Conjunctive Queries on Probabilistic Graphs: Combined Complexity - - PowerPoint PPT Presentation

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Conjunctive Queries on Probabilistic Graphs: Combined Complexity - - PowerPoint PPT Presentation

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Conjunctive Queries on Probabilistic Graphs: Combined Complexity Antoine Amarilli 1 , Mikal Monet 1 , 2 , Pierre Senellart 2 , 3 May 16th, 2017 1 LTCI, Tlcom ParisTech, Universit Paris-Saclay; Paris, France 2 Inria Paris; Paris, France 3

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SLIDE 1

Conjunctive Queries on Probabilistic Graphs: Combined Complexity

Antoine Amarilli1, Mikaël Monet1,2, Pierre Senellart2,3

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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SLIDE 2

Tuple-independent databases (TID)

  • Probabilistic databases: model uncertainty about data
  • Simplest model: tuple-independent databases (TID)
  • A relational database I
  • A probability valuation π mapping each fact of I to [0, 1]
  • Semantics of a TID (I, π): a probability distribution on I′ ⊆ I:
  • Each fact F ∈ I is either present or absent with probability π(F)
  • Assume independence across facts

1/17

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SLIDE 3

Example: TID

S a b .5 a c .2

2/17

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Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution:

2/17

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SLIDE 5

Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution: .5 × .2 S a b a c

2/17

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Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution: .5 × .2 S a b a c .5 × (1 − .2) S a b

2/17

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SLIDE 7

Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability distribution: .5 × .2 S a b a c .5 × (1 − .2) S a b (1 − .5) × .2 S a c

2/17

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SLIDE 8

Example: TID

S a b .5 a c .2 This TID (I, π) represents the following probability 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

2/17

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SLIDE 9

Probabilistic query evaluation (PQE)

Let us fix:

  • Relational signature σ
  • Class I of relational instances on σ (e.g., acyclic, treelike)
  • Class Q of Boolean queries (e.g., paths, trees)

3/17

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SLIDE 10

Probabilistic query evaluation (PQE)

Let us fix:

  • Relational signature σ
  • Class I of relational instances on σ (e.g., acyclic, treelike)
  • Class Q of Boolean queries (e.g., paths, trees)

Probabilistic query evaluation (PQE) problem for Q and I:

  • Given a query q ∈ Q
  • Given an instance I ∈ I and a probability valuation π
  • Compute the probability that (I, π) satisfies q

3/17

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SLIDE 11

Probabilistic query evaluation (PQE)

Let us fix:

  • Relational signature σ
  • Class I of relational instances on σ (e.g., acyclic, treelike)
  • Class Q of Boolean queries (e.g., paths, trees)

Probabilistic query evaluation (PQE) problem for Q and I:

  • Given a query q ∈ Q
  • Given an instance I ∈ I and a probability valuation π
  • Compute the probability that (I, π) satisfies q

→ Pr((I, π) | = q) =

J⊆I, J| =q Pr(J)

3/17

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SLIDE 12

Complexity of probabilistic query evaluation (PQE)

Question: what is the (data, combined) complexity of PQE depending on the class Q of queries and class I of instances?

4/17

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SLIDE 13

Data complexity results

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • Q = UCQs
  • I is all instances
  • There is a class S ⊆ Q of safe queries

5/17

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SLIDE 14

Data complexity results

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • Q = UCQs
  • I is all instances
  • There is a class S ⊆ Q of safe queries

→ PQE is PTIME for any q ∈ S

5/17

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SLIDE 15

Data complexity results

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • Q = UCQs
  • I is all instances
  • There is a class S ⊆ Q of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ Q\S

5/17

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SLIDE 16

Data complexity results

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • Q = UCQs
  • I is all instances
  • There is a class S ⊆ Q of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ Q\S

  • Existing data dichotomy result on instances

5/17

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SLIDE 17

Data complexity results

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • Q = UCQs
  • I is all instances
  • There is a class S ⊆ Q of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ Q\S

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015]

5/17

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SLIDE 18

Data complexity results

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • Q = UCQs
  • I is all instances
  • There is a class S ⊆ Q of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ Q\S

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015] → There is an FO query for which PQE is #P-hard on any unbounded-treewidth graph family I (under some assumptions) [Amarilli, Bourhis, & Senellart, 2016]

5/17

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SLIDE 19

Data complexity results

  • Existing data dichotomy result on queries [Dalvi & Suciu, 2012]
  • Q = UCQs
  • I is all instances
  • There is a class S ⊆ Q of safe queries

→ PQE is PTIME for any q ∈ S → PQE is #P-hard for any q ∈ Q\S

  • Existing data dichotomy result on instances

→ PQE for MSO on bounded-treewidth instances has linear data complexity [Amarilli, Bourhis, & Senellart, 2015] → There is an FO query for which PQE is #P-hard on any unbounded-treewidth graph family I (under some assumptions) [Amarilli, Bourhis, & Senellart, 2016]

What about combined complexity?

5/17

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SLIDE 20

Restrict to CQs on graph signatures

∃x y z t R(x, y) ∧ S(y, z) ∧ S(t, z) R a b .1 b c .1 c d .05 d a 1. d b .8 S b d .7

6/17

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SLIDE 21

Restrict to CQs on graph signatures

∃x y z t R(x, y) ∧ S(y, z) ∧ S(t, z) → x y z t R S S R a b .1 b c .1 c d .05 d a 1. d b .8 S b d .7

6/17

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SLIDE 22

Restrict to CQs on graph signatures

∃x y z t R(x, y) ∧ S(y, z) ∧ S(t, z) → x y z t R S S R a b .1 b c .1 c d .05 d a 1. d b .8 S b d .7 → d c b a 1. R .1 R R .1 R .05 S a .7 R a .8

6/17

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SLIDE 23

Restrict instances to trees

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

7/17

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Restrict instances to trees

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

7/17

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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 T T S S S S S S T S T

7/17

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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 T T S S S S S S T S T Proposition PQE of 1WP on PT is #P-hard

7/17

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

  • What if we do not have labels?

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T

8/17

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SLIDE 28

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

  • What if we do not have labels?

Q: I: + prob. for each edge

8/17

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SLIDE 29

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|

Q: I: + prob. for each edge

8/17

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SLIDE 30

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

Q: I: + prob. for each edge

8/17

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SLIDE 31

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
  • Labels have an impact!

Q: I: + prob. for each edge

8/17

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SLIDE 32

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

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

Q: I: + prob. for each edge

9/17

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SLIDE 33

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

  • Q = two-way paths (2WP), I = polytrees (PT)

Q: I: + prob. for each edge

9/17

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SLIDE 34

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

  • Q = two-way paths (2WP), I = polytrees (PT)
  • #P-hard

Q: I: + prob. for each edge

9/17

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SLIDE 35

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

  • Q = two-way paths (2WP), I = polytrees (PT)
  • #P-hard
  • Global orientation of the query has an impact

Q: I: + prob. for each edge

9/17

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SLIDE 36

Q = one-way paths, I = downwards trees

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

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T

10/17

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SLIDE 37

Q = one-way paths, I = downwards trees

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

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T

10/17

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SLIDE 38

Q = one-way paths, I = downwards trees

  • Q = one-way paths (1WP), I = downwards trees (DWT)
  • PTIME also: β-acyclicity of the lineage

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T

10/17

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Q = one-way paths, I = downwards trees

  • Q = one-way paths (1WP), I = downwards trees (DWT)
  • PTIME also: β-acyclicity of the lineage
  • Global orientation of the instance also has an impact!

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T

10/17

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SLIDE 40

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

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

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T

11/17

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SLIDE 41

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

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

Q: T S S S I: + prob. for each edge T T T T S S S S S S T S T

11/17

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SLIDE 42

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

  • Q = downwards trees (DWT), I = downwards trees
  • #P-hard

Q: T S S S I: + prob. for each edge T T T T S S S S S S T S T

11/17

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SLIDE 43

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

  • Q = downwards trees (DWT), I = downwards trees
  • #P-hard
  • Branching has an impact!

Q: T S S S I: + prob. for each edge T T T T S S S S S S T S T

11/17

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SLIDE 44

Our graph classes

1WP 2WP R S S T R S S T R DWT PT 1WP 2WP DWT PT Connected All ⊆ ⊆ ⊆ ⊆ ⊆ ⊆

12/17

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SLIDE 45

Results

↓Q I→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected 2 labels

13/17

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Results

↓Q I→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected 2 labels ↓Q I→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected No labels

13/17

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SLIDE 47

Results

↓Q I→ 1WP 2WP DWT PT Connected 1WP

  • 2WP
  • DWT

PTIME

  • PT

#P-hard Connected

  • 2 labels

↓Q I→ 1WP 2WP DWT PT Connected 1WP

  • 2WP
  • DWT

PTIME

  • PT

#P-hard Connected

  • No labels

13/17

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SLIDE 48

Results

↓Q I→ 1WP 2WP DWT PT Connected 1WP

  • 2WP

DWT PTIME PT #P-hard Connected 2 labels ↓Q I→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected No labels

13/17

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SLIDE 49

Reduction for Q = one-way paths, I = polytrees

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T

14/17

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Reduction for Q = one-way paths, I = polytrees

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T Reduction from #P-hard problem #PP2DNF:

  • INPUT: Boolean formula ϕ =

j=1...m(Xxj ∧ Yyj) on variables

{X1, . . . , Xn1} ⊔ {Y1, . . . , Yn2}

14/17

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SLIDE 51

Reduction for Q = one-way paths, I = polytrees

Q: T S S S T I: + prob. for each edge T T T T S S S S S S T S T Reduction from #P-hard problem #PP2DNF:

  • INPUT: Boolean formula ϕ =

j=1...m(Xxj ∧ Yyj) on variables

{X1, . . . , Xn1} ⊔ {Y1, . . . , Yn2}

  • OUTPUT: number of satisfying assignments of ϕ

14/17

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SLIDE 52

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2

15/17

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SLIDE 53

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I: Q:

15/17

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SLIDE 54

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • Q:

15/17

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SLIDE 55

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 S S Q:

15/17

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SLIDE 56

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S Q:

15/17

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SLIDE 57

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S Q:

15/17

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SLIDE 58

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S Q:

15/17

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SLIDE 59

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T Q:

15/17

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SLIDE 60

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T Q:

15/17

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SLIDE 61

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T Q:

15/17

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SLIDE 62

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T Q:

15/17

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SLIDE 63

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T Q:

15/17

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SLIDE 64

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

15/17

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SLIDE 65

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

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SLIDE 66

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

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SLIDE 67

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

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SLIDE 68

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

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SLIDE 69

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

slide-70
SLIDE 70

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

slide-71
SLIDE 71

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

slide-72
SLIDE 72

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

slide-73
SLIDE 73

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

slide-74
SLIDE 74

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

slide-75
SLIDE 75

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

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SLIDE 76

Reduction for Q = one-way paths, I = polytrees

ϕ = X1Y2 ∨ X1Y1 ∨ X2Y2 #ϕ = Pr((I, π) | = Q) × 2|vars(ϕ)| I:

  • X1

X2 Y1 Y2 S S S S S S S S S S S S S S S S T T T T T T Q:

T

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

S

− − →

T

− − →

15/17

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SLIDE 77

Disconnected graphs

We also introduce the classes 1WP (resp., 2WP, DWT, PT) of graphs that are disjoint unions of 1WP (resp., 2WP, DWT, PT)

16/17

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SLIDE 78

Disconnected graphs

We also introduce the classes 1WP (resp., 2WP, DWT, PT) of graphs that are disjoint unions of 1WP (resp., 2WP, DWT, PT) ↓G H→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PT All No labels

16/17

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SLIDE 79

Disconnected graphs

We also introduce the classes 1WP (resp., 2WP, DWT, PT) of graphs that are disjoint unions of 1WP (resp., 2WP, DWT, PT) ↓G H→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PT All No labels With labels, PQE of 1WP on 1WP is already #P-hard!

16/17

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SLIDE 80

Conclusion

Contributions:

  • Detailed study of the combined complexity of PQE

17/17

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SLIDE 81

Conclusion

Contributions:

  • Detailed study of the combined complexity of PQE
  • Focus on CQs on arity-two signatures

17/17

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SLIDE 82

Conclusion

Contributions:

  • Detailed study of the combined complexity of PQE
  • Focus on CQs on arity-two signatures
  • Showed the importance of various features on the problem:

labels, global orientation, branching, connectedness

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slide-83
SLIDE 83

Conclusion

Contributions:

  • Detailed study of the combined complexity 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

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slide-84
SLIDE 84

Conclusion

Contributions:

  • Detailed study of the combined complexity 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”

17/17

slide-85
SLIDE 85

Conclusion

Contributions:

  • Detailed study of the combined complexity 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?

17/17

slide-86
SLIDE 86

Conclusion

Contributions:

  • Detailed study of the combined complexity 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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