A Dichotomy for Homomorphism-Closed Queries on Probabilistic Graphs

A Dichotomy for Homomorphism-Closed Queries on Probabilistic Graphs

Antoine Amarilli¹and İsmail İlkan Ceylan² September 16, 2020

1Télécom Paris

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud Benny Technion İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

Antoine

Benny

İsmail

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

2/7

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud Benny Technion İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

Antoine

Benny

İsmail

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud Benny Technion İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

Antoine

Benny

İsmail

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

2/7

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud Benny Technion İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

Antoine

Benny

İsmail

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud Benny Technion İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

Antoine

Benny

İsmail

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

2/7

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud Benny Technion İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

Antoine

Benny

İsmail

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data

Uncertain data management

In this talk, we managedatarepresented as alabeled graph

WorksAt

Antoine Télécom Paris Antoine Paris Sud

Benny Paris Sud Benny Technion İsmail U. Oxford

MemberOf

Télécom Paris ParisTech Télécom Paris IP Paris

Paris Sud IP Paris Paris Sud Paris-Saclay Technion CESAER

Antoine

Benny

İsmail

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

→ Problem:we are notcertainabout the true state of the data ^2/7

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

80% 10%

40% 80%

100%

90% 90% 50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

ProbabilityofW: Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

ProbabilityofW: Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

3/7

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

ProbabilityofW: Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

ProbabilityofW: Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

3/7

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER

80% 10%

40% 80%

100%

90% 90% 50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

ProbabilityofW: Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

ProbabilityofW:

Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

3/7

Uncertain data model A.

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

Uncertain data model:TID, for tuple-independent database

•

Each fact (edge) carries aprobability

•

Each fact exists with its givenprobability

•

All facts areindependent

•

Possible worldW:subset of facts

•

ProbabilityofW: Pr(W) = Y

F∈W

Pr(F)

!

× Y

F∈W/

1−Pr(F)

!

Homomorphism-closed queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

•

Union of conjunctive queries(UCQ): doesone of the CQsmatch?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

They generalizeCQsandUCQs, but alsoregular path queries(RPQs),Datalog, etc.

4/7

Homomorphism-closed queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

•

Union of conjunctive queries(UCQ): doesone of the CQsmatch?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

They generalizeCQsandUCQs, but alsoregular path queries(RPQs),Datalog, etc.

Homomorphism-closed queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

•

Union of conjunctive queries(UCQ): doesone of the CQsmatch?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

They generalizeCQsandUCQs, but alsoregular path queries(RPQs),Datalog, etc.

4/7

Homomorphism-closed queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

•

Union of conjunctive queries(UCQ): doesone of the CQsmatch?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

They generalizeCQsandUCQs, but alsoregular path queries(RPQs),Datalog, etc.

Homomorphism-closed queries

•

^Query:maps a graph (without probabilities) to YES/NO

•

Conjunctive query(CQ): can I find a match of apattern? e.g., x y z

•

Union of conjunctive queries(UCQ): doesone of the CQsmatch?

→ Homomorphism-closed queryQ: ifGsatisfiesQandGhas a homomorphism toG⁰ thenG⁰ also satisfiesQ

They generalizeCQsandUCQs, but alsoregular path queries(RPQs),Datalog, etc.

4/7

Problem statement: Probabilistic query evaluation (PQE) Here is the problemPQE(Q):

•

^{Wefixa queryQ: x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40% 80%

100%

90% 90% 50%

90%

100%

•

^{Theoutputis the}probabilitythat the query is true

→ Question: What is the complexity ofPQE(Q)depending on the queryQ?

Problem statement: Probabilistic query evaluation (PQE) Here is the problemPQE(Q):

•

^{Wefixa queryQ: x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

^{Theoutputis the}probabilitythat the query is true

→ Question: What is the complexity ofPQE(Q)depending on the queryQ?

5/7

Problem statement: Probabilistic query evaluation (PQE) Here is the problemPQE(Q):

•

^{Wefixa queryQ: x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

^{Theoutputis the}probabilitythat the query is true

→ Question: What is the complexity ofPQE(Q)depending on the queryQ?

Problem statement: Probabilistic query evaluation (PQE) Here is the problemPQE(Q):

•

^{Wefixa queryQ: x y z}

•

^{Theinputis a TIDD: A.}

B.

İ.

Télécom Paris

Paris Sud

Technion

U. Oxford

ParisTech

IP Paris

Paris-Saclay

CESAER 80%

10%

40%

80%

100%

90%

90%

50%

90%

100%

•

^{Theoutputis the}probabilitythat the query is true

→ Question: What is the complexity ofPQE(Q)depending on the queryQ?

5/7

Results on PQE

Existing dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

We study PQE forhomomorphism-closed queriesand show: Theorem [Amarilli and Ceylan, 2020]

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQandPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

Sobad news: all homomorphism-closed queries arehardexcept safe UCQs

Results on PQE

Existing dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

We study PQE forhomomorphism-closed queriesand show:

Theorem [Amarilli and Ceylan, 2020]

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQandPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

Sobad news: all homomorphism-closed queries arehardexcept safe UCQs

6/7

Results on PQE

Existing dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

We study PQE forhomomorphism-closed queriesand show:

Theorem [Amarilli and Ceylan, 2020]

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQandPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

Sobad news: all homomorphism-closed queries arehardexcept safe UCQs

Results on PQE

Existing dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

We study PQE forhomomorphism-closed queriesand show:

Theorem [Amarilli and Ceylan, 2020]

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQandPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

Sobad news: all homomorphism-closed queries arehardexcept safe UCQs

6/7

Results on PQE

Existing dichotomy on theunions of conjunctive queries(UCQs):

Theorem [Dalvi and Suciu, 2012]

•

^{Some UCQsQaresafeandPQE(Q)is inPTIME}

•

All others areunsafeandPQE(Q)is#P-hard

We study PQE forhomomorphism-closed queriesand show:

Theorem [Amarilli and Ceylan, 2020]

For anyquery Q closed under homomorphisms:

•

^EitherQis equivalent to asafe UCQandPQE(Q)is inPTIME

•

In all other cases,PQE(Q)is#P-hard

What’s next?

•

The result only applies tographs, not higher-arity databases

• Weconjecturethat it holds for arbitrary arity

•

Adapting tounweightedPQE, where all probabilities are1/2?

• We have a recent result onnon-hierarchical self-join-free CQs [Amarilli and Kimelfeld, 2020]

• Recent paper by Suciu and Kenig [Kenig and Suciu, 2020]

Thanks for your attention!

7/7

What’s next?

•

The result only applies tographs, not higher-arity databases

• Weconjecturethat it holds for arbitrary arity

•

Adapting tounweightedPQE, where all probabilities are1/2?

• We have a recent result onnon-hierarchical self-join-free CQs [Amarilli and Kimelfeld, 2020]

• Recent paper by Suciu and Kenig [Kenig and Suciu, 2020]

Thanks for your attention!

What’s next?

•

The result only applies tographs, not higher-arity databases

• Weconjecturethat it holds for arbitrary arity

•

Adapting tounweightedPQE, where all probabilities are1/2?

• We have a recent result onnon-hierarchical self-join-free CQs [Amarilli and Kimelfeld, 2020]

• Recent paper by Suciu and Kenig [Kenig and Suciu, 2020]

Thanks for your attention!

7/7

References i

Amarilli, A. and Ceylan, I. I. (2020).

A dichotomy for homomorphism-closed queries on probabilistic graphs.

InICDT.

Amarilli, A. and Kimelfeld, B. (2020).

Uniform reliability of self-join-free conjunctive queries.

arXiv preprint arXiv:1908.07093.

Dalvi, N. and Suciu, D. (2012).

The dichotomy of probabilistic inference for unions of conjunctive queries.

J. ACM, 59(6).

References ii

Kenig, B. and Suciu, D. (2020).

A dichotomy for the generalized model counting problem for unions of conjunctive queries.

arXiv preprint arXiv:2008.00896.