classes may seem “arbitrary”

•

Not yet a dichotomy, just starting to understand the problem

•

Practical applications?

Thanks for your attention!

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Conclusion

Contributions:

•

Showed the importance of various features on the problem: labels, global orientation, branching, connectedness