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

Probabilistic Graph Homomorphism Antoine Amarilli 1 , Mikaël Monet 1 , 2 , Pierre Senellart 2 , 3 September 15th, 2017 1 LTCI, Télécom ParisTech, Université Paris-Saclay; Paris, France 2 Inria Paris; Paris, France 3 École normale supérieure, PSL Research University; Paris, France

Probabilistic Graph A directed graph H = ( V , E ) 1/13

Probabilistic Graph A directed graph H = ( V , E ) With labels λ : E → Σ R S 1/13

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 1/13

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: 1/13

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 S R 1/13

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 . 5 × ( 1 − . 2 ) S R R 1/13

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 . 5 × ( 1 − . 2 ) ( 1 − . 5 ) × . 2 S R S R 1/13

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 . 5 × ( 1 − . 2 ) ( 1 − . 5 ) × . 2 ( 1 − . 5 ) × ( 1 − . 2 ) S R S R 1/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . 2/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . h : V G → V H is a homomorphism iff: 2/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . h : V G → V H is a homomorphism iff: • ( x , y ) ∈ E G = ⇒ ( h ( x ) , h ( y )) ∈ E H • ( x , y ) ∈ E G = ⇒ λ G (( x , y )) = λ H (( h ( x ) , h ( y ))) 2/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . h : V G → V H is a homomorphism iff: • ( x , y ) ∈ E G = ⇒ ( h ( x ) , h ( y )) ∈ E H • ( x , y ) ∈ E G = ⇒ λ G (( x , y )) = λ H (( h ( x ) , h ( y ))) R S S y x z G = t R S R H = • • • • 2/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . h : V G → V H is a homomorphism iff: • ( x , y ) ∈ E G = ⇒ ( h ( x ) , h ( y )) ∈ E H • ( x , y ) ∈ E G = ⇒ λ G (( x , y )) = λ H (( h ( x ) , h ( y ))) R S S y x z G = t R S R H = • • • • 2/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . h : V G → V H is a homomorphism iff: • ( x , y ) ∈ E G = ⇒ ( h ( x ) , h ( y )) ∈ E H • ( x , y ) ∈ E G = ⇒ λ G (( x , y )) = λ H (( h ( x ) , h ( y ))) R S S y x z G = t R S R H = • • • • 2/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . h : V G → V H is a homomorphism iff: • ( x , y ) ∈ E G = ⇒ ( h ( x ) , h ( y )) ∈ E H • ( x , y ) ∈ E G = ⇒ λ G (( x , y )) = λ H (( h ( x ) , h ( y ))) R S S y x z G = t R S R H = • • • • 2/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . h : V G → V H is a homomorphism iff: • ( x , y ) ∈ E G = ⇒ ( h ( x ) , h ( y )) ∈ E H • ( x , y ) ∈ E G = ⇒ λ G (( x , y )) = λ H (( h ( x ) , h ( y ))) R S S y x z G = t R S R H = • • • • 2/13

Graph Homomorphism G = ( V G , E G , λ G ) H = ( V H , E H , λ H ) . h : V G → V H is a homomorphism iff: • ( x , y ) ∈ E G = ⇒ ( h ( x ) , h ( y )) ∈ E H • ( x , y ) ∈ E G = ⇒ λ G (( x , y )) = λ H (( h ( x ) , h ( y ))) R S S y x z G = t R S R H = • • • • We write G ❀ H if there exists a homomorphism from G to H 2/13

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 3/13

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 3/13

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 ) 3/13

Example R S S y x z t G = R S R H = • • • • . 2 . 5 . 8 4/13

Example R S S y x z t G = R S R H = • • • • . 2 . 5 . 8 Pr ( G ❀ H ) = . 2 × . 5 4/13

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 ? 5/13

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! 5/13

Fix one side • 1.0 1.0 1.0 • Fix the instance graph H = • • NP-hard 6/13

Fix one side • 1.0 1.0 1.0 • Fix the instance graph H = • • NP-hard • Fix the query graph G = #P-hard 6/13

Fix one side • 1.0 1.0 1.0 • Fix the instance graph H = • • NP-hard • Fix the query graph G = #P-hard To make PHom tractable , we must restrict both sides 6/13

Restrict instance graphs to trees G = one-way paths ( 1WP ), H = polytrees ( PT ) 7/13

Restrict instance graphs to trees G = one-way paths ( 1WP ), H = polytrees ( PT ) T S S S T G : 7/13

Restrict instance graphs to trees G = one-way paths ( 1WP ), H = polytrees ( PT ) T T S S T T H : S S S T S S S T G : S T T S + prob. for each edge 7/13

Restrict instance graphs to trees G = one-way paths ( 1WP ), H = polytrees ( PT ) T T S S T T H : S S S T S S S T G : S T T S + prob. for each edge PHom of 1WP on PT is #P-hard ! 7/13

G = one-way paths , H = polytrees , without labels • What if we do not have labels ? T T S S T T H : S S S T S S S T G : S T T S + prob. for each edge 8/13

G = one-way paths , H = polytrees , without labels • What if we do not have labels ? H : G : + prob. for each edge 8/13

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 | H : G : + prob. for each edge 8/13

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 H : G : + prob. for each edge 8/13

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! H : G : + prob. for each edge 8/13

G = two-way paths , H = polytrees , without labels • G = one-way paths ( 1WP ), H = polytrees ( PT ) H : G : + prob. for each edge 9/13

G = two-way paths , H = polytrees , without labels • G = two -way paths ( 2WP ), H = polytrees ( PT ) H : G : + prob. for each edge 9/13

G = two-way paths , H = polytrees , without labels • G = two -way paths ( 2WP ), H = polytrees ( PT ) • #P-hard H : G : + prob. for each edge 9/13

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 H : G : + prob. for each edge 9/13

G = one-way paths , H = downwards trees • G = one-way paths ( 1WP ), H = polytrees ( PT ) T T S S T T H : S S S T S S S T G : S T T S + prob. for each edge 10/13

G = one-way paths , H = downwards trees • G = one-way paths ( 1WP ), H = downwards trees ( DWT ) T T S S T T H : S S S T S S S T G : S T T S + prob. for each edge 10/13

G = one-way paths , H = downwards trees • G = one-way paths ( 1WP ), H = downwards trees ( DWT ) • PTIME also: β - acyclicity of the lineage T T S S T T H : S S S T S S S T G : S T T S + prob. for each edge 10/13