FAQ: Questions Asked Frequently
Mahmoud Abo Khamis
Hung Q. Ngo
.
Atri Rudra
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Table of Contents
The FAQ problem
Summary of Results and Approach
The OutsideIn algorithm
The InsideOut algorithm
Choosing a variable ordering
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Table of Contents
The FAQ problem
Summary of Results and Approach
The OutsideIn algorithm
The InsideOut algorithm
Choosing a variable ordering
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Notations
▶
Xi denote variables, i ∈ [n] = {1, . . . , n},
. . . .... .... .... . . . . .
Notations
▶
Xi denote variables, i ∈ [n] = {1, . . . , n},
▶
xi are values in discrete domain Dom(Xi )
. . . .... .... .... . . . . .
Notations
▶
Xi denote variables, i ∈ [n] = {1, . . . , n},
▶
xi are values in discrete domain Dom(Xi )
▶
x = (x1 , . . . , xn ) ∈ Dom(X1 ) × · · · × Dom(Xn )
. . . .... .... .... . . . . .
Notations
▶
Xi denote variables, i ∈ [n] = {1, . . . , n},
▶
xi are values in discrete domain Dom(Xi )
▶
x = (x1 , . . . , xn ) ∈ Dom(X1 ) × · · · × Dom(Xn )
▶
For any S ⊆ [n],
xS = (xi )i∈S ∈
∏
Dom(Xi )
i∈S
e.g. x{2,5,8} = (x2 , x5 , x8 ) ∈ Dom(X2 ) × Dom(X5 ) × Dom(X8 )
. . . .... .... .... . . . . .
Function Aggregate Query: the Problem
ψ14 (X1 , X4 )
FAQ-expression
ψ13 (X1 , X3 )
ψ23 (X2 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
ψ124 (X1 , X2 , X4 )
All functions have the same range D
. . . .... .... .... . . . . .
Function Aggregate Query: the Input
ψ14 (X1 , X4 )
FAQ-expression
ψ13 (X1 , X3 )
ψ23 (X2 , X3 )
φ(x3 ) =
∑∏
x1
x2
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
ψ124 (X1 , X2 , X4 )
▶
All functions have the same range D
φ(X3 )
n=4
n variables X1 , . . . , Xn
. . . .... .... .... . . . . .
Function Aggregate Query: the Input
ψ14 (X1 , X4 )
ψ23 (X2 , X3 )
φ(x3 ) =
∑∏
x1
ψ124 (X1 , X2 , X4 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
V = {1, 2, 3, 4}
E = {{1, 4}, {1, 3}, {2, 3}, {1, 2, 4}}
▶
▶
n variables X1 , . . . , Xn
a multi-hypergraph H = (V, E)
. . . .... .... .... . . . . .
Function Aggregate Query: the Input
ψ14 (X1 , X4 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
ψ23 (X2 , X3 )
φ(x3 ) =
∑∏
x1
ψ124 (X1 , X2 , X4 )
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
V = {1, 2, 3, 4}
E = {{1, 4}, {1, 3}, {2, 3}, {1, 2, 4}}
▶
▶
n variables X1 , . . . , Xn
a multi-hypergraph H = (V, E)
▶
Each vertex is a variable (notation overload: V = [n])
. . . .... .... .... . . . . .
Function Aggregate Query: the Input
ψ14 (X1 , X4 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
ψ23 (X2 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
V = {1, 2, 3, 4}
ψ124 (X1 , X2 , X4 )
E = {{1, 4}, {1, 3}, {2, 3}, {1, 2, 4}}
▶
▶
n variables X1 , . . . , Xn
a multi-hypergraph H = (V, E)
▶
▶
Each vertex is a variable (notation overload: V = [n])
To each hyperedge S ∈ E there corresponds a factor ψS
∏
ψS :
Dom(Xi ) → D
i∈S
. . . .... .... .... . . . . .
Function Aggregate Query: the Input
ψ14 (X1 , X4 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
ψ23 (X2 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
V = {1, 2, 3, 4}
ψ124 (X1 , X2 , X4 )
E = {{1, 4}, {1, 3}, {2, 3}, {1, 2, 4}}
▶
▶
n variables X1 , . . . , Xn
a multi-hypergraph H = (V, E)
▶
▶
Each vertex is a variable (notation overload: V = [n])
To each hyperedge S ∈ E there corresponds a factor ψS
∏
ψS :
Dom(Xi ) → D
i∈S
R+ , {true, false}, {0, 1}, 2U , etc.
. . . .... .... .... . . . . .
Function Aggregate Query: the Input
ψ14 (X1 , X4 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
ψ23 (X2 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
F = {3}
ψ124 (X1 , X2 , X4 )
▶
▶
n variables X1 , . . . , Xn
a multi-hypergraph H = (V, E)
▶
▶
Each vertex is a variable (notation overload: V = [n])
To each hyperedge S ∈ E there corresponds a factor ψS
∏
ψS :
Dom(Xi ) → D
i∈S
R+ , {true, false}, {0, 1}, 2U , etc.
▶
a set F ⊆ V of free variables (wlog, F = [f ] = {1, . . . , f })
. . . .... .... .... . . . . .
Function Aggregate Query: the Output
ψ14 (X1 , X4 )
ψ23 (X2 , X3 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
ψ124 (X1 , X2 , X4 )
▶
Compute the function φ :
∏
Dom(Xi ) → D.
i∈F
. . . .... .... .... . . . . .
Function Aggregate Query: the Output
ψ14 (X1 , X4 )
ψ23 (X2 , X3 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
ψ124 (X1 , X2 , X4 )
▶
Compute the function φ :
∏
Dom(Xi ) → D.
i∈F
▶
φ defined by the FAQ-expression
⊕(f +1)
⊕(n−1)
φ(x[f ] ) =
···
xf +1 ∈Dom(Xf +1 )
⊕(n)
⊗
xn−1 ∈Dom(Xn−1 ) xn ∈Dom(Xn ) S∈E
ψS (xS )
. . . .... .... .... . . . . .
Function Aggregate Query: the Output
ψ14 (X1 , X4 )
ψ23 (X2 , X3 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
ψ124 (X1 , X2 , X4 )
▶
Compute the function φ :
∏
Dom(Xi ) → D.
i∈F
▶
φ defined by the FAQ-expression
⊕(f +1)
⊕(n−1)
φ(x[f ] ) =
···
xf +1 ∈Dom(Xf +1 )
▶
For each
⊕(n)
⊗
xn−1 ∈Dom(Xn−1 ) xn ∈Dom(Xn ) S∈E
ψS (xS )
⊕(i)
. . . .... .... .... . . . . .
Function Aggregate Query: the Output
ψ14 (X1 , X4 )
ψ23 (X2 , X3 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
ψ124 (X1 , X2 , X4 )
▶
Compute the function φ :
∏
Dom(Xi ) → D.
i∈F
▶
φ defined by the FAQ-expression
⊕(f +1)
⊕(n−1)
φ(x[f ] ) =
···
xf +1 ∈Dom(Xf +1 )
▶
⊕
For each ((i)
⊕(i) ⊗)
▶
Either D,
,
⊕(n)
⊗
xn−1 ∈Dom(Xn−1 ) xn ∈Dom(Xn ) S∈E
ψS (xS )
is a commutative semiring
. . . .... .... .... . . . . .
Function Aggregate Query: the Output
ψ14 (X1 , X4 )
ψ23 (X2 , X3 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
ψ124 (X1 , X2 , X4 )
▶
Compute the function φ :
∏
Dom(Xi ) → D.
i∈F
▶
φ defined by the FAQ-expression
⊕(f +1)
⊕(n−1)
φ(x[f ] ) =
···
xf +1 ∈Dom(Xf +1 )
▶
⊕
For each ((i)
⊕(i) ⊗)
▶
▶
Either D,
⊕(i) ⊗
Or
=
,
⊕(n)
⊗
xn−1 ∈Dom(Xn−1 ) xn ∈Dom(Xn ) S∈E
ψS (xS )
is a commutative semiring
. . . .... .... .... . . . . .
SumProd = FAQ-SS without free variables
▶
SumProd: compute the constant
⊕⊕
⊕⊗
⊕⊗
φ=
···
ψS (xS ) =
ψS (xS )
x1
x2
xn S∈E
x S∈E
. . . .... .... .... . . . . .
SumProd = FAQ-SS without free variables
▶
SumProd: compute the constant
⊕⊕
⊕⊗
⊕⊗
φ=
···
ψS (xS ) =
ψS (xS )
x1
▶
x2
xn S∈E
x S∈E
Where (D, ⊕, ⊗) is a commutative semiring
Additive identity 0 ∈ D : 0 ⊕ e = e ⊕ 0 = e
Multiplicative identity 1 ∈ D : 1 ⊗ e = e ⊗ 1 = e
Annihilation by 0 0 ⊗ e = e ⊗ 0 = 0
Distributive law a ⊗ b ⊕ a ⊗ c = a ⊗ (b ⊕ c)
. . . .... .... .... . . . . .
SumProd = FAQ-SS without free variables
▶
SumProd: compute the constant
⊕⊕
⊕⊗
⊕⊗
φ=
···
ψS (xS ) =
ψS (xS )
x1
x2
xn S∈E
x S∈E
▶
Where (D, ⊕, ⊗) is a commutative semiring
Additive identity 0 ∈ D : 0 ⊕ e = e ⊕ 0 = e
Multiplicative identity 1 ∈ D : 1 ⊗ e = e ⊗ 1 = e
Annihilation by 0 0 ⊗ e = e ⊗ 0 = 0
Distributive law a ⊗ b ⊕ a ⊗ c = a ⊗ (b ⊕ c)
▶
Common examples (there are many more!)
Boolean ({true, false}, ∨, ∧)
sum-product (R, +, ×)
max-product (R+ , max, ×)
set (2U , ∪, ∩)
. . . .... .... .... . . . . .
Problem (CSP = Boolean Join Query)
is there a “truth assignment” x satisfying all constraints?
. . . .... .... .... . . . . .
Problem (CSP = Boolean Join Query)
is there a “truth assignment” x satisfying all constraints?
▶
Boolean semiring ({true, false}, ∨, ∧)
⊕⊗
x S∈E
ψS (xS ) =
∨ ∧
x
ψS (xS )
S∈E
. . . .... .... .... . . . . .
Problem (CSP = Boolean Join Query)
is there a “truth assignment” x satisfying all constraints?
▶
Boolean semiring ({true, false}, ∨, ∧)
⊕⊗
ψS (xS ) =
x S∈E
▶
constraint ψS :
∏
∨ ∧
x
ψS (xS )
S∈E
Dom(Xi ) → {true, false} with support S
i∈S
. . . .... .... .... . . . . .
Problem (CSP = Boolean Join Query)
is there a “truth assignment” x satisfying all constraints?
▶
Boolean semiring ({true, false}, ∨, ∧)
⊕⊗
ψS (xS ) =
x S∈E
▶
constraint ψS :
∏
∨ ∧
x
ψS (xS )
S∈E
Dom(Xi ) → {true, false} with support S
i∈S
▶
truth assignment satisfying all constraints
. . . .... .... .... . . . . .
Problem (CSP = Boolean Join Query)
is there a “truth assignment” x satisfying all constraints?
▶
Boolean semiring ({true, false}, ∨, ∧)
⊕⊗
ψS (xS ) =
x S∈E
▶
constraint ψS :
∏
∨ ∧
x
ψS (xS )
S∈E
Dom(Xi ) → {true, false} with support S
i∈S
▶
truth assignment satisfying all constraints
▶
over all possible truth assignments
. . . .... .... .... . . . . .
Problem (#CSP = Join cardinality query)
count number of satisfying assignments x
. . . .... .... .... . . . . .
Problem (#CSP = Join cardinality query)
count number of satisfying assignments x
▶
Sum-product semiring ({0, 1}, +, ×)
⊕⊗
x S∈E
ψS (xS ) =
∑ ∏
x
ψS (xS )
S∈E
. . . .... .... .... . . . . .
Problem (#CSP = Join cardinality query)
count number of satisfying assignments x
▶
Sum-product semiring ({0, 1}, +, ×)
⊕⊗
ψS (xS ) =
x S∈E
▶
constraint ψS :
∏
∑ ∏
x
ψS (xS )
S∈E
Dom(Xi ) → {0, 1}, where
i∈S
{
ψS (x) =
1
0
xS satisfies constraint
otherwise
. . . .... .... .... . . . . .
Problem (#CSP = Join cardinality query)
count number of satisfying assignments x
▶
Sum-product semiring ({0, 1}, +, ×)
⊕⊗
ψS (xS ) =
x S∈E
▶
constraint ψS :
∏
∑ ∏
x
ψS (xS )
S∈E
Dom(Xi ) → {0, 1}, where
i∈S
{
ψS (x) =
▶
1
0
xS satisfies constraint
otherwise
count 1 only if x satisfies all constraints
. . . .... .... .... . . . . .
Problem (#CSP = Join cardinality query)
count number of satisfying assignments x
▶
Sum-product semiring ({0, 1}, +, ×)
⊕⊗
ψS (xS ) =
x S∈E
▶
constraint ψS :
∏
∑ ∏
x
ψS (xS )
S∈E
Dom(Xi ) → {0, 1}, where
i∈S
{
ψS (x) =
1
0
xS satisfies constraint
otherwise
▶
count 1 only if x satisfies all constraints
▶
sum over all possible truth assignments
. . . .... .... .... . . . . .
Many examples for FAQ-SS w/o free variables
▶
Boolean conjunctive query evaluation (Boolean semiring)
▶
SAT (Boolean semiring)
▶
Quantifier-free conjunctive query evaluation (set semiring)
▶
k-colorability (Boolean)
▶
Permanent (Sum-Product semiring)
▶
Partition function (Sum-Product semiring)
▶
etc.
▶
Sum-Prod = Marginalize a Product Function problem
▶
▶
Dechter (Artificial Intelligence 1999)
Aji and McEliece (IEEE Trans. Inform. Theory 2000)
. . . .... .... .... . . . . .
Adding free variables
Problem (FAQ-SS with free variables)
Compute the function
⊕⊗
⊕ ⊗
⊕
φ(x[f ] ) =
···
ψS (xS ) =
ψS (xS )
xf +1
xn S∈E
x[n]−[f ] S∈E
. . . .... .... .... . . . . .
Adding free variables
Problem (FAQ-SS with free variables)
Compute the function
⊕⊗
⊕ ⊗
⊕
φ(x[f ] ) =
···
ψS (xS ) =
ψS (xS )
xf +1
▶
xn S∈E
x[n]−[f ] S∈E
In DB jargons: we have Group By!
. . . .... .... .... . . . . .
Problem (Conjunctive Query Evaluation – CQE)
∧
Φ(x[f ] ) = ∃xf +1 · · · ∃xn
R(vars(R))
R∈atoms(Φ)
▶
Boolean Semiring ({true, false}), ∨, ∧)
▶
FAQ-SS instance:
φ(x[f ] ) =
∨
xf +1
···
∨∧
φS (xS ).
xn S∈E
. . . .... .... .... . . . . .
Problem (Matrix Chain Multiplication – MCM)
Let Ai = (a(i)
x,y ), compute
A = A1 × A2 × · · · Ak .
|{z}
|{z} |{z}
|{z}
p0 ×pk
p0 ×p1
p1 ×p2
pk−1 ×pk
. . . .... .... .... . . . . .
Problem (Matrix Chain Multiplication – MCM)
Let Ai = (a(i)
x,y ), compute
A = A1 × A2 × · · · Ak .
|{z}
|{z} |{z}
|{z}
p0 ×pk
▶
p0 ×p1
p1 ×p2
pk−1 ×pk
Sum-product semiring (R, +, ×), (C, +, ×), (Z, +, ×)
. . . .... .... .... . . . . .
Problem (Matrix Chain Multiplication – MCM)
Let Ai = (a(i)
x,y ), compute
A = A1 × A2 × · · · Ak .
|{z}
|{z} |{z}
|{z}
p0 ×pk
▶
▶
p0 ×p1
p1 ×p2
pk−1 ×pk
Sum-product semiring (R, +, ×), (C, +, ×), (Z, +, ×)
FAQ-SS instance:
. . . .... .... .... . . . . .
Problem (Matrix Chain Multiplication – MCM)
Let Ai = (a(i)
x,y ), compute
A = A1 × A2 × · · · Ak .
|{z}
|{z} |{z}
|{z}
p0 ×pk
▶
▶
p0 ×p1
p1 ×p2
pk−1 ×pk
Sum-product semiring (R, +, ×), (C, +, ×), (Z, +, ×)
FAQ-SS instance:
▶
Variables: Dom(Xi ) = [pi ], i ∈ {0, . . . , k}
. . . .... .... .... . . . . .
Problem (Matrix Chain Multiplication – MCM)
Let Ai = (a(i)
x,y ), compute
A = A1 × A2 × · · · Ak .
|{z}
|{z} |{z}
|{z}
p0 ×pk
▶
▶
p0 ×p1
p1 ×p2
pk−1 ×pk
Sum-product semiring (R, +, ×), (C, +, ×), (Z, +, ×)
FAQ-SS instance:
▶
▶
Variables: Dom(Xi ) = [pi ], i ∈ {0, . . . , k}
Factors: ψi,i+1 (xi , xi+1 ) = a(i)
xi ,xi+1
. . . .... .... .... . . . . .
Problem (Matrix Chain Multiplication – MCM)
Let Ai = (a(i)
x,y ), compute
A = A1 × A2 × · · · Ak .
|{z}
|{z} |{z}
|{z}
p0 ×pk
▶
▶
p0 ×p1
p1 ×p2
pk−1 ×pk
Sum-product semiring (R, +, ×), (C, +, ×), (Z, +, ×)
FAQ-SS instance:
▶
▶
▶
Variables: Dom(Xi ) = [pi ], i ∈ {0, . . . , k}
Factors: ψi,i+1 (xi , xi+1 ) = a(i)
xi ,xi+1
Compute new function φ : [p0 ] × [pk ] → D
φ(x0 , xk ) =
∑
x1
···
∑ k−1
∏
ψi,i+1 (xi , xi+1 ).
xk−1 i=0
. . . .... .... .... . . . . .
Probabilistic Graphical Models
▶
In PGMs, factors are also called potential functions
. . . .... .... .... . . . . .
Probabilistic Graphical Models
▶
▶
In PGMs, factors are also called potential functions
Three typical inference/learning tasks on PGMs
▶
▶
▶
Compute some marginal distribution
Compute p(xA | xB )
Compute argmaxxA p(xA | xB ) (e.g. MAP queries)
. . . .... .... .... . . . . .
Probabilistic Graphical Models
▶
▶
In PGMs, factors are also called potential functions
Three typical inference/learning tasks on PGMs
▶
▶
▶
▶
Compute some marginal distribution
Compute p(xA | xB )
Compute argmaxxA p(xA | xB ) (e.g. MAP queries)
These problems are precisely FAQ-SS (i.e. SumProd) on
▶
▶
(R+ , +, ×)
(R+ , max, ×)
. . . .... .... .... . . . . .
Many more examples
▶
Discrete Fourier Transform
▶
Hollant Problem (as in Holographic algorithms)
▶
Graph Homomorphism Problem
▶
Weighted CSP
▶
List recoverable codes
▶
LDPC codes
With a squint: also called Aggregate over Factorized DB
▶
▶
▶
Bakibayev et al. (VLDB 2014), Olteanu-Zavodny (TODS
2015)
etc.
. . . .... .... .... . . . . .
Why the generality of FAQ again?
ψ14 (X1 , X4 )
ψ23 (X2 , X3 )
All functions have the same range D
FAQ-expression
ψ13 (X1 , X3 )
φ(x3 ) =
∑∏
x1
x2
φ(X3 )
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4
x4
ψ124 (X1 , X2 , X4 )
▶
Compute the function φ :
∏
Dom(Xi ) → D.
i∈F
▶
φ defined by the FAQ-expression
⊕(f +1)
⊕(n−1)
φ(x[f ] ) =
···
xf +1 ∈Dom(Xf +1 )
▶
⊕
For each ((i)
⊕(i) ⊗)
▶
▶
Either D,
⊕(i) ⊗
Or
=
,
⊕(n)
⊗
xn−1 ∈Dom(Xn−1 ) xn ∈Dom(Xn ) S∈E
ψS (xS )
is a commutative semiring
. . . .... .... .... . . . . .
Quantified Conjunctive Queries
Problem (QCQ with free variables)
Given Qi ∈ {∃, ∀}, for i > f .

Φ(X1 , . . . , Xf ) = Qf +1 Xf +1 · · · Qn Xn 
∧

R ,
R∈atoms(Φ)
. . . .... .... .... . . . . .
Quantified Conjunctive Queries
Problem (QCQ with free variables)
Given Qi ∈ {∃, ∀}, for i > f .


∧
Φ(X1 , . . . , Xf ) = Qf +1 Xf +1 · · · Qn Xn 
R ,
R∈atoms(Φ)
▶
FAQ instance:
▶
▶
({0, 1}, {max, ×}, ×)
Compute the function
φ(x1 , · · · , xf ) =
where
⊕(i)
⊕(f +1)
xf +1 ∈{0,1}
{
=
max
×
···
⊕(n) ∏
xn ∈{0,1} S∈E
ψS (xS )
if Qi = ∃
if Qi = ∀
. . . .... .... .... . . . . .
# Quantified Conjunctive Queries
Problem (#QCQ)
Given Qi ∈ {∃, ∀}, for i > f , and expression

∧
Φ(X1 , . . . , Xf ) = Qf +1 Xf +1 · · · Qn Xn 

R ,
R∈atoms(Φ)
Count the number of x[f ] for which Φ(x[f ] ) = true
. . . .... .... .... . . . . .
# Quantified Conjunctive Queries
Problem (#QCQ)
Given Qi ∈ {∃, ∀}, for i > f , and expression

∧
Φ(X1 , . . . , Xf ) = Qf +1 Xf +1 · · · Qn Xn 

R ,
R∈atoms(Φ)
Count the number of x[f ] for which Φ(x[f ] ) = true
▶
FAQ instance:
▶
▶
({0, 1} ∪ R+ , {max, ×, +}, ×)
Compute the constant
∑ ∑ ⊕(f +1)
⊕(n) ∏
φ=
···
···
ψS (xS )
x1
xf xf +1 ∈{0,1}
{
where
⊕
(i)
=
max
×
xn ∈{0,1} S∈E
if Qi = ∃
if Qi = ∀
. . . .... .... .... . . . . .
Table of Contents
The FAQ problem
Summary of Results and Approach
The OutsideIn algorithm
The InsideOut algorithm
Choosing a variable ordering
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
The Algorithm
InsideOut
FAQ-expression σ
OutsideIn1
to compute φ
OutsideIn2
φ
..
.
OutsideInn
. . . .... .... .... . . . . .
The Algorithm
InsideOut
FAQ-expression σ
OutsideIn1
to compute φ
OutsideIn2
φ
..
.
OutsideInn
▶
InsideOut represents dynamic programming
. . . .... .... .... . . . . .
The Algorithm
InsideOut
FAQ-expression σ
OutsideIn1
to compute φ
OutsideIn2
φ
..
.
OutsideInn
▶
InsideOut represents dynamic programming
▶
is variable elimination with new twists
▶
▶
Variable Elimination (Zhang-Poole, JAIR 1996)
Bucket Elimination (Dechter, Artif. Intell. 1999)
. . . .... .... .... . . . . .
The Algorithm
InsideOut
FAQ-expression σ
OutsideIn1
to compute φ
OutsideIn2
φ
..
.
OutsideInn
▶
InsideOut represents dynamic programming
▶
is variable elimination with new twists
▶
▶
▶
Variable Elimination (Zhang-Poole, JAIR 1996)
Bucket Elimination (Dechter, Artif. Intell. 1999)
OutsideIn represents backtracking search
. . . .... .... .... . . . . .
The Algorithm
InsideOut
FAQ-expression σ
OutsideIn1
to compute φ
OutsideIn2
φ
..
.
OutsideInn
▶
InsideOut represents dynamic programming
▶
is variable elimination with new twists
▶
▶
▶
Variable Elimination (Zhang-Poole, JAIR 1996)
Bucket Elimination (Dechter, Artif. Intell. 1999)
OutsideIn represents backtracking search
▶
any worst-case optimal join algorithm, such as
▶
▶
NPRR (Ngo et al., PODS 2012)
Leapfrog-Triejoin (Veldhuizen, ICDT’2014)
. . . .... .... .... . . . . .
The Algorithm
InsideOut
FAQ-expression σ
OutsideIn1
to compute φ
OutsideIn2
Runtime = Õ(N faqw(σ) + |φ|)
φ
..
.
OutsideInn
▶
InsideOut represents dynamic programming
▶
is variable elimination with new twists
▶
▶
▶
Variable Elimination (Zhang-Poole, JAIR 1996)
Bucket Elimination (Dechter, Artif. Intell. 1999)
OutsideIn represents backtracking search
▶
any worst-case optimal join algorithm, such as
▶
▶
NPRR (Ngo et al., PODS 2012)
Leapfrog-Triejoin (Veldhuizen, ICDT’2014)
. . . .... .... .... . . . . .
The Algorithm
InsideOut
FAQ-expression σ
OutsideIn1
to compute φ
OutsideIn2
..
.
OutsideInn
▶
faqw(σ) is the “width” of σ
φ
FAQ-analog of
fractional hypertree width
InsideOut represents dynamic programming
▶
is variable elimination with new twists
▶
▶
▶
Runtime = Õ(N faqw(σ) + |φ|)
Variable Elimination (Zhang-Poole, JAIR 1996)
Bucket Elimination (Dechter, Artif. Intell. 1999)
OutsideIn represents backtracking search
▶
any worst-case optimal join algorithm, such as
▶
▶
NPRR (Ngo et al., PODS 2012)
Leapfrog-Triejoin (Veldhuizen, ICDT’2014)
. . . .... .... .... . . . . .
Technical Contributions
FAQ-expr. σ
for φ, hypergraph H
. . . .... .... .... . . . . .
Technical Contributions
EVO(φ) = set of expressions
“semantically equivalent” to φ
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
. . . .... .... .... . . . . .
Technical Contributions
EVO(φ) = set of expressions
“semantically equivalent” to φ
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
φ(x3 ) =
φ(x3 ) =
∑∑∑
x2 x1
∑
∑
x1
x2
max ψ1,2,4 ψ2,3 ψ1,3 ψ1,4 ψ2,5
x5
max
x4
x4
∑
ψ1,2,4 ψ2,3 ψ1,3 ψ1,4 ψ2,5
x5
. . . .... .... .... . . . . .
Technical Contributions
σ ∗ = arg min faqw(τ )
τ ∈EVO(φ)
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ ∗
for φ
. . . .... .... .... . . . . .
Technical Contributions
σ ∗ = arg min faqw(τ )
τ ∈EVO(φ)
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ
for φ
∗
∗
Runtime = Õ(N faqw(σ ) + |φ|)
= Õ(N opt + |φ|)
InsideOut
φ
. . . .... .... .... . . . . .
Technical Contributions
∗
Runtime = Õ(N faqw(σ ) + |φ|)
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ
for φ
∗
= Õ(N opt + |φ|)
InsideOut
φ
poly(|H|)
∑
x1 ,x4
∏
max
x3
x2 ,x7
∑
x5
max
x6
Expression Tree
Precedence Poset P
. . . .... .... .... . . . . .
Technical Contributions
∗
Runtime = Õ(N faqw(σ ) + |φ|)
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ
for φ
∗
= Õ(N opt + |φ|)
InsideOut
φ
EVO(φ) = CWE(LinEx(P ))
poly(|H|)
LinEx(P ) ⊆ EVO(φ)
∑
x1 ,x4
∏
max
x3
x2 ,x7
∑
x5
max
x6
Expression Tree
Precedence Poset P
. . . .... .... .... . . . . .
Technical Contributions
∗
Runtime = Õ(N faqw(σ ) + |φ|)
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ
for φ
∗
= Õ(N opt + |φ|)
InsideOut
φ
EVO(φ) = CWE(LinEx(P ))
poly(|H|)
LinEx(P ) ⊆ EVO(φ)
min faqw(τ ) = min
τ ∈LinEx(P )
τ ∈EVO(φ)
faqw(τ )
∑
x1 ,x4
∏
max
x3
x2 ,x7
∑
x5
max
x6
Expression Tree
Precedence Poset P
. . . .... .... .... . . . . .
Technical Contributions
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ ∗
for φ
InsideOut
φ
EVO(φ) = CWE(LinEx(P ))
poly(|H|)
LinEx(P ) ⊆ EVO(φ)
min faqw(τ ) = min
τ ∈LinEx(P )
τ ∈EVO(φ)
faqw(τ )
∑
x1 ,x4
poly(|H|)
∏
max
x3
x2 ,x7
∑
x5
Expression Tree
Precedence Poset P
max
x6
Tree Decomposition of H
. . . .... .... .... . . . . .
Technical Contributions
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ ∗
for φ
InsideOut
φ
EVO(φ) = CWE(LinEx(P ))
poly(|H|)
LinEx(P ) ⊆ EVO(φ)
min faqw(τ ) = min
τ ∈LinEx(P )
τ ∈EVO(φ)
faqw(τ )
∑
x1 ,x4
poly(|H|)
∏
max
x3
x2 ,x7
∑
x5
Expression Tree
Precedence Poset P
poly(|H|)
max
x6
Tree Decomposition of H
FAQ-expr. σ̄
for φ
faqw(σ̄) ≤ opt + g(opt)
g = approx. factor
for fractional
hypertree width of H
. . . .... .... .... . . . . .
Technical Contributions
Runtime = Õ(N opt+g(opt) + |φ|)
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ ∗
for φ
InsideOut
φ
EVO(φ) = CWE(LinEx(P ))
poly(|H|)
LinEx(P ) ⊆ EVO(φ)
min faqw(τ ) = min
τ ∈LinEx(P )
τ ∈EVO(φ)
faqw(τ )
∑
x1 ,x4
poly(|H|)
∏
max
x3
x2 ,x7
∑
x5
Expression Tree
Precedence Poset P
poly(|H|)
max
x6
Tree Decomposition of H
FAQ-expr. σ̄
for φ
faqw(σ̄) ≤ opt + g(opt)
g = approx. factor
for fractional
hypertree width of H
. . . .... .... .... . . . . .
Some Corollaries
Problem
Previous Algo.
#QCQ
No non-trivial algo
QCQ
Õ(N
PW(H)
+ |φ|)
InsideOut
Õ(N
Õ(N
faqw(φ)
+ |φ|)
faqw(φ)
+ |φ|)
faqw(φ)
+ |φ|)
Chen-Dalmau (LICS 2012)
DM(H)
+ |φ|)
#CQ
Õ(N
Joins
Durand-Mengel (ICDT 2013)
(
)
fhtw(H)
Õ N
+ |φ|
Marginal Distrib.
Õ(N
MAP query
Õ(N
Õ(N
(
)
fhtw(H)
Õ N
+ |φ|
Grohe-Marx (SODA’06)
htw(φ)
+ |φ|)
Õ(N
htw(φ)
+ |φ|)
Õ(N
faqw(φ)
+ |φ|)
faqw(φ)
+ |φ|)
Kask et al. (Artif. Intel. 2005)
Matrix Chain Mult.
DP bound
DFT
O(N logp N )
DP Bound
O(N logp N )
Aji-McEliece (IEEE Trans. IT 2000)
Dechter (Artif. Intell. 1999)
Textbook
. . . .... .... .... . . . . .
Table of Contents
The FAQ problem
Summary of Results and Approach
The OutsideIn algorithm
The InsideOut algorithm
Choosing a variable ordering
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
How are factors accessed?
▶
A factor oracle for each input factor ψS
. . . .... .... .... . . . . .
How are factors accessed?
▶
A factor oracle for each input factor ψS
▶
Oracle = (complexity people’s) fancy name for an API
. . . .... .... .... . . . . .
How are factors accessed?
▶
A factor oracle for each input factor ψS
▶
▶
Oracle = (complexity people’s) fancy name for an API
Output factor: a table of tuples
⟨xF , φ[xF ]⟩
but only store tuples for which φ[xF ] ̸= 0
. . . .... .... .... . . . . .
How are factors accessed?
▶
A factor oracle for each input factor ψS
▶
▶
Oracle = (complexity people’s) fancy name for an API
Output factor: a table of tuples
⟨xF , φ[xF ]⟩
but only store tuples for which φ[xF ] ̸= 0
▶
Called the listing representation
. . . .... .... .... . . . . .
How are factors accessed?
▶
A factor oracle for each input factor ψS
▶
▶
Oracle = (complexity people’s) fancy name for an API
Output factor: a table of tuples
⟨xF , φ[xF ]⟩
but only store tuples for which φ[xF ] ̸= 0
▶
Called the listing representation
Example (Salary : {ID, Name} → R+ )
. . . .... .... .... . . . . .
How are factors accessed?
▶
A factor oracle for each input factor ψS
▶
▶
Oracle = (complexity people’s) fancy name for an API
Output factor: a table of tuples
⟨xF , φ[xF ]⟩
but only store tuples for which φ[xF ] ̸= 0
▶
Called the listing representation
Example (Salary : {ID, Name} → R+ )
ID
1234
5678
9999
Name
AnHai Doan
Current Hung Ngo
Future Hung Ngo
Salary
$400,000
$100,000
$1,000,000
. . . .... .... .... . . . . .
Notation: conditional factor
▶
Let ψS be a factor
▶
Let K ⊆ [n], yK ∈
∏
Dom(Xi )
i∈K
▶
Conditional factor
ψS (· | yK ) :
∏
Dom(Xi ) → D
i∈S
where
ψS (xS | yK ) =
{
0
ψS (xS )
if S ∩ K ̸= ∅ and xS∩K ̸= yS∩K
otherwise.
. . . .... .... .... . . . . .
Notation: conditional factor
▶
Let ψS be a factor
▶
Let K ⊆ [n], yK ∈
∏
Dom(Xi )
i∈K
▶
Conditional factor
ψS (· | yK ) :
∏
Dom(Xi ) → D
i∈S
where
ψS (xS | yK ) =
▶
{
0
ψS (xS )
if S ∩ K ̸= ∅ and xS∩K ̸= yS∩K
otherwise.
Salary(ID=9999 | Name = “Future Hung Ngo”) = $1, 000, 000.
. . . .... .... .... . . . . .
Notation: conditional factor
▶
Let ψS be a factor
▶
Let K ⊆ [n], yK ∈
∏
Dom(Xi )
i∈K
▶
Conditional factor
ψS (· | yK ) :
∏
Dom(Xi ) → D
i∈S
where
ψS (xS | yK ) =
{
0
ψS (xS )
if S ∩ K ̸= ∅ and xS∩K ̸= yS∩K
otherwise.
▶
Salary(ID=9999 | Name = “Future Hung Ngo”) = $1, 000, 000.
▶
Salary(ID=9999 | Name = “Paris Koutris”) = $0.
. . . .... .... .... . . . . .
OutsideIn for FAQ-SS without free variables
φ =
⊕
x1
···
⊕⊗
ψS (xS )
xn S∈E
. . . .... .... .... . . . . .
OutsideIn for FAQ-SS without free variables
φ =
⊕
x1
=
···
⊕⊗
ψS (xS )
xn S∈E
⊕
x1 ∈ Dom(X1 )


⊕ ⊗

ψS (xS | x1 ) 
x[n]−{1} S∈E
. . . .... .... .... . . . . .
OutsideIn for FAQ-SS without free variables
φ =
⊕
x1
=
···
⊕⊗
Subproblem on H − {1}
ψS (xS )
xn S∈E
⊕
x1 ∈ Dom(X1 )


⊕ ⊗

ψS (xS | x1 ) 
x[n]−{1} S∈E
. . . .... .... .... . . . . .
OutsideIn for FAQ-SS without free variables
φ =
⊕
x1
=
Runtime O(Dn )
···
⊕⊗
Subproblem on H − {1}
ψS (xS )
xn S∈E
⊕
x1 ∈ Dom(X1 )


⊕ ⊗

ψS (xS | x1 ) 
x[n]−{1} S∈E
. . . .... .... .... . . . . .
OutsideIn for FAQ-SS without free variables
φ =
⊕
···
x1
xn S∈E
⊕
=
Runtime O(Dn )
=
⊕⊗
Subproblem on H − {1}
ψS (xS )

⊕ ⊗


ψS (xS | x1 ) 
x[n]−{1} S∈E
x1 ∈ Dom(X1 )


⊕
⊕ ⊗

ψS (xS | x1 ) .
x1 ∈ I
1
x[n]−{1} S∈E
. . . .... .... .... . . . . .
OutsideIn for FAQ-SS without free variables
φ =
⊕
···
x1
xn S∈E
⊕
=
Runtime O(Dn )
=
⊕⊗
Subproblem on H − {1}
ψS (xS )

⊕ ⊗


ψS (xS | x1 ) 
x[n]−{1} S∈E
x1 ∈ Dom(X1 )


⊕
⊕ ⊗

ψS (xS | x1 ) .
x1 ∈ I
1
x[n]−{1} S∈E
{
}
I1 = x1 | ψS (· | x1 ) ̸≡ 0, ∀S ∈ E
. . . .... .... .... . . . . .
OutsideIn = Backtracking Search
. . . .... .... .... . . . . .
OutsideIn = Backtracking Search
▶
Question: How good is OutsideIn?
. . . .... .... .... . . . . .
OutsideIn = Backtracking Search
▶
▶
Question: How good is OutsideIn?
Answer: Worst-case optimal for joins!
. . . .... .... .... . . . . .
OutsideIn = Backtracking Search
▶
▶
Question: How good is OutsideIn?
Answer: Worst-case optimal for joins!
▶
▶
Grohe, Marx (SODA’06)
Atserias, Grohe, Marx (FOCS’08)
AGM-bound
. . . .... .... .... . . . . .
OutsideIn = Backtracking Search
▶
▶
Question: How good is OutsideIn?
Answer: Worst-case optimal for joins!
▶
▶
▶
Grohe, Marx (SODA’06)
Atserias, Grohe, Marx (FOCS’08)
Ngo, Ré, Porat, Rudra (PODS’12)
AGM-bound
NPRR algorithm
. . . .... .... .... . . . . .
OutsideIn = Backtracking Search
▶
▶
Question: How good is OutsideIn?
Answer: Worst-case optimal for joins!
▶
▶
▶
▶
Grohe, Marx (SODA’06)
Atserias, Grohe, Marx (FOCS’08)
AGM-bound
Ngo, Ré, Porat, Rudra (PODS’12)
NPRR algorithm
Veldhuizen (ICDT’14)
Leapfrog-Triejoin algorithm
. . . .... .... .... . . . . .
OutsideIn = Backtracking Search
▶
▶
Question: How good is OutsideIn?
Answer: Worst-case optimal for joins!
▶
▶
▶
▶
▶
Grohe, Marx (SODA’06)
Atserias, Grohe, Marx (FOCS’08)
AGM-bound
Ngo, Ré, Porat, Rudra (PODS’12)
NPRR algorithm
Veldhuizen (ICDT’14)
Leapfrog-Triejoin algorithm
Ngo, Ré, Rudra (SIGMOD Records’13)
Generic-Join
. . . .... .... .... . . . . .
Background: fractional edge cover of H
X1
X2
X3
. . . .... .... .... . . . . .
Background: fractional edge cover of H
X1
λ=1
λ=1
X2
X3
λ=0
. . . .... .... .... . . . . .
Background: fractional edge cover of H
X1
λ=
1
2
λ=
X2
1
2
X3
λ=
1
2
. . . .... .... .... . . . . .
Background: fractional edge cover of B = {1, 2, 3}
X4
X1
X5
X8
X7
X6
X2
X3
. . . .... .... .... . . . . .
Background: fractional edge cover of B = {1, 2, 3}
X4
X1
X5
X8
λ=1
X7
λ=1
X6
X2
X3
λ=0
. . . .... .... .... . . . . .
Background: fractional edge cover of B = {1, 2, 3}
λ=1
X4
X1
X5
X8
X7
X6
X2
X3
λ=1
. . . .... .... .... . . . . .
Background: fractional edge cover of B = {1, 2, 3}
λ=
X4
X1
X5
λ=
X6
1
3
X8
X7
2
3
λ=
X2
2
3
X3
λ=
1
3
. . . .... .... .... . . . . .
Background: fractional edge cover
▶
H = (V, E) a multi hypergraph
▶
B⊆V
▶
a fractional edge cover of B is a vector (λS )S∈E , s.t.
∑
λS ≥ 1, ∀v ∈ B
S:v∈S
λS
▶
≥ 0, ∀S ∈ E.
called fractional edge cover of H when B = V
. . . .... .... .... . . . . .
Pros of OutsideIn
Theorem (Runtime of OutsideIn)
Let (λS )S∈E be any fractional edge cover of H, then OutsideIn
runs in time
(
)
∏
λS
|ψS |
Õ mn
S∈E
. . . .... .... .... . . . . .
Pros of OutsideIn
Theorem (Runtime of OutsideIn)
Let (λS )S∈E be any fractional edge cover of H, then OutsideIn
runs in time
(
)
∏
λS
|ψS |
Õ mn
AGM-bound
S∈E
. . . .... .... .... . . . . .
Pros of OutsideIn
Theorem (Runtime of OutsideIn)
Let (λS )S∈E be any fractional edge cover of H, then OutsideIn
runs in time
(
)
∏
λS
|ψS |
Õ mn
AGM-bound
S∈E
▶
OutsideIn is worst-case optimal for joins
▶
Very simple to implement
. . . .... .... .... . . . . .
Pros of OutsideIn
Theorem (Runtime of OutsideIn)
Let (λS )S∈E be any fractional edge cover of H, then OutsideIn
runs in time
(
)
∏
λS
|ψS |
Õ mn
AGM-bound
S∈E
▶
OutsideIn is worst-case optimal for joins
▶
Very simple to implement
▶
Works for FAQ-SS (i.e. Sum-Prod) with free variables!
. . . .... .... .... . . . . .
Pros of OutsideIn
Theorem (Runtime of OutsideIn)
Let (λS )S∈E be any fractional edge cover of H, then OutsideIn
runs in time
(
)
∏
λS
|ψS |
Õ mn
AGM-bound
S∈E
▶
OutsideIn is worst-case optimal for joins
▶
Very simple to implement
▶
Works for FAQ-SS (i.e. Sum-Prod) with free variables!
▶
Works well for dense queries (such as cliques)
. . . .... .... .... . . . . .
Pros of OutsideIn
Theorem (Runtime of OutsideIn)
Let (λS )S∈E be any fractional edge cover of H, then OutsideIn
runs in time
(
)
∏
λS
|ψS |
Õ mn
AGM-bound
S∈E
▶
OutsideIn is worst-case optimal for joins
▶
Very simple to implement
▶
Works for FAQ-SS (i.e. Sum-Prod) with free variables!
▶
Works well for dense queries (such as cliques)
▶
Works with any variable ordering
. . . .... .... .... . . . . .
Pros of OutsideIn
Theorem (Runtime of OutsideIn)
Let (λS )S∈E be any fractional edge cover of H, then OutsideIn
runs in time
(
)
∏
λS
|ψS |
Õ mn
AGM-bound
S∈E
▶
OutsideIn is worst-case optimal for joins
▶
Very simple to implement
▶
Works for FAQ-SS (i.e. Sum-Prod) with free variables!
▶
Works well for dense queries (such as cliques)
▶
Works with any variable ordering
▶
Virtually zero memory footprint (backtracking search!).
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
–
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
–
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
▶
N 3/2 is the AGM-bound AGM(H)
–
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
▶
N 3/2 is the AGM-bound AGM(H)
▶
3/2 is the fractional edge cover number ρ∗H
–
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
▶
▶
N 3/2 is the AGM-bound AGM(H)
▶
3/2 is the fractional edge cover number ρ∗H
Any join-project plan requires Ω(N 2 ) for some input
–
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
▶
N 3/2 is the AGM-bound AGM(H)
▶
3/2 is the fractional edge cover number ρ∗H
▶
Any join-project plan requires Ω(N 2 ) for some input
▶
Q = R(A, B, C) 1 S(A, B, D) 1 T (A, C, D) 1 W (B, C, D)
–
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
▶
N 3/2 is the AGM-bound AGM(H)
▶
3/2 is the fractional edge cover number ρ∗H
▶
Any join-project plan requires Ω(N 2 ) for some input
▶
Q = R(A, B, C) 1 S(A, B, D) 1 T (A, C, D) 1 W (B, C, D)
▶
OutsideIn runs in time Õ(N 4/3 ) for any input
–
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
▶
N 3/2 is the AGM-bound AGM(H)
▶
3/2 is the fractional edge cover number ρ∗H
▶
Any join-project plan requires Ω(N 2 ) for some input
▶
Q = R(A, B, C) 1 S(A, B, D) 1 T (A, C, D) 1 W (B, C, D)
▶
OutsideIn runs in time Õ(N 4/3 ) for any input
–
▶
N 4/3 is the AGM-bound AGM(H)
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
▶
N 3/2 is the AGM-bound AGM(H)
▶
3/2 is the fractional edge cover number ρ∗H
▶
Any join-project plan requires Ω(N 2 ) for some input
▶
Q = R(A, B, C) 1 S(A, B, D) 1 T (A, C, D) 1 W (B, C, D)
▶
OutsideIn runs in time Õ(N 4/3 ) for any input
–
▶
N 4/3 is the AGM-bound AGM(H)
▶
4/3 is the fractional edge cover number ρ∗H
. . . .... .... .... . . . . .
Examples of AGM-bound
▶
Q = R(A, B) 1 S(B, C) 1 T (A, C), |R| = |S| = |T | = N
▶
OutsideIn runs in time Õ(N 3/2 ) for any input
▶
N 3/2 is the AGM-bound AGM(H)
▶
3/2 is the fractional edge cover number ρ∗H
▶
Any join-project plan requires Ω(N 2 ) for some input
▶
Q = R(A, B, C) 1 S(A, B, D) 1 T (A, C, D) 1 W (B, C, D)
▶
OutsideIn runs in time Õ(N 4/3 ) for any input
–
▶
▶
N 4/3 is the AGM-bound AGM(H)
▶
4/3 is the fractional edge cover number ρ∗H
Any join-project plan requires Ω(N 2 ) for some input
. . . .... .... .... . . . . .
Cons of OutsideIn
▶
Horrible for large acyclic queries
. . . .... .... .... . . . . .
Cons of OutsideIn
▶
Horrible for large acyclic queries
▶
Yannakakis algorithm runs in time
Õ(N + |output|)
on any (α-)acyclic query
. . . .... .... .... . . . . .
Cons of OutsideIn
▶
Horrible for large acyclic queries
▶
Yannakakis algorithm runs in time
Õ(N + |output|)
on any (α-)acyclic query
▶
On a 2k-path query, AGM-bound is O(N k ).
X1
X2
X3
X4
X5
X6
X7
X8
X9
X10
. . . .... .... .... . . . . .
Cons of OutsideIn
▶
Horrible for large acyclic queries
▶
Yannakakis algorithm runs in time
Õ(N + |output|)
on any (α-)acyclic query
▶
On a 2k-path query, AGM-bound is O(N k ).
λ=1
X1
λ=1
X2
X3
λ=1
X4
X5
λ=1
X6
X7
λ=1
X8
X9
X10
. . . .... .... .... . . . . .
Cons of OutsideIn
▶
Horrible for large acyclic queries
▶
Yannakakis algorithm runs in time
Õ(N + |output|)
on any (α-)acyclic query
▶
On a 2k-path query, AGM-bound is O(N k ).
λ=1
X1
▶
λ=1
X2
X3
λ=1
X4
X5
λ=1
X6
X7
λ=1
X8
X9
X10
More generally, horrible whenever fhtw < AGM-bound.
. . . .... .... .... . . . . .
Table of Contents
The FAQ problem
Summary of Results and Approach
The OutsideIn algorithm
The InsideOut algorithm
Choosing a variable ordering
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-SS without free variables
φ =
⊕
x1
···
⊕⊗
ψS (xS )
xn S∈E
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
x1
=
⊕
x1
⊕⊗
ψS (xS )
xn S∈E
···
⊕
(
⊕⊗
xn−1
xn S∈E
)
ψS (xS )
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
x1
=
⊕
x1
⊕⊗
ψS (xS )
xn S∈E
···
⊕
(
⊕⊗
xn−1
xn S∈E
)
ψS (xS )
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
x1
=
⊕
=
x1
ψS (xS )
xn S∈E
···
x1
⊕
⊕⊗
···
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
xn−1 n∈S
/
)
ψS (xS )
ψS (xS )
(
⊕⊗
)
ψS (xS )
xn n∈S
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
x1
=
⊕
=
x1
ψS (xS )
xn S∈E
···
x1
⊕
⊕⊗
···
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
xn−1 n∈S
/
)
ψS (xS )
ψS (xS )
(
⊕⊗
)
ψS (xS )
xn n∈S
⊗ distributes over ⊕
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
x1
=
⊕
=
x1
ψS (xS )
xn S∈E
···
x1
⊕
⊕⊗
···
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
xn−1 n∈S
/
)
ψS (xS )
ψS (xS )
(
⊕⊗
)
ψS (xS )
xn n∈S
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
x1
=
⊕
=
x1
ψS (xS )
xn S∈E
···
x1
⊕
⊕⊗
···
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
xn−1 n∈S
/
)
ψS (xS )
ψS (xS )
(
⊕⊗
∪
Un =
S
) S:n∈S
ψS (xS )
xn n∈S
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
x1
=
⊕
=
x1
ψS (xS )
xn S∈E
···
x1
⊕
⊕⊗
···
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
xn−1 n∈S
/
)
ψS (xS )
ψS (xS )
(
⊕⊗
∪
Un =
S
) S:n∈S
ψS (xS )
xn n∈S
ψUn −{n} (xUn −{n} )
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
⊕⊗
x1
=
⊕
···
x1
=
⊕
···
x1
=
⊕
x1
ψS (xS )
xn S∈E
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
xn−1 n∈S
/
···
⊕⊗
)
ψS (xS )
ψS (xS )
(
⊕⊗
∪
Un =
S
) S:n∈S
ψS (xS )
xn n∈S
ψS (xS ) ⊗ ψUn −{n} (xUn −{n} )
xn−1 n∈S
/
ψUn −{n} (xUn −{n} )
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
⊕⊗
x1
=
⊕
···
x1
=
⊕
···
x1
=
⊕
x1
ψS (xS )
xn S∈E
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
xn−1 n∈S
/
···
⊕⊗
)
ψS (xS )
ψS (xS )
(
⊕⊗
∪
Un =
S
) S:n∈S
ψS (xS )
xn n∈S
ψS (xS ) ⊗ ψUn −{n} (xUn −{n} )
xn−1 n∈S
/
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
⊕⊗
x1
=
⊕
···
x1
=
⊕
···
x1
=
⊕
x1
ψS (xS )
xn S∈E
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
xn−1 n∈S
/
···
⊕⊗
)
ψS (xS )
ψS (xS )
(
⊕⊗
∪
Un =
S
) S:n∈S
ψS (xS )
xn n∈S
ψS (xS ) ⊗ ψUn −{n} (xUn −{n} )
xn−1 n∈S
/
. . . .... .... .... . . . . .
FAQ-SS without free variables
φ =
⊕
···
⊕⊗
x1
=
⊕
···
x1
=
⊕
···
x1
=
⊕
x1
ψS (xS )
xn S∈E
⊕
(
⊕⊗
xn−1
xn S∈E
⊕ ⊗
)
ψS (xS )
ψS (xS )
(
⊕⊗
xn−1 n∈S
/
···
⊕⊗
∪
Un =
S
) S:n∈S
ψS (xS )
xn n∈S
ψS (xS ) ⊗ ψUn −{n} (xUn −{n} )
xn−1 n∈S
/
New instance of FAQ-SS
Hn−1 = ([n − 1], En−1 )
En−1 = E + (Un − {n}) − {S | n ∈ S}
. . . .... .... .... . . . . .
Naïve InsideOut = Variable Elimination
φ =
⊕
x1
···
⊕⊗
xn−1 n∈S
/


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn S∈∂(n)
. . . .... .... .... . . . . .
Naïve InsideOut = Variable Elimination
φ =
⊕
x1
···
⊕⊗
xn−1 n∈S
/


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn S∈∂(n)
. . . .... .... .... . . . . .
Naïve InsideOut = Variable Elimination
φ =
⊕
x1
···
⊕⊗
xn−1 n∈S
/


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn S∈∂(n)
ψUn −{n} (xUn −{n} )
. . . .... .... .... . . . . .
Naïve InsideOut = Variable Elimination
φ =
⊕
x1
···
⊕⊗


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn−1 n∈S
/
But how do we compute this?
xn S∈∂(n)
ψUn −{n} (xUn −{n} )
. . . .... .... .... . . . . .
Naïve InsideOut = Variable Elimination
φ =
⊕
x1
···
⊕⊗


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn−1 n∈S
/
Run the OutsideIn algorithm!
xn S∈∂(n)
ψUn −{n} (xUn −{n} )
. . . .... .... .... . . . . .
Naïve InsideOut = Variable Elimination
φ =
⊕
x1
···
⊕⊗


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn−1 n∈S
/
xn S∈∂(n)
ψUn −{n} (xUn −{n} )
Run the OutsideIn algorithm!
∏
(n)
Runtime ≈ |Un | · |∂(n)| ·
|ψS |λS
S∈∂(n)
. . . .... .... .... . . . . .
Naïve InsideOut = Variable Elimination
φ =
⊕
x1
···
⊕⊗


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn−1 n∈S
/
xn S∈∂(n)
Theorem (Runtime of Naïve InsideOut)
For a given variable ordering v1 , . . . , vn , the runtime of Naïve
InsideOut makes is Õ of
n
∑
k=1
|Uk | · |∂(vk )| ·
∏
S∈∂(vk )
(k)
|ψS |λS ≤ mn
n
∑
∏
(k)
|ψS |λS
k=1 S∈∂(vk )
. . . .... .... .... . . . . .
Naïve InsideOut = Variable Elimination
φ =
⊕
x1
···
⊕⊗


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn−1 n∈S
/
xn S∈∂(n)
Theorem (Runtime of Naïve InsideOut)
For a given variable ordering v1 , . . . , vn , the runtime of Naïve
InsideOut makes is Õ of
n
∑
k=1
|Uk | · |∂(vk )| ·
∏
S∈∂(vk )
(k)
|ψS |λS ≤ mn
n
∑
∏
(k)
|ψS |λS
k=1 S∈∂(vk )
Corollary (Textbook result – Zhang-Poole’95, Dechter’96)
For a “good” variable ordering, PGM inference can be done in
time O(mn2 Dw+1 ) where w = tree-width(H) = tw(H).
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
1
2
3
4
5
6
7
8
9
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
1
2
3
4
5
6
7
8
9
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
1
2
3
4
5
6
7
8
9
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
λ{i,9} = 1, ∀i ∈ {5, 6, 7, 8}
1
2
3
4
5
6
7
8
9
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
λ{i,9} = 1, ∀i ∈ {5, 6, 7, 8}
1
▶
2
3
4
5
6
7
8
9
Say |ψS | = N , then runtime O(N 4 ).
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
1
▶
2
3
4
5
6
7
8
9
Say |ψS | = N , then runtime O(N 4 ).
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
1
▶
2
3
4
5
6
7
8
9
Say |ψS | = N , then runtime O(N 4 ).
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
Runtime O(N 2 )?
1
▶
2
3
4
5
6
7
8
9
Say |ψS | = N , then runtime O(N 4 ).
. . . .... .... .... . . . . .
Pros and Cons of Naïve InsideOut
▶
The good: O(kN )-time for k-path query!
▶
The bad: consider the following (part of a) graph
Runtime O(N 2 )?
1
▶
▶
2
3
4
5
6
7
8
9
Say |ψS | = N , then runtime O(N 4 ).
Good idea, but naïve application increases U9
. . . .... .... .... . . . . .
01-projection
▶
Projection onto U9 :
1
2
3
4
5
6
7
8
9
. . . .... .... .... . . . . .
01-projection
▶
Projection onto U9 :
1
2
3
4
5
6
7
8
9
. . . .... .... .... . . . . .
01-projection
▶
Projection onto U9 :
1
▶
2
3
4
5
6
7
8
9
Formally, S ∈ E, U ⊆ V s.t. S ∩ U ̸= ∅,
∏
ψS|U :
Dom(Xi ) → {0, 1},
i∈S∩U
where
{
1
ψS|U (xS∩U ) =
0
if ∃yS s.t. ψS (yS ) ̸= 0 and yS∩U = xS∩U
otherwise
. . . .... .... .... . . . . .
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn S∈∂(n)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/


⊕ ⊗
ψS (xS ) 
ψS (xS )
xn S∈∂(n)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/
ψS (xS )
⊕ ⊗
⊗
ψS (xS ) ⊗
ψS|Un (xS∩Un )
S ∈∂(n)
/
S∩Un ̸=∅
xn S∈∂(n)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/
ψS (xS )
⊕ ⊗
⊗
ψS (xS ) ⊗
ψS|Un (xS∩Un )
S ∈∂(n)
/
S∩Un ̸=∅
xn S∈∂(n)
∏
Time ≈ m|Un | ·
(k)
|ψS |λS = AGM(Un )
S∩Un ̸=∅
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/
ψS (xS )
⊕ ⊗
⊗
ψS (xS ) ⊗
ψS|Un (xS∩Un )
S ∈∂(n)
/
S∩Un ̸=∅
xn S∈∂(n)
Theorem (Runtime of InsideOut)
For FAQ-SS, InsideOut runs in time Õ of
n
∑
|{S | S ∩ Uk ̸= ∅}| · |Uk | · AGM(Uk ).
k=1
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/
ψS (xS )
⊕ ⊗
⊗
ψS (xS ) ⊗
ψS|Un (xS∩Un )
S ∈∂(n)
/
S∩Un ̸=∅
xn S∈∂(n)
Theorem (Runtime of InsideOut)
For FAQ-SS, InsideOut runs in time Õ of
n
∑
|{S | S ∩ Uk ̸= ∅}| · |Uk | · AGM(Uk ).
k=1
Corollary (New (textbook?) result)
For a “good” variable ordering v1 , . . . , vn , PGM inference can
be done in time O(mn2 N w ) where w = fhtw(H).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/
ψS (xS )
⊕ ⊗
⊗
ψS (xS ) ⊗
ψS|Un (xS∩Un )
S ∈∂(n)
/
S∩Un ̸=∅
xn S∈∂(n)
Theorem (Runtime of InsideOut)
For FAQ-SS, InsideOut runs in time Õ of
n
∑
|{S | S ∩ Uk ̸= ∅}| · |Uk | · AGM(Uk ).
k=1
Corollary (Grohe-Marx (SODA’06))
Join can be computed in time Õ(N fhtw(H) + |output|).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/
ψS (xS )
⊕ ⊗
⊗
ψS (xS ) ⊗
ψS|Un (xS∩Un )
S ∈∂(n)
/
S∩Un ̸=∅
xn S∈∂(n)
Theorem (Runtime of InsideOut)
For FAQ-SS, InsideOut runs in time Õ of
n
∑
|{S | S ∩ Uk ̸= ∅}| · |Uk | · AGM(Uk ).
k=1
Corollary (Grohe-Marx (SODA’06))
CSP on instances with bounded fhtw are fixed-parameter
tractable.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/
ψS (xS )
⊕ ⊗
⊗
ψS (xS ) ⊗
ψS|Un (xS∩Un )
S ∈∂(n)
/
S∩Un ̸=∅
xn S∈∂(n)
Theorem (Runtime of InsideOut)
For FAQ-SS, InsideOut runs in time Õ of
n
∑
|{S | S ∩ Uk ̸= ∅}| · |Uk | · AGM(Uk ).
k=1
Corollary (New?)
Join cardinality can be computed in time Õ(N fhtw(H) )
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut = VE + 01-projections
φ=
⊕
x1
···
⊕⊗
xn−1 n∈S
/
ψS (xS )
⊕ ⊗
⊗
ψS (xS ) ⊗
ψS|Un (xS∩Un )
S ∈∂(n)
/
S∩Un ̸=∅
xn S∈∂(n)
Theorem (Runtime of InsideOut)
For FAQ-SS, InsideOut runs in time Õ of
n
∑
|{S | S ∩ Uk ̸= ∅}| · |Uk | · AGM(Uk ).
k=1
Corollary (Pichler-Skritek 2011, Durand-Mengel (ICDT’2013))
For query graphs H with bounded fhtw, quantifier-free #CQ is
polynomial-time solvable.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
InsideOut for general FAQ
φ = ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
. . . .... .... .... . . . . .
InsideOut for general FAQ
φ = ···
= ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
⊕(n−1)
xn−1
(
⊕(n) ⊗
xn
)
ψS (xS )
S∈E
. . . .... .... .... . . . . .
InsideOut for general FAQ
φ = ···
= ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
⊕(n−1)
xn−1
(
⊕(n) ⊗
xn
)
ψS (xS )
S∈E
. . . .... .... .... . . . . .
InsideOut for general FAQ
φ = ···
= ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
⊕(n−1)
(
⊕(n) ⊗
xn−1
xn
)
ψS (xS )
S∈E
If (D, ⊕(n) , ⊗) is a semiring
. . . .... .... .... . . . . .
InsideOut for general FAQ
φ = ···
= ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
⊕(n−1)
(
⊕(n) ⊗
xn−1
xn
)
ψS (xS )
S∈E
If (D, ⊕(n) , ⊗) is a semiring
φ = ···
⊕(n−1)
xn−1
⊗
S ∈∂(n)
/

⊕
ψS (xS )  (n)
xn
⊗

ψS (xS ) 
S∈∂(n)
. . . .... .... .... . . . . .
InsideOut for general FAQ
φ = ···
= ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
⊕(n−1)
xn−1
(
⊕(n) ⊗
xn
)
ψS (xS )
S∈E
If ⊕(n) = ⊗
. . . .... .... .... . . . . .
InsideOut for general FAQ
φ = ···
= ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
⊕(n−1)
(
⊕(n) ⊗
xn
xn−1
)
ψS (xS )
S∈E
If ⊕(n) = ⊗
φ = ···
⊕(n−1)
xn−1
(
⊗ ⊕(n)
S∈E
xn
)
ψS (xS )
. . . .... .... .... . . . . .
InsideOut for general FAQ
φ = ···
= ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
⊕(n−1)
(
⊕(n) ⊗
xn
xn−1
)
ψS (xS )
S∈E
If ⊕(n) = ⊗
φ = ···
= ···
⊕(n−1)
xn−1
(
⊕(n−1)
⊗
S ∈∂(n)
/
(
xn−1
[ψS (xS )]
⊗ ⊕(n)
S∈E
xn
|Dom(Xn )|
)
ψS (xS )
) 
⊗
⊗ ⊕(n)
S∈∂(n)
xn

ψS (xS )
. . . .... .... .... . . . . .
InsideOut for general FAQ
φ = ···
= ···
⊕(n) ⊗
xn
ψS (xS )
S∈E
⊕(n−1)
(
⊕(n) ⊗
xn
xn−1
)
ψS (xS )
S∈E
If ⊕(n) = ⊗ and ⊕(n) is not-idempotent
φ = ···
= ···
⊕(n−1)
xn−1
(
⊕(n−1)
⊗
S ∈∂(n)
/
(
xn−1
[ψS (xS )]
⊗ ⊕(n)
S∈E
xn
|Dom(Xn )|
)
ψS (xS )
) 
⊗
⊗ ⊕(n)
S∈∂(n)
xn

ψS (xS )
. . . .... .... .... . . . . .
Table of Contents
The FAQ problem
Summary of Results and Approach
The OutsideIn algorithm
The InsideOut algorithm
Choosing a variable ordering
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a variable ordering
Time ≈ incarnations of OutsideIn + output size .
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a variable ordering
Time ≈ incarnations of OutsideIn + output size .
[
≤ Õ(mn)
∑
]
AGM(Uk ) + |φ|
k∈K
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a variable ordering
Time ≈ incarnations of OutsideIn + output size .
[
]
∑
≤ Õ(mn)
AGM(Uk ) + |φ|
k∈K
Dynamic Programming for
for Matrix Chain Multiplication
and The FFT
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a variable ordering
Time ≈ incarnations of OutsideIn + output size .
[
≤ Õ(mn)
[
≤ Õ(mn)
∑
]
AGM(Uk ) + |φ|
k∈K
∑
]
N
ρ∗H (Uk )
+ |φ|
k∈K
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a variable ordering
Time ≈ incarnations of OutsideIn + output size .
[
≤ Õ(mn)
[
≤ Õ(mn)
∑
]
AGM(Uk ) + |φ|
k∈K
∑
]
N
ρ∗H (Uk )
+ |φ|
k∈K
Definition (FAQ-width of a variable ordering σ)
faqw(σ) := max ρ∗H (Ukσ ).
k∈K
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a variable ordering
Time ≈ incarnations of OutsideIn + output size .
[
≤ Õ(mn)
[
≤ Õ(mn)
∑
]
AGM(Uk ) + |φ|
k∈K
∑
]
N
ρ∗H (Uk )
+ |φ|
k∈K
[
]
≤ Õ(mn) nN faqw(σ) + |φ| .
Definition (FAQ-width of a variable ordering σ)
faqw(σ) := max ρ∗H (Ukσ ).
k∈K
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a query
Definition (φ-equivalent variable ordering)
Let φ be an FAQ-query. A φ-equivalent variable ordering is a
vertex ordering σ = (v1 , . . . , vn ) that satisfies:
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a query
Definition (φ-equivalent variable ordering)
Let φ be an FAQ-query. A φ-equivalent variable ordering is a
vertex ordering σ = (v1 , . . . , vn ) that satisfies:
(a) The set {v1 , . . . , vf } is exactly F = [f ].
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a query
Definition (φ-equivalent variable ordering)
Let φ be an FAQ-query. A φ-equivalent variable ordering is a
vertex ordering σ = (v1 , . . . , vn ) that satisfies:
(a) The set {v1 , . . . , vf } is exactly F = [f ].
(b) The function φ′ defined by
⊕ f +1 )
⊕ n) ⊗
φ′ (xF ) := (v
· · · (v
xv
xvn
S∈E ψS (xS )
f +1
is identical to the function φ no matter what the input
factors are.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a query
Definition (φ-equivalent variable ordering)
Let φ be an FAQ-query. A φ-equivalent variable ordering is a
vertex ordering σ = (v1 , . . . , vn ) that satisfies:
(a) The set {v1 , . . . , vf } is exactly F = [f ].
(b) The function φ′ defined by
⊕ f +1 )
⊕ n) ⊗
φ′ (xF ) := (v
· · · (v
xv
xvn
S∈E ψS (xS )
f +1
is identical to the function φ no matter what the input
factors are.
EVO(φ) denotes the set of all φ-equivalent variable orderings.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ-width of a query
Definition (φ-equivalent variable ordering)
Let φ be an FAQ-query. A φ-equivalent variable ordering is a
vertex ordering σ = (v1 , . . . , vn ) that satisfies:
(a) The set {v1 , . . . , vf } is exactly F = [f ].
(b) The function φ′ defined by
⊕ f +1 )
⊕ n) ⊗
φ′ (xF ) := (v
· · · (v
xv
xvn
S∈E ψS (xS )
f +1
is identical to the function φ no matter what the input
factors are.
EVO(φ) denotes the set of all φ-equivalent variable orderings.
Definition (FAQ-width of an FAQ-query φ)
faqw(φ) :=
min
faqw(σ).
σ∈EVO(φ)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Roadmap Reminder
FAQ-expr. σ
for φ, hypergraph H
EVO(φ)
FAQ-expr. σ ∗
for φ
InsideOut
φ
EVO(φ) = CWE(LinEx(P ))
poly(|H|)
LinEx(P ) ⊆ EVO(φ)
min faqw(τ ) = min
τ ∈LinEx(P )
τ ∈EVO(φ)
faqw(τ )
∑
x1 ,x4
poly(|H|)
∏
max
x3
x2 ,x7
∑
x5
Expression Tree
Precedence Poset P
poly(|H|)
max
x6
Tree Decomposition of H
FAQ-expr. σ̄
for φ
faqw(σ̄) ≤ opt + g(opt)
g = approx. factor
for fractional
hypertree width of H
. . . .... .... .... . . . . .
Characterizing EVO(φ)
We define a partially ordered set P on the variables of φ, called
the Precedence Poset.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Characterizing EVO(φ)
We define a partially ordered set P on the variables of φ, called
the Precedence Poset.
Soundness
LinEx(P ) ⊆ EVO(φ).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Characterizing EVO(φ)
We define a partially ordered set P on the variables of φ, called
the Precedence Poset.
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness Every σ ∈ EVO(φ) has the same faqw as some
σ ′ ∈ LinEx(P ).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Characterizing EVO(φ)
We define a partially ordered set P on the variables of φ, called
the Precedence Poset.
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness Every σ ∈ EVO(φ) has the same faqw as some
σ ′ ∈ LinEx(P ).
faqw(φ) :=
faqw(σ).
min
σ ∈ EVO(φ)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Characterizing EVO(φ)
We define a partially ordered set P on the variables of φ, called
the Precedence Poset.
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness Every σ ∈ EVO(φ) has the same faqw as some
σ ′ ∈ LinEx(P ).
faqw(φ) :=
faqw(σ).
min
σ ∈ LinEx(P )
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with semiring variable aggregates
φ(x[f ] ) =
⊕(f +1)
xf +1
···
⊕(n) ⊗
xn
S∈E
.
ψS (xS )
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with semiring variable aggregates
φ(x[f ] ) =
▶
⊕(f +1)
xf +1
···
⊕(n) ⊗
xn
S∈E
ψS (xS )
Expression Tree defines the precedence poset
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with semiring variable aggregates
φ(x[f ] ) =
▶
⊕(f +1)
xf +1
···
⊕(n) ⊗
xn
S∈E
ψS (xS )
Expression Tree defines the precedence poset
▶
▶
Compartmentalization Step.
Compression Step.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
φ=
∑∑
x1
x2
max
x3
∑∑
x4
x5
)
max max ψ12 ψ135 ψ14 ψ246 ψ27 ψ37
x6
5
x7
1
4
2
6
3
7
. . . .... .... .... . . . . .
φ=
∑∑
x1
x2
max
x3
∑∑
x4
x5
)
max max ψ12 ψ135 ψ14 ψ246 ψ27 ψ37
x6
5
x7
1
4
2
6
3
7
. . . .... .... .... . . . . .
φ=
∑∑
x1
x2
max
x3
∑∑
x4
x5
)
max max ψ12 ψ135 ψ14 ψ246 ψ27 ψ37
x6
5
x7
1
4
2
6
3
7
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
ψ12
x2
max
x3
∑
)(
max ψ135 ψ27 ψ37
x7
x5
5
∑
x4
)
max ψ14 ψ246
x6
1
4
2
6
3
7
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
ψ12
x2
max
x3
∑
)(
max ψ135 ψ27 ψ37
x7
x5
5
∑
x4
)
max ψ14 ψ246
x6
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3, 5, 7
4, 6
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
ψ12
x2
max
x3
∑
)(
max ψ135 ψ27 ψ37
x7
x5
5
∑
x4
)
max ψ14 ψ246
x6
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3, 5, 7
4, 6
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
ψ12
x2
max
x3
∑
)(
max ψ135 ψ27 ψ37
x7
x5
5
∑
x4
)
max ψ14 ψ246
x6
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3, 5, 7
4, 6
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
ψ12
x2
max
x3
∑
)(
max ψ135 ψ27 ψ37
x7
x5
5
∑
x4
)
max ψ14 ψ246
x6
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3, 5, 7
4, 6
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
max
ψ12
x3
x2
∑
)(
ψ135 max ψ27 ψ37
x7
x5
∑
x4
5
)
max ψ14 ψ246
x6
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3
5
∑
4, 6
max
7
max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
max
ψ12
x3
x2
∑
)(
ψ135 max ψ27 ψ37
x7
x5
∑
x4
5
)
max ψ14 ψ246
x6
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3
5
∑
4, 6
max
7
max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
max
ψ12
x3
x2
∑
)(
ψ135 max ψ27 ψ37
x7
x5
∑
x4
5
)
max ψ14 ψ246
x6
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3
5
∑
4, 6
max
7
max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
max
ψ12
x3
x2
∑
)(
ψ135 max ψ27 ψ37
x7
x5
∑
x4
5
)
max ψ14 ψ246
x6
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3
5
∑
4, 6
max
7
max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
max
ψ12
x3
x2
∑
)(
ψ135 max ψ27 ψ37
x7
x5
∑
)
ψ14 max ψ246
x6
x4
5
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization...]
3
5
∑
4
max
7
max
∑
6 max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
max
ψ12
x3
x2
∑
)(
ψ135 max ψ27 ψ37
x7
x5
∑
)
ψ14 max ψ246
x6
x4
5
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compartmentalization!]
3
5
∑
4
max
7
max
∑
6 max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
max
ψ12
x3
x2
∑
)(
ψ135 max ψ27 ψ37
x7
x5
∑
)
ψ14 max ψ246
x6
x4
5
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compression...]
3
5
∑
4
max
7
max
∑
6 max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
max max ψ27 ψ37
ψ12
x3
x2
x7
∑
)(
ψ135
x5
5
∑
)
ψ14 max ψ246
x6
x4
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compression...]
3
5
∑
4
max
7
max
∑
6 max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
ψ12
x2
max max ψ27 ψ37
x3
x7
∑
)(
ψ135
x5
5
∑
)
ψ14 max ψ246
x6
x4
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compression...]
3, 7
5
max
∑
4
∑
6 max
. . . .... .... .... . . . . .
φ=
∑∑
x1
(
ψ12
x2
max max ψ27 ψ37
x3
x7
∑
)(
ψ135
x5
5
∑
)
ψ14 max ψ246
x6
x4
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compression...]
3, 7
5
max
∑
4
∑
6 max
. . . .... .... .... . . . . .
φ=
∑∑∑
x1
x2
(
max max ψ27 ψ37
ψ12 ψ14
x3
x4
∑
x7
5
)(
)
max ψ246
ψ135
x6
x5
1
4
2
6
3
7
Expression Tree
1, 2
∑
[Compression...]
3, 7
5
max
∑
4
∑
6 max
. . . .... .... .... . . . . .
φ=
∑∑∑
x1
x2
(
max max ψ27 ψ37
ψ12 ψ14
x3
x4
x7
5
∑
)(
)
max ψ246
ψ135
x6
x5
1
4
2
6
3
7
Expression Tree
1, 2, 4
∑
[Compression...]
3, 7
5
max
6
max
∑
. . . .... .... .... . . . . .
φ=
∑∑∑
x1
x2
(
max max ψ27 ψ37
ψ12 ψ14
x3
x4
x7
5
∑
)(
)
max ψ246
ψ135
x6
x5
1
4
2
6
3
7
Expression Tree
1, 2, 4
∑
[Compression!]
3, 7
5
max
6
max
∑
. . . .... .... .... . . . . .
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
faqw(φ) + g(faqw(φ)).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
faqw(φ) + g(faqw(φ)).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
OPT + g(OPT).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
OPT + g(OPT).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
OPT + g(OPT).
Best known g(x)=O( x3 ) [Marx’12]
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
O(fhtw(H)3 ),
we have an approximation for faqw(φ) within abound of
OPT + O(OPT3 ).
Best known g(x)=O( x3 ) [Marx’12]
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Corollaries
Strictly subsumes a result by Durand-Mengel ([ICDT’13])
Corollary
(
(
))
O faqw3 (φ)
#CQ is solvable in time O poly(m, n) · N
.
Strictly subsumes a result by Chen-Dalmau (LICS’12).
Corollary (QCQ tractability)
QCQ is tractable when faqw is bounded.
Answered an open question posed by Durand-Mengel (ICDT’13)
Corollary (#QCQ tractability)
#QCQ is tractable when faqw is bounded.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Many Thanks!
Any FAQ?
. . . .... .... .... . . . . .
FAQ with two blocks of semiring aggregates
φ=
⊕ ⊕
¯
xL
x[n]−L
⊗
S∈E ψS (xS ).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with two blocks of semiring aggregates
φ=
⊕ ⊕
¯
xL
x[n]−L
⊗
S∈E ψS (xS ).
Example: #CQ
φ=
∑
x1
···
∑
xl
max · · · max
xl+1
xn
∏
ψS (xS ).
S∈E
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with two blocks of semiring aggregates
φ=
⊕ ⊕
¯
xL
x[n]−L
⊗
S∈E ψS (xS ).
Example: #CQ
φ=
∑
x1
···
∑
xl
max · · · max
xl+1
xn
∏
ψS (xS ).
S∈E
Precedence Poset:
u ≺ v whenever u ∈ L and v ∈
/ L.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
faqw(φ) + g(faqw(φ)).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
faqw(φ) + g(faqw(φ)).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
OPT + g(OPT).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
OPT + g(OPT).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
g(fhtw(H)),
we have an approximation for faqw(φ) within abound of
OPT + g(OPT).
Best known g(x)=O( x3 ) [Marx’12]
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Theorem
Given an approximation algorithm for fhtw(H) within a bound
O(fhtw(H)3 ),
we have an approximation for faqw(φ) within abound of
OPT + O(OPT3 ).
Best known g(x)=O( x3 ) [Marx’12]
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
C1
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
C1
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
C1
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
C1
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
C1
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
U (C1 )
C2
C1
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
OPT
C2
g(OPT)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
C2
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
U (C2 )
C2
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
OPT
g(OPT)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
L
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
g(OPT)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Corollary
FAQ with two block of semiring aggregates is solvable in time
(
(
))
O faqw3 (φ)
O poly(m, n) · N
.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Corollary
#CQ is solvable in time
(
(
))
O faqw3 (φ)
O poly(m, n) · N
.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Approximating faqw(φ)
Corollary
#CQ is solvable in time
(
(
))
O faqw3 (φ)
O poly(m, n) · N
.
Strictly subsumes Durand-Mengel
Theorem (Durand-Mengel [ICDT’13])
#CQ is solvable in time
(
)
O poly(|H|) · N poly(L-ss(H),fhtw(H)) .
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
¯ are commutative iff
Two aggregates ⊕, ⊕
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
¯ are commutative iff
Two aggregates ⊕, ⊕
a
c
⊕
¯
⊕
⊕
b
d
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
¯ are commutative iff
Two aggregates ⊕, ⊕
a
c
⊕
¯
⊕
⊕
b
=
d
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
¯ are commutative iff
Two aggregates ⊕, ⊕
a
c
⊕
¯
⊕
⊕
b
a
b
¯ ⊕ ⊕
¯
= ⊕
d
c
d
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
¯ are commutative iff
Two aggregates ⊕, ⊕
a
0
⊕
¯
⊕
⊕
0
a
0
¯ ⊕ ⊕
¯
= ⊕
d
0
d
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
¯ are commutative iff
Two aggregates ⊕, ⊕
a
a
¯
⊕
=
d
⊕
d
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
¯ are commutative iff
Two aggregates ⊕, ⊕
a
a
¯
⊕
=
d
⊕
d
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Completeness Background
¯ are commutative iff
Two aggregates ⊕, ⊕
a
a
¯
⊕
=
d
⊕
d
¯ are identical!
Hence, ⊕ and ⊕
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) = EVO(φ) ?
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) = EVO(φ) ?
∑
φ =
x2
x1
.
∑
max ψ12 ψ13
.
.
.
.
.
. . . .
x3
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) = EVO(φ) ?
∑
φ =
x2
x1
.
∑
max ψ12 ψ13
.
.
.
.
.
. . . .
x3
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) = EVO(φ) ?
1
∑
∑
φ =
x2
x1
2
max
3
∑
∑
max ψ12 ψ13
x3
∑∑
=
ψ13 maxψ12
x2
x1 x3
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) = EVO(φ) ?
1, 3
∑
∑
φ =
x2
x1
2
max
∑
max ψ12 ψ13
x3
∑∑
=
ψ13 maxψ12
x2
x1 x3
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) = EVO(φ) ?
1, 3
∑
∑
φ =
x2
x1
2
max
∑
max ψ12 ψ13
x3
∑∑
=
ψ13 maxψ12
x2
x1 x3
(1, 2, 3) ∈
/ LinEx(P ) !
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) ̸= EVO(φ) .
1, 3
∑
∑
φ =
x2
x1
2
max
∑
max ψ12 ψ13
x3
∑∑
=
ψ13 maxψ12
x2
x1 x3
(1, 2, 3) ∈
/ LinEx(P ) !
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) ̸= EVO(φ) .
1, 3
∑
∑
φ =
x2
x1
2
∑
max ψ12 ψ13
x3
∑∑
=
ψ13 maxψ12
max
x2
x1 x3
(1, 2, 3) ∈
/ LinEx(P ) !
faqw(φ) :=
faqw(σ).
min
σ ∈ EVO(φ)
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
Soundness and Completeness
Soundness
LinEx(P ) ⊆ EVO(φ).
Completeness
LinEx(P ) ̸= EVO(φ) .
1, 3
∑
∑
φ =
∑
max ψ12 ψ13
x2
x1
2
x3
∑∑
=
ψ13 maxψ12
max
x2
x1 x3
(1, 2, 3) ∈
/ LinEx(P ) !
Theorem faqw(φ) :=
faqw(σ).
min
σ ∈ LinEx(P )
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
Let DI ⊇ {0, 1} be a set of idempotent elements of ⊗.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
Let DI ⊇ {0, 1} be a set of idempotent elements of ⊗.
φ(x[f ] ) =
⊕(f +1)
|
xf +1
···
{z
⊕(f +ℓ) ⊕(f +ℓ+1)
xf +ℓ
Semiring aggregates
}|
···
{z
xf +ℓ+1
⊕(n) ⊗
xn
S∈E ψS (xS )
} | {z } | {z }
Products or semirings
Closed
closed under DI
under DI
.
.
.
.
.
.
. . . .
. . . .
∈DI
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
Let DI ⊇ {0, 1} be a set of idempotent elements of ⊗.
φ(x[f ] ) =
⊕(f +1)
|
xf +1
···
{z
⊕(f +ℓ) ⊕(f +ℓ+1)
xf +ℓ
Semiring aggregates
}|
···
{z
xf +ℓ+1
⊕(n) ⊗
xn
S∈E ψS (xS )
} | {z } | {z }
Products or semirings
Closed
closed under DI
under DI
.
.
.
.
.
.
. . . .
. . . .
∈DI
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
Let DI ⊇ {0, 1} be a set of idempotent elements of ⊗.
φ(x[f ] ) =
⊕(f +1)
|
xf +1
···
{z
⊕(f +ℓ) ⊕(f +ℓ+1)
xf +ℓ
Semiring aggregates
}|
···
{z
xf +ℓ+1
⊕(n) ⊗
xn
S∈E ψS (xS )
} | {z } | {z }
Products or semirings
Closed
closed under DI
under DI
.
.
.
.
.
.
. . . .
. . . .
∈DI
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
Let DI ⊇ {0, 1} be a set of idempotent elements of ⊗.
φ(x[f ] ) =
⊕(f +1)
|
xf +1
···
{z
⊕(f +ℓ) ⊕(f +ℓ+1)
xf +ℓ
Semiring aggregates
}|
···
{z
xf +ℓ+1
⊕(n) ⊗
xn
S∈E ψS (xS )
} | {z } | {z }
Products or semirings
Closed
closed under DI
under DI
.
.
.
.
.
.
. . . .
. . . .
∈DI
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
Let DI ⊇ {0, 1} be a set of idempotent elements of ⊗.
φ(x[f ] ) =
⊕(f +1)
|
xf +1
···
{z
⊕(f +ℓ) ⊕(f +ℓ+1)
xf +ℓ
Semiring aggregates
}|
···
{z
xf +ℓ+1
⊕(n) ⊗
xn
S∈E ψS (xS )
} | {z } | {z }
Products or semirings
Closed
closed under DI
under DI
.
.
.
.
.
.
. . . .
. . . .
∈DI
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
Let DI ⊇ {0, 1} be a set of idempotent elements of ⊗.
φ(x[f ] ) =
⊕(f +1)
|
xf +1
···
{z
⊕(f +ℓ) ⊕(f +ℓ+1)
xf +ℓ
}|
Semiring aggregates
···
{z
xf +ℓ+1
⊕(n) ⊗
xn
S∈E ψS (xS )
} | {z } | {z }
Products or semirings
Closed
closed under DI
under DI
∈DI
Examples: QCQ,
φ(x[f ] ) =
⊕(f +1)
xf +1
···
⊕(n) ∏
xn
S∈E ψS (xS ).
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
FAQ with idempotent product aggregates
Let DI ⊇ {0, 1} be a set of idempotent elements of ⊗.
φ(x[f ] ) =
⊕(f +1)
|
xf +1
···
{z
⊕(f +ℓ) ⊕(f +ℓ+1)
xf +ℓ
}|
Semiring aggregates
···
{z
xf +ℓ+1
⊕(n) ⊗
xn
S∈E ψS (xS )
} | {z } | {z }
Products or semirings
Closed
closed under DI
under DI
∈DI
Examples: QCQ, #QCQ
φ(x[f ] ) =
∑
∑
φ=
···
x1
⊕(f +1)
xf +1
···
xl+1
···
⊕(l+1)
⊕(n) ∏
xn
S∈E ψS (xS ).
xn
S∈E ψS (xS ).
⊕(n) ∏
xl
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
φ=
∑
x1
max
x2
∏
max max ψ123 ψ125 ψ14 ψ34
x3
2
x4
x5
∏
)
x′3
1
4
5
3
. . . .... .... .... . . . . .
φ=
∑
x1
max
x2
∏
max max ψ123 ψ125 ψ14 ψ34
x3
2
x4
x5
∏
)
x′3
1
4
5
3
. . . .... .... .... . . . . .
φ=
∑
x1
max
x2
∏
max max ψ123 ψ125 ψ14 ψ34
x3
2
x4
x5
∏
)
x′3
1
4
5
3
. . . .... .... .... . . . . .
φ=
∑
x1
max
x2
∏
max ψ123 ψ125 max ψ14 ψ34
x3
2
x5
x4
∏
)
x′3
1
4
5
3
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
x3
)(
max ψ123 ψ125
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
x3
)(
max ψ123 ψ125
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
x3
)(
max ψ123 ψ125
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
Expression Tree
3
1∑
[Compartmentalization...]
2, 3, 5
3, 4
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
x3
)(
max ψ123 ψ125
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
Expression Tree
3
1∑
[Compartmentalization...]
2, 3, 5
3, 4
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
x3
)(
max ψ123 ψ125
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
Expression Tree
3
1∑
[Compartmentalization...]
2, 3, 5
3, 4
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
x3
)(
max ψ123 ψ125
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
Expression Tree
3
1∑
[Compartmentalization...]
2, 3, 5
3, 4
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
)(
ψ123 max ψ125
x3
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compartmentalization...]
3, 4
2 max
5
max
3
∏
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
)(
ψ123 max ψ125
x3
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compartmentalization...]
3, 4
2 max
5
max
3
∏
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
)(
ψ123 max ψ125
x3
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compartmentalization...]
3, 4
2 max
5
max
3
∏
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
)(
ψ123 max ψ125
x3
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compartmentalization...]
3∏
2 max
5
max
3
∏
4 max
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
)(
ψ123 max ψ125
x3
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compartmentalization!]
3∏
2 max
5
max
3
∏
4 max
. . . .... .... .... . . . . .
φ=
∑
x1
(
max
x2
∏
)(
ψ123 max ψ125
x3
x5
∏
x′3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compression...]
3∏
2 max
5
max
3
∏
4 max
. . . .... .... .... . . . . .
φ=
∑
x1
(
max max ψ125
x2
x5
∏
)(
ψ123
∏
x′3
x3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compression...]
3∏
2 max
5
max
3
∏
4 max
. . . .... .... .... . . . . .
φ=
∑
x1
(
max max ψ125
x2
x5
∏
)(
ψ123
∏
x′3
x3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compression...]
2, 5 max
3
∏
3∏
4 max
. . . .... .... .... . . . . .
φ=
∑
x1
(
max max ψ125
x2
x5
∏
)(
ψ123
∏
x′3
x3
)
max ψ14 ψ34
x4
1
2
4
5
3
3
1∑
Expression Tree
[Compression!]
2, 5 max
3
∏
3∏
4 max
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V8 = V
1
2
3
4
1
E8 = E
1
5
6
1
8
1
1
1
1
1
7
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V8 = V
1
2
3
4
1
E8 = E
1
5
6
1
8
1
1
1
1
1
7
∂(8)
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V8 = V
1
2
3
4
1
E8 = E
1
5
6
1
8
1
1
1
1
1
U8
7
1
∂(8)
1
1
1
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V7
1
2
3
1
4
5
6
7
1
1
1
E7
1
U8 − {8}
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V7
1
2
3
1
4
5
6
7
1
1
1
E7
1
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V7
1
2
3
1
4
5
6
7
1
1
1
E7
∂(7)
1
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V7
1
2
3
1
4
5
6
7
1
1
1
E7
∂(7)
1
U7
1
1
1
1
1
1
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V6
1
2
3
1
4
5
6
1
E6
U7 − {7}
1
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V6
1
2
3
1
4
5
6
1
E6
1
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V6
1
2
3
1
4
5
6
1
E6
1
1
1
1
1
1
1
1
∂(6)
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V6
1
2
3
1
4
5
6
1
E6
U6
1
1
1
1
1
1
1
1
1
1
1
1
1
∂(6)
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
V5
1
2
3
1
4
5
1
E5
U6 − {6}
1
1
1
1
1
1
1
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
▶
Given H = (V, E), and a variable ordering v1 , . . . , vn
▶
Hn = H
▶
For k = n − 1, . . . , 1, define Hk = (Vk , Ek )
∂(vk+1 ) = {S | S ∈ Ek+1 and vk+1 ∈ S}
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
▶
Given H = (V, E), and a variable ordering v1 , . . . , vn
▶
Hn = H
▶
For k = n − 1, . . . , 1, define Hk = (Vk , Ek )
∂(vk+1 ) = {S | S ∈ Ek+1 and vk+1 ∈ S}
∪
Uk+1 =
S
S∈∂(vk+1 )
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
▶
Given H = (V, E), and a variable ordering v1 , . . . , vn
▶
Hn = H
▶
For k = n − 1, . . . , 1, define Hk = (Vk , Ek )
∂(vk+1 ) = {S | S ∈ Ek+1 and vk+1 ∈ S}
∪
Uk+1 =
S
S∈∂(vk+1 )
Vk = Vk+1 − {vk+1 }
. . . .... .... .... . . . . .
Variable ordering and hypergraph sequence
▶
Given H = (V, E), and a variable ordering v1 , . . . , vn
▶
Hn = H
▶
For k = n − 1, . . . , 1, define Hk = (Vk , Ek )
∂(vk+1 ) = {S | S ∈ Ek+1 and vk+1 ∈ S}
∪
Uk+1 =
S
S∈∂(vk+1 )
Vk = Vk+1 − {vk+1 }
Ek = (Ek+1 − ∂(vk+1 )) ∪ {Uk+1 − {vk+1 }}
. . . .... .... .... . . . . .
Tree decomposition of H = (V, E)
{a,b,d}, {c, d}, {b, c, d), {b, e}, {c, e}, {b, e, f},
{b, e, g}, {g, f, h}, {i, j, h}, {e, g}, {e, g, b}, {e, b, h}.
b c
e
bags
b e
f
g h
b c
a d
i
h
j
e h
g b
. . . .... .... .... . . . . .
Every hyperedge is covered by some bag
{a,b,d}, {c, d}, {b, c, d), {b, e}, {c, e}, {b, e, f},
{b, e, g}, {g, f, h}, {i, j, h}, {e, g}, {e, g, b}, {e, b, h}.
b c
e
b e
f
g h
b c
a d
i
h
j
e h
g b
. . . .... .... .... . . . . .
Every hyperedge is covered by some bag
{a,b,d}, {c, d}, {b, c, d), {b, e}, {c, e}, {b, e, f},
{b, e, g}, {g, f, h}, {i, j, h}, {e, g}, {e, g, b}, {e, b, h}.
b c
e
b e
f
g h
b c
a d
i
h
j
e h
g b
. . . .... .... .... . . . . .
Every hyperedge is covered by some bag
{a,b,d}, {c, d}, {b, c, d), {b, e}, {c, e}, {b, e, f},
{b, e, g}, {g, f, h}, {i, j, h}, {e, g}, {e, g, b}, {e, b, h}.
b c
e
b e
f
g h
b c
a d
i
h
j
e h
g b
. . . .... .... .... . . . . .
Every hyperedge is covered by some bag
{a,b,d}, {c, d}, {b, c, d), {b, e}, {c, e}, {b, e, f},
{b, e, g}, {g, f, h}, {i, j, h}, {e, g}, {e, g, b}, {e, b, h}.
b c
e
b e
f
g h
b c
a d
i
h
j
e h
g b
. . . .... .... .... . . . . .
Running intersection property (RIP)
{a,b,d}, {c, d}, {b, c, d), {b, e}, {c, e}, {b, e, f},
{b, e, g}, {g, f, h}, {i, j, h}, {e, g}, {e, g, b}, {e, b, h}.
b c
e
b e
f
g h
b c
a d
i
h
j
e h
g b
. . . .... .... .... . . . . .
Running intersection property (RIP)
{a,b,d}, {c, d}, {b, c, d), {b, e}, {c, e}, {b, e, f},
{b, e, g}, {g, f, h}, {i, j, h}, {e, g}, {e, g, b}, {e, b, h}.
b c
e
b e
f
g h
b c
a d
i
h
j
e h
g b
. . . .... .... .... . . . . .
Variable Ordering and Tree decomposition
▶
Given H = (V, E), and a variable ordering v1 , . . . , vn
▶
Hn = H
▶
For k = n − 1, . . . , 1, define Hk = (Vk , Ek )
∂(vk+1 ) = {S | S ∈ Ek+1 and vk+1 ∈ S}
∪
Uk+1 =
S
S∈∂(vk+1 )
Vk = Vk+1 − {vk+1 }
Ek = (Ek+1 − ∂(vk+1 )) ∪ {Uk+1 − {vk+1 }}
Proposition (Folklorish)
There is a tree decomposition whose bags are the sets Uk and
vice versa.
. . . .... .... .... . . . . .