Probabilistic Graph Homomorphism Antoine Amarilli 1 , Mikal Monet 1 , - - PowerPoint PPT Presentation

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Probabilistic Graph Homomorphism Antoine Amarilli 1 , Mikal Monet 1 , - - PowerPoint PPT Presentation

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Probabilistic Graph Homomorphism Antoine Amarilli 1 , Mikal Monet 1 , 2 , Pierre Senellart 2 , 3 September 15th, 2017 1 LTCI, Tlcom ParisTech, Universit Paris-Saclay; Paris, France 2 Inria Paris; Paris, France 3 cole normale suprieure,

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Probabilistic Graph Homomorphism

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

September 15th, 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

Probabilistic Graph

A directed graph H = (V, E)

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Probabilistic Graph

A directed graph H = (V, E) With labels λ : E → Σ R S

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SLIDE 4

Probabilistic Graph

A directed graph H = (V, E) With labels λ : E → Σ With independent probability annotations π : E → [0, 1] on edges R S .5 .2

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SLIDE 5

Probabilistic Graph

A directed graph H = (V, E) With labels λ : E → Σ With independent probability annotations π : E → [0, 1] on edges R S .5 .2 This probabilistic graph represents the following probability distribution:

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SLIDE 6

Probabilistic Graph

A directed graph H = (V, E) With labels λ : E → Σ With independent probability annotations π : E → [0, 1] on edges R S .5 .2 This probabilistic graph represents the following probability distribution: .5 × .2 R S

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SLIDE 7

Probabilistic Graph

A directed graph H = (V, E) With labels λ : E → Σ With independent probability annotations π : E → [0, 1] on edges R S .5 .2 This probabilistic graph represents the following probability distribution: .5 × .2 R S .5 × (1 − .2) R

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SLIDE 8

Probabilistic Graph

A directed graph H = (V, E) With labels λ : E → Σ With independent probability annotations π : E → [0, 1] on edges R S .5 .2 This probabilistic graph represents the following probability distribution: .5 × .2 R S .5 × (1 − .2) R (1 − .5) × .2 S

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SLIDE 9

Probabilistic Graph

A directed graph H = (V, E) With labels λ : E → Σ With independent probability annotations π : E → [0, 1] on edges R S .5 .2 This probabilistic graph represents the following probability distribution: .5 × .2 R S .5 × (1 − .2) R (1 − .5) × .2 S (1 − .5) × (1 − .2)

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Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH).

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SLIDE 11

Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH). h : VG → VH is a homomorphism iff:

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SLIDE 12

Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH). h : VG → VH is a homomorphism iff:

  • (x, y) ∈ EG =

⇒ (h(x), h(y)) ∈ EH

  • (x, y) ∈ EG =

⇒ λG((x, y)) = λH((h(x), h(y)))

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SLIDE 13

Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH). h : VG → VH is a homomorphism iff:

  • (x, y) ∈ EG =

⇒ (h(x), h(y)) ∈ EH

  • (x, y) ∈ EG =

⇒ λG((x, y)) = λH((h(x), h(y))) G = x y z t R S S H =

  • R

S R

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SLIDE 14

Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH). h : VG → VH is a homomorphism iff:

  • (x, y) ∈ EG =

⇒ (h(x), h(y)) ∈ EH

  • (x, y) ∈ EG =

⇒ λG((x, y)) = λH((h(x), h(y))) G = x y z t R S S H =

  • R

S R

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SLIDE 15

Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH). h : VG → VH is a homomorphism iff:

  • (x, y) ∈ EG =

⇒ (h(x), h(y)) ∈ EH

  • (x, y) ∈ EG =

⇒ λG((x, y)) = λH((h(x), h(y))) G = x y z t R S S H =

  • R

S R

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SLIDE 16

Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH). h : VG → VH is a homomorphism iff:

  • (x, y) ∈ EG =

⇒ (h(x), h(y)) ∈ EH

  • (x, y) ∈ EG =

⇒ λG((x, y)) = λH((h(x), h(y))) G = x y z t R S S H =

  • R

S R

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SLIDE 17

Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH). h : VG → VH is a homomorphism iff:

  • (x, y) ∈ EG =

⇒ (h(x), h(y)) ∈ EH

  • (x, y) ∈ EG =

⇒ λG((x, y)) = λH((h(x), h(y))) G = x y z t R S S H =

  • R

S R

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SLIDE 18

Graph Homomorphism

G = (VG, EG, λG) H = (VH, EH, λH). h : VG → VH is a homomorphism iff:

  • (x, y) ∈ EG =

⇒ (h(x), h(y)) ∈ EH

  • (x, y) ∈ EG =

⇒ λG((x, y)) = λH((h(x), h(y))) G = x y z t R S S H =

  • R

S R We write G ❀ H if there exists a homomorphism from G to H

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SLIDE 19

Probabilistic Graph Homomorphism (PHom)

Let us fix:

  • Finite set of labels Σ
  • Class G of query graphs on Σ (e.g., paths, trees)
  • Class H of instance graphs on Σ

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SLIDE 20

Probabilistic Graph Homomorphism (PHom)

Let us fix:

  • Finite set of labels Σ
  • Class G of query graphs on Σ (e.g., paths, trees)
  • Class H of instance graphs on Σ

Probabilistic Graph Homomorphism (PHom) problem for G and H:

  • Given a query graph G ∈ G
  • Given an instance graph H ∈ H and a probability valuation π
  • Compute the probability that G has a homomorphism to H

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SLIDE 21

Probabilistic Graph Homomorphism (PHom)

Let us fix:

  • Finite set of labels Σ
  • Class G of query graphs on Σ (e.g., paths, trees)
  • Class H of instance graphs on Σ

Probabilistic Graph Homomorphism (PHom) problem for G and H:

  • Given a query graph G ∈ G
  • Given an instance graph H ∈ H and a probability valuation π
  • Compute the probability that G has a homomorphism to H

→ Pr(G ❀ H) =

J⊆H, G❀J Pr(J)

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SLIDE 22

Example

G = x y z t R S S H =

  • R

.2 S .5 R .8

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SLIDE 23

Example

G = x y z t R S S H =

  • R

.2 S .5 R .8 Pr(G ❀ H) = .2 × .5

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Complexity of Probabilistic Graph Homomorphism

Question: what is the complexity of PHom depending on the class G of query graphs and class H of instance graphs?

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Complexity of Probabilistic Graph Homomorphism

Question: what is the complexity of PHom depending on the class G of query graphs and class H of instance graphs? Like CSP but with probabilities!

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SLIDE 26

Fix one side

  • Fix the instance graph H = •
  • 1.0

1.0 1.0 NP-hard

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Fix one side

  • Fix the instance graph H = •
  • 1.0

1.0 1.0 NP-hard

  • Fix the query graph G =

#P-hard

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SLIDE 28

Fix one side

  • Fix the instance graph H = •
  • 1.0

1.0 1.0 NP-hard

  • Fix the query graph G =

#P-hard To make PHom tractable, we must restrict both sides

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SLIDE 29

Restrict instance graphs to trees

G = one-way paths (1WP), H = polytrees (PT)

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Restrict instance graphs to trees

G = one-way paths (1WP), H = polytrees (PT) G: T S S S T

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Restrict instance graphs to trees

G = one-way paths (1WP), H = polytrees (PT) G: T S S S T H: + prob. for each edge T T T T S S S S S S T S T

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SLIDE 32

Restrict instance graphs to trees

G = one-way paths (1WP), H = polytrees (PT) G: T S S S T H: + prob. for each edge T T T T S S S S S S T S T PHom of 1WP on PT is #P-hard!

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SLIDE 33

G = one-way paths, H = polytrees, without labels

  • What if we do not have labels?

G: T S S S T H: + prob. for each edge T T T T S S S S S S T S T

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G = one-way paths, H = polytrees, without labels

  • What if we do not have labels?

G: H: + prob. for each edge

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G = one-way paths, H = polytrees, without labels

  • What if we do not have labels?
  • Probability that the instance graph has a path of length |G|

G: H: + prob. for each edge

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SLIDE 36

G = one-way paths, H = polytrees, without labels

  • What if we do not have labels?
  • Probability that the instance graph has a path of length |G|
  • PTIME: Bottom-up, e.g., tree automaton

G: H: + prob. for each edge

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SLIDE 37

G = one-way paths, H = polytrees, without labels

  • What if we do not have labels?
  • Probability that the instance graph has a path of length |G|
  • PTIME: Bottom-up, e.g., tree automaton
  • Labels have an impact!

G: H: + prob. for each edge

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SLIDE 38

G = two-way paths, H = polytrees, without labels

  • G = one-way paths (1WP), H = polytrees (PT)

G: H: + prob. for each edge

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SLIDE 39

G = two-way paths, H = polytrees, without labels

  • G = two-way paths (2WP), H = polytrees (PT)

G: H: + prob. for each edge

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G = two-way paths, H = polytrees, without labels

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

G: H: + prob. for each edge

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G = two-way paths, H = polytrees, without labels

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

G: H: + prob. for each edge

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G = one-way paths, H = downwards trees

  • G = one-way paths (1WP), H = polytrees (PT)

G: T S S S T H: + prob. for each edge T T T T S S S S S S T S T

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G = one-way paths, H = downwards trees

  • G = one-way paths (1WP), H = downwards trees (DWT)

G: T S S S T H: + prob. for each edge T T T T S S S S S S T S T

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G = one-way paths, H = downwards trees

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

G: T S S S T H: + prob. for each edge T T T T S S S S S S T S T

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G = one-way paths, H = downwards trees

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

G: T S S S T H: + prob. for each edge T T T T S S S S S S T S T

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G = downwards trees, H = downwards trees, with labels

  • G = one-way paths (1WP), H = downwards trees

G: T S S S T H: + prob. for each edge T T T T S S S S S S T S T

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SLIDE 47

G = downwards trees, H = downwards trees, with labels

  • G = downwards trees (DWT), H = downwards trees

G: T S S S H: + prob. for each edge T T T T S S S S S S T S T

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SLIDE 48

G = downwards trees, H = downwards trees, with labels

  • G = downwards trees (DWT), H = downwards trees
  • #P-hard

G: T S S S H: + prob. for each edge T T T T S S S S S S T S T

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G = downwards trees, H = downwards trees, with labels

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

G: T S S S H: + prob. for each edge T T T T S S S S S S T S T

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Results

↓G H→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected 2 labels ↓G H→ 1WP 2WP DWT PT Connected 1WP 2WP DWT PTIME PT #P-hard Connected No labels

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Conclusion

  • We identified the complexity of PHom for a variety of graph

classes

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SLIDE 52

Conclusion

  • We identified the complexity of PHom for a variety of graph

classes

  • Showed the importance of various features on the problem:

labels, global orientation, branching, connectedness

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SLIDE 53

Conclusion

  • We identified the complexity of PHom for a variety of graph

classes

  • Showed the importance of various features on the problem:

labels, global orientation, branching, connectedness Future work:

  • What is the hidden logic behind these tables?
  • Can we get a dichotomy?

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SLIDE 54

Conclusion

  • We identified the complexity of PHom for a variety of graph

classes

  • Showed the importance of various features on the problem:

labels, global orientation, branching, connectedness Future work:

  • What is the hidden logic behind these tables?
  • Can we get a dichotomy?

Thank you for your attention!

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