Conjunctive Queries on Probabilistic Graphs: Combined Complexity Antoine Amarilli 1 , Mikaël Monet 1 , 2 , Pierre Senellart 2 , 3 May 16th, 2017 1 LTCI, Télécom ParisTech, Université Paris-Saclay; Paris, France 2 Inria Paris; Paris, France 3 École normale supérieure, PSL Research University; Paris, France
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
Example: TID S a b . 5 a c . 2 2/17
Example: TID S a b . 5 a c . 2 This TID ( I , π ) represents the following probability distribution: 2/17
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
Example: TID S a b . 5 a c . 2 This TID ( I , π ) represents the following probability distribution: . 5 × . 2 . 5 × ( 1 − . 2 ) S S a b a b a c 2/17
Example: TID S a b . 5 a c . 2 This TID ( I , π ) represents the following probability distribution: . 5 × . 2 . 5 × ( 1 − . 2 ) ( 1 − . 5 ) × . 2 S S S a b a b a c a c 2/17
Example: TID S a b . 5 a c . 2 This TID ( I , π ) represents the following probability distribution: . 5 × . 2 . 5 × ( 1 − . 2 ) ( 1 − . 5 ) × . 2 ( 1 − . 5 ) × ( 1 − . 2 ) S S S S a b a b a c a c 2/17
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
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
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 ) = � = q Pr ( J ) J ⊆ I , J | 3/17
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
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
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
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
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
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
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
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
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
Restrict to CQs on graph signatures R S S ∃ x y z t R ( x , y ) ∧ S ( y , z ) ∧ S ( t , z ) → y x z t R a b . 1 b c . 1 c d . 05 d a 1 . d b . 8 S b d . 7 6/17
Restrict to CQs on graph signatures R S S ∃ x y z t R ( x , y ) ∧ S ( y , z ) ∧ S ( t , z ) → y x z t R b a b . 1 . 1 R R . 1 b c . 1 R S c d . 05 a a c a → d a 1 . . 8 . 7 . 05 R d b . 8 1 . R d S b d . 7 6/17
Restrict instances to trees Q = one-way paths ( 1WP ), I = polytrees ( PT ) 7/17
Restrict instances to trees Q = one-way paths ( 1WP ), I = polytrees ( PT ) S S S T T Q : 7/17
Restrict instances to trees Q = one-way paths ( 1WP ), I = polytrees ( PT ) T T S S T T I : S S S S S S T T Q : S T T S + prob. for each edge 7/17
Restrict instances to trees Q = one-way paths ( 1WP ), I = polytrees ( PT ) T T S S T T I : S S S S S S T T Q : S T T S + prob. for each edge Proposition PQE of 1WP on PT is #P-hard 7/17
Q = one-way paths , I = polytrees , without labels • What if we do not have labels ? T T S S T T I : S S S T S S S T Q : S T T S + prob. for each edge 8/17
Q = one-way paths , I = polytrees , without labels • What if we do not have labels ? I : Q : + prob. for each edge 8/17
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 | I : Q : + prob. for each edge 8/17
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 I : Q : + prob. for each edge 8/17
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! I : Q : + prob. for each edge 8/17
Q = two-way paths , I = polytrees , without labels • Q = one-way paths ( 1WP ), I = polytrees ( PT ) I : Q : + prob. for each edge 9/17
Q = two-way paths , I = polytrees , without labels • Q = two -way paths ( 2WP ), I = polytrees ( PT ) I : Q : + prob. for each edge 9/17
Q = two-way paths , I = polytrees , without labels • Q = two -way paths ( 2WP ), I = polytrees ( PT ) • #P-hard I : Q : + prob. for each edge 9/17
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 I : Q : + prob. for each edge 9/17
Q = one-way paths , I = downwards trees • Q = one-way paths ( 1WP ), I = polytrees ( PT ) T T S S T T I : S S S T S S S T Q : S T T S + prob. for each edge 10/17
Q = one-way paths , I = downwards trees • Q = one-way paths ( 1WP ), I = downwards trees ( DWT ) T T S S T T I : S S S T S S S T Q : S T T S + prob. for each edge 10/17
Q = one-way paths , I = downwards trees • Q = one-way paths ( 1WP ), I = downwards trees ( DWT ) • PTIME also: β - acyclicity of the lineage T T S S T T I : S S S T S S S T Q : S T T S + prob. for each edge 10/17
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! T T S S T T I : S S S T S S S T Q : S T T S + prob. for each edge 10/17
Q = downwards trees, I = downwards trees, with labels • Q = one-way paths ( 1WP ), I = downwards trees T T S S T T I : S S S T S S S T Q : S T T S + prob. for each edge 11/17
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