A Faster Algorithm for Propositional Model Counting Parameterized - - PowerPoint PPT Presentation

A Faster Algorithm for Propositional Model Counting Parameterized by Incidence Treewidth

Friedrich Slivovsky and Stefan Szeider

1

Propositional Model Counting (#SAT)

2

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰 𝖮𝖰

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰 𝖮𝖰 𝖣𝗉𝖮𝖰

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Propositional Model Counting (#SAT)

2

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

Π𝖰

3

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

Σ𝖰

3

Π𝖰

3

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

Σ𝖰

3

Π𝖰

3

Δ𝖰

3

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

Σ𝖰

3

Π𝖰

3

Δ𝖰

3

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

…

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

Σ𝖰

3

Π𝖰

3

Δ𝖰

3

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

…

𝖰𝖨 = ⋃

i

Σ𝖰

i

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

Σ𝖰

3

Π𝖰

3

Δ𝖰

3

Toda’s Theorem Every problem in the polynomial hierarchy can be solved in polynomial time using a #SAT oracle.

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

…

𝖰𝖨 = ⋃

i

Σ𝖰

i

Propositional Model Counting (#SAT)

3

SAT

Does have a satisfying assignment?

φ

#SAT

Count the satisfying assignments of .

φ

Σ𝖰

3

Π𝖰

3

Δ𝖰

3

Toda’s Theorem Every problem in the polynomial hierarchy can be solved in polynomial time using a #SAT oracle. #SAT #P-complete even for

• monotone 2CNF and
• Horn formulas.

𝖰 𝖮𝖰 Σ𝖰

2

𝖣𝗉𝖮𝖰 Π𝖰

2

Δ𝖰

2

…

𝖰𝖨 = ⋃

i

Σ𝖰

i

Applications and Solvers

4

Applications and Solvers

Probabilistic Inference

4

Applications and Solvers

Pr(A|B) = Pr(A ∧ B) Pr(B) Probabilistic Inference

4

Applications and Solvers

Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference

4

Applications and Solvers

Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Applications and Solvers

Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Applications and Solvers

Model Counters Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Applications and Solvers

Model Counters Exact Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Applications and Solvers

Model Counters Exhaustive DPLL Exact Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Applications and Solvers

Model Counters Exhaustive DPLL Knowledge Compilation Exact Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Applications and Solvers

Model Counters Exhaustive DPLL Knowledge Compilation Dynamic Programming Exact Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Applications and Solvers

Model Counters Exhaustive DPLL Knowledge Compilation Dynamic Programming Approximate Exact Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Applications and Solvers

Model Counters Exhaustive DPLL Knowledge Compilation Dynamic Programming Approximate Exact Hashing-Based Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Applications and Solvers

Model Counters Exhaustive DPLL Knowledge Compilation Dynamic Programming Approximate Exact Hashing-Based Pr(A|B) = Pr(A ∧ B) Pr(B) #(A ∧ B) #B Probabilistic Inference Circuit Design

4

Product Configuration

Decomposition-Based

Treewidth of Formulas

5

Treewidth of Formulas

treewidth measures the “tree-likeness” of a graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

treewidth measures the “tree-likeness” of a graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

treewidth measures the “tree-likeness” of a graph primal graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x2

treewidth measures the “tree-likeness” of a graph primal graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2

treewidth measures the “tree-likeness” of a graph primal graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x3

treewidth measures the “tree-likeness” of a graph primal graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

x1 x2 x4 x3

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

x1 x2 x4 x3

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

x1 x2 x4 x3

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

x1 x2 x4 x3

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

x1 x2 x4 x3

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

x1 x2 x4 x3

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

x1 x2 x4 x3

5

Treewidth of Formulas

(x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ (¬x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x4)

x1 x2 x4 x3

treewidth measures the “tree-likeness” of a graph primal graph incidence graph

x1 x2 x4 x3

5

#SAT and Treewidth

#SAT and Treewidth

primal treewidth k

#SAT and Treewidth

primal treewidth k incidence treewidth k*

#SAT and Treewidth

primal treewidth k incidence treewidth k*

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F)

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F)

Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F) 2O(k) ⋅ p(F)

Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003 Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F) 4k* ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003 Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Samer and Szeider 2007

4k* ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003 Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Samer and Szeider 2007

4k* ⋅ p(F) 4k ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003 Fischer, Makowsky, Ravve 2006 Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Samer and Szeider 2007

4k* ⋅ p(F) 4k ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003

2k ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Samer and Szeider 2007

4k* ⋅ p(F) 4k ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003

2k ⋅ p(F)

Samer and Szeider 2007 Fischer, Makowsky, Ravve 2006 Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Samer and Szeider 2007

4k* ⋅ p(F) 4k ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003

2k ⋅ p(F)

Samer and Szeider 2007

2k* ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Samer and Szeider 2007

4k* ⋅ p(F) 4k ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003

2k ⋅ p(F)

Samer and Szeider 2007

2k* ⋅ p(F)

Our result Fischer, Makowsky, Ravve 2006 Courcelle, Makowsky, Rotics 2001

#SAT and Treewidth

primal treewidth k incidence treewidth k*

k* ≤ k + 1 f(k) ⋅ p(F) f(k*) ⋅ p(F)

Fischer, Makowsky, Ravve 2006 Samer and Szeider 2007

4k* ⋅ p(F) 4k ⋅ p(F) 2O(k) ⋅ p(F)

Bacchus, Dalmao, Pitassi 2003

2k ⋅ p(F)

Samer and Szeider 2007

2k* ⋅ p(F)

Our result

O p t i m a l u n d e r E T H

Fischer, Makowsky, Ravve 2006 Courcelle, Makowsky, Rotics 2001

Tree Decompositions

7

Tree Decompositions

7

Tree Decompositions

“node”

7

Tree Decompositions

“node”

7

Tree Decompositions

“node” “bag”

7

Tree Decompositions

• 1. Each vertex appears in a bag.

“node” “bag”

7

Tree Decompositions

• 1. Each vertex appears in a bag.
• 2. Each edge

is contained in a bag.

vw

“node” v w “bag”

7

Tree Decompositions

• 1. Each vertex appears in a bag.
• 2. Each edge

is contained in a bag.

vw

• 3. The set of nodes in whose bags

appears form a connected subtree.

v

“node” v w “bag”

7

Tree Decompositions

• 1. Each vertex appears in a bag.
• 2. Each edge

is contained in a bag.

vw

• 3. The set of nodes in whose bags

appears form a connected subtree.

v

“node” v v w “bag”

7

Tree Decompositions

• 1. Each vertex appears in a bag.
• 2. Each edge

is contained in a bag.

vw

• 3. The set of nodes in whose bags

appears form a connected subtree.

v

“node” v v w v “bag”

7

Tree Decompositions

• 1. Each vertex appears in a bag.
• 2. Each edge

is contained in a bag.

vw

• 3. The set of nodes in whose bags

appears form a connected subtree.

v

“node” v v w v The width of a tree decomposition is the size of its largest bag - .

1

“bag”

7

Dynamic Programming for #SAT

8

x2 x2 x4 x3 C2 C2 C2 x3 C1 x1

Dynamic Programming for #SAT

8

x2 x2 x4 x3 C2 C2 C2 x3 C1

t

x1

Dynamic Programming for #SAT

8

x2 x2 x4 x3 C2 C2 C2 x3 C1

t

x1

n(t, α, A)

Dynamic Programming for #SAT

8

x2 x2 x4 x3 C2 C2 C2 x3 C1

t

x1

n(t, α, A)

assignment of bag variables

Dynamic Programming for #SAT

8

x2 x2 x4 x3 C2 C2 C2 x3 C1

t

x1

n(t, α, A)

assignment of bag variables subset of bag clauses

Join Nodes

9

x4 x3 C2 x4 x3 C2 x4 x3 C2

… …

Join Nodes

9

x4 x3 C2 x4 x3 C2 x4 x3 C2

t

… …

Join Nodes

9

x4 x3 C2 x4 x3 C2 x4 x3 C2

t t1

… …

Join Nodes

9

x4 x3 C2 x4 x3 C2 x4 x3 C2

t t1 t2

… …

Join Nodes

9

x4 x3 C2 x4 x3 C2 x4 x3 C2

t t1 t2

n(t, α, A) = ∑

A1, A2 ⊆ χc(t) A1 ∩ A2 = A

n(t1, α, A1) n(t2, α, A2)

… …

Covering Product

10

n(t, α, A) = ∑

A1, A2 ⊆ χc(t) A1 ∩ A2 = A

n(t1, α, A1) n(t2, α, A2)

Covering Product

10

f, g : 2S → ℤ

n(t, α, A) = ∑

A1, A2 ⊆ χc(t) A1 ∩ A2 = A

n(t1, α, A1) n(t2, α, A2)

Covering Product

10

f, g : 2S → ℤ

n(t, α, A) = ∑

A1, A2 ⊆ χc(t) A1 ∩ A2 = A

n(t1, α, A1) n(t2, α, A2)

|S| = k

Covering Product

10

f, g : 2S → ℤ (f *c g)(Y) = ∑

A∪B=Y

f(A) g(B)

n(t, α, A) = ∑

A1, A2 ⊆ χc(t) A1 ∩ A2 = A

n(t1, α, A1) n(t2, α, A2)

|S| = k

Covering Product

10

f, g : 2S → ℤ (f *c g)(Y) = ∑

A∪B=Y

f(A) g(B)

n(t, α, A) = ∑

A1, A2 ⊆ χc(t) A1 ∩ A2 = A

n(t1, α, A1) n(t2, α, A2)

|S| = k Y ⊆ S

Covering Product

10

f, g : 2S → ℤ (f *c g)(Y) = ∑

A∪B=Y

f(A) g(B)

n(t, α, A) = ∑

A1, A2 ⊆ χc(t) A1 ∩ A2 = A

n(t1, α, A1) n(t2, α, A2)

|S| = k Y ⊆ S

All values of the covering product can be computed with arithmetic operations.

f *c g O(2kk)

Theorem (Björklund, Husfeldt, Kaski, Koivisto 2007)

Summary

4k* ⋅ p(F)

#SAT with incidence treewidth k*

Summary

4k* ⋅ p(F)

#SAT with incidence treewidth k*

Fast Zeta- and Möbius Transform Covering Product

Summary

4k* ⋅ p(F)

#SAT with incidence treewidth k*

Fast Zeta- and Möbius Transform Covering Product

Summary

4k* ⋅ p(F)

#SAT with incidence treewidth k*

Fast Zeta- and Möbius Transform Covering Product

Summary

4k* ⋅ p(F) 2k* ⋅ p(F)

#SAT with incidence treewidth k*

Fast Zeta- and Möbius Transform Covering Product