Hardness for unweighted PQE

Problem statement

•

We restrict back toCQs

•

^{We impose}self-join-freeness: every edge color is different

x y z w x y

z w

•

Existing dichotomy:

• IfQonly consists ofstars, then it is safe andPQE(Q)is inPTIME

• In all other cases,PQE(Q)is#P-hard

But what if all facts of the TIDs hadprobability 1/2?

→ Equivalently: given a graphG, how manysubgraphssatisfyQ

•

We call this problemMC(Q): model countingforQ Theorem

For anyself-join-free CQQ, ifQis unsafe thenMC(Q)is#P-hard.

Problem statement

•

We restrict back toCQs

•

^{We impose}self-join-freeness: every edge color is different

x y z w x y

z w

•

Existing dichotomy:

• IfQonly consists ofstars, then it is safe andPQE(Q)is inPTIME

• In all other cases,PQE(Q)is#P-hard

But what if all facts of the TIDs hadprobability 1/2?

→ Equivalently: given a graphG, how manysubgraphssatisfyQ

•

We call this problemMC(Q): model countingforQ Theorem

For anyself-join-free CQQ, ifQis unsafe thenMC(Q)is#P-hard.

Problem statement

•

We restrict back toCQs

•

^{We impose}self-join-freeness: every edge color is different

x y z w x y

z w

•

Existing dichotomy:

• IfQonly consists ofstars, then it is safe andPQE(Q)is inPTIME

• In all other cases,PQE(Q)is#P-hard

But what if all facts of the TIDs hadprobability 1/2?

→ Equivalently: given a graphG, how manysubgraphssatisfyQ

•

We call this problemMC(Q): model countingforQ Theorem

For anyself-join-free CQQ, ifQis unsafe thenMC(Q)is#P-hard.

Problem statement

•

We restrict back toCQs

•

^{We impose}self-join-freeness: every edge color is different

x y z w x y

z w

•

Existing dichotomy:

• IfQonly consists ofstars, then it is safe andPQE(Q)is inPTIME

• In all other cases,PQE(Q)is#P-hard

But what if all facts of the TIDs hadprobability 1/2?

→ Equivalently: given a graphG, how manysubgraphssatisfyQ

•

We call this problemMC(Q): model countingforQ

Theorem

For anyself-join-free CQQ, ifQis unsafe thenMC(Q)is#P-hard.

Problem statement

•

We restrict back toCQs

•

^{We impose}self-join-freeness: every edge color is different

x y z w x y

z w

•

Existing dichotomy:

• IfQonly consists ofstars, then it is safe andPQE(Q)is inPTIME

• In all other cases,PQE(Q)is#P-hard

But what if all facts of the TIDs hadprobability 1/2?

→ Equivalently: given a graphG, how manysubgraphssatisfyQ

•

We call this problemMC(Q): model countingforQ Theorem

For anyself-join-free CQQ, ifQis unsafe thenMC(Q)is#P-hard.

First step: Restricting to a simpler query

For any unsafe query, we can reduce fromsimpler queries, essentially:

x y z w

→We must show thatMC(Q)is #P-hard forthis query

Can we use ourearlier reductionfor #P-hardness of PQE?

a⁰₁

a⁰₂

a⁰₃

a₁

a₂

a3

1/2 1/2 1/2

b₁

b₂

b⁰₁

b⁰₂ 1/2

1/2 1

1 1

1

1

→Problem: this reduction crucially usesprobability 1

First step: Restricting to a simpler query

For any unsafe query, we can reduce fromsimpler queries, essentially:

x y z w

→We must show thatMC(Q)is #P-hard forthis query Can we use ourearlier reductionfor #P-hardness of PQE?

a⁰₁

a⁰₂

a⁰₃

a₁

a₂

a3

1/2 1/2 1/2

b₁

b₂

b⁰₁

b⁰₂ 1/2

1/2 1

1 1

1

1

→Problem: this reduction crucially usesprobability 1

First step: Restricting to a simpler query

For any unsafe query, we can reduce fromsimpler queries, essentially:

x y z w

→We must show thatMC(Q)is #P-hard forthis query Can we use ourearlier reductionfor #P-hardness of PQE?

a⁰₁

a⁰₂

a⁰₃

a₁

a₂

a3

1/2 1/2 1/2

b₁

b₂

b⁰₁

b⁰₂ 1/2

1/2 1

1 1

1

1

Getting to an equation system We want to reduce fromPQE(Q), on some graphGwith probabilities

a⁰₁

Task: count the numberXofred-blue edge subsetsthatviolateQ

•

^{Split thesubsetson someparameter}e.g., the number of nodes

→ X=X₁, . . . ,X_k

•

Create unweighted copies ofGmodified with somegadgets e.g., replace each edge by multiple copies of a path

→ CreatedG₁, . . . ,G_k

Getting to an equation system We want to reduce fromPQE(Q), on some graphGwith probabilities

a⁰₁

Task: count the numberXofred-blue edge subsetsthatviolateQ

•

^{Split thesubsetson someparameter}e.g., the number of nodes

→ X=X₁, . . . ,X_k

•

Create unweighted copies ofGmodified with somegadgets e.g., replace each edge by multiple copies of a path

→ CreatedG₁, . . . ,G_k

Getting to an equation system We want to reduce fromPQE(Q), on some graphGwith probabilities

a⁰₁

Task: count the numberXofred-blue edge subsetsthatviolateQ

•

^{Split thesubsetson someparameter}e.g., the number of nodes

→ X=X₁, . . . ,X_k

•

Create unweighted copies ofGmodified with somegadgets e.g., replace each edge by multiple copies of a path

→ CreatedG₁, . . . ,G_k

Getting to an equation system We want to reduce fromPQE(Q), on some graphGwith probabilities

a⁰₁

Task: count the numberXofred-blue edge subsetsthatviolateQ

•

^{Split thesubsetson someparameter}e.g., the number of nodes

→ X=X₁, . . . ,X_k

•

Create unweighted copies ofGmodified with somegadgets e.g., replace each edge by multiple copies of a path

→ CreatedG₁, . . . ,G_k

Using the equation system

We have obtained the system:



→ If the matrix isinvertible, then we have succeeded

We can choose gadgets and parameters to get aVandermonde matrix, and show invertibility via severalarithmetical tricks

Using the equation system

We have obtained the system:



→ If the matrix isinvertible, then we have succeeded

We can choose gadgets and parameters to get aVandermonde matrix, and show invertibility via severalarithmetical tricks

Using the equation system

We have obtained the system:



→ If the matrix isinvertible, then we have succeeded

We can choose gadgets and parameters to get aVandermonde matrix, and show invertibility via severalarithmetical tricks

Using the equation system

We have obtained the system:



→ If the matrix isinvertible, then we have succeeded

We can choose gadgets and parameters to get aVandermonde matrix, and show invertibility via severalarithmetical tricks

Using the equation system

We have obtained the system:



→ If the matrix isinvertible, then we have succeeded

We can choose gadgets and parameters to get aVandermonde matrix, and show invertibility via severalarithmetical tricks

Dans le document Query Evaluation on Probabilistic Data: New Hard Cases (Page 76-94)