exploiting treewidth for projected model counting and its
play

Exploiting Treewidth for Projected Model Counting and its Limits - PowerPoint PPT Presentation

Nov 28, 2023 •382 likes •1.56k views

Exploiting Treewidth for Projected Model Counting and its Limits Gnther Charwat, Johannes K. Fichte 1 , Markus Hecher 2 , 3 , Michael Morak 4 , Andreas Pfandler, and Stefan Woltran 2 MII Shonan Meeting 144 Shonan Village March 7 th , 2019 1 TU

  1. Tree Decompositions Tree Decompositions Tree Decomposition T of G b , c G : x T : b , c b , c c a y b , x , c b , c , y b b , x , a � �� � width Definition A tree decomposition is a tree obtained from an arbitrary graph s.t. 1. Each vertex must occur in some bag 2. For each edge, there is a bag containing both endpoints 3. Connected : If vertex v appears in bags of nodes t 0 and t 1 , then v is also in the bag of each node on the path between t 0 and t 1 5 / 22

  2. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x c y a b 6 / 22

  3. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x c y a b 1. Decompose graph b , c b , c b , c b , x , c b , c , y b , x , a 6 / 22

  4. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x c y a b 1. Decompose graph 2. Solve problems via A b , c b , c b , c b , x , c b , c , y b , x , a 6 / 22

  5. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x b c · · · · · · c y a a · · · · · · · · · · · · · · · · · · b b c b c 1. Decompose graph · · · · · · · · · · · · · · · · · · · · · · · · 2. Solve problems via A · · · · · · · · · · · · · · · · · · · · · · · · b x c b c y b , c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b , c b , c · · · · · · · · · · · · · · · · · · b x a b , x , c b , c , y · · · · · · · · · · · · · · · · · · b , x , a b , x , a 6 / 22

  6. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x b c · · · · · · c y a · · · · · · · · · · · · · · · · · · b b c b c 1. Decompose graph · · · · · · · · · · · · · · · · · · · · · · · · 2. Solve problems via A · · · · · · · · · · · · · · · · · · · · · · · · b x c b c y b , c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b , c b , c · · · · · · · · · · · · · · · · · · b x a b , x , c b , x , c b , c , y · · · · · · · · · · · · · · · · · · b , x , a 6 / 22

  7. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x b c · · · · · · c y a · · · · · · · · · · · · · · · · · · b b c b c 1. Decompose graph · · · · · · · · · · · · · · · · · · · · · · · · 2. Solve problems via A · · · · · · · · · · · · · · · · · · · · · · · · b x c b c y b , c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b , c b , c b , c · · · · · · · · · · · · · · · · · · b x a b , x , c b , c , y · · · · · · · · · · · · · · · · · · b , x , a 6 / 22

  8. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x b c · · · · · · c y a · · · · · · · · · · · · · · · · · · b b c b c 1. Decompose graph · · · · · · · · · · · · · · · · · · · · · · · · 2. Solve problems via A · · · · · · · · · · · · · · · · · · · · · · · · b x c b c y b , c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b , c b , c · · · · · · · · · · · · · · · · · · b x a b , x , c b , c , y b , c , y · · · · · · · · · · · · · · · · · · b , x , a 6 / 22

  9. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x b c · · · · · · c y a · · · · · · · · · · · · · · · · · · b b c b c 1. Decompose graph · · · · · · · · · · · · · · · · · · · · · · · · 2. Solve problems via A · · · · · · · · · · · · · · · · · · · · · · · · b x c b c y b , c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b , c b , c b , c b , c · · · · · · · · · · · · · · · · · · b x a b , x , c b , c , y · · · · · · · · · · · · · · · · · · b , x , a 6 / 22

  10. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x b c · · · · · · c y a · · · · · · · · · · · · · · · · · · b b c b c 1. Decompose graph · · · · · · · · · · · · · · · · · · · · · · · · 2. Solve problems via A · · · · · · · · · · · · · · · · · · · · · · · · b x c b c y b , c b , c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b , c b , c · · · · · · · · · · · · · · · · · · b x a b , x , c b , c , y · · · · · · · · · · · · · · · · · · b , x , a 6 / 22

  11. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x b c · · · · · · c y a · · · · · · · · · · · · · · · · · · b b c b c 1. Decompose graph · · · · · · · · · · · · · · · · · · · · · · · · 2. Solve problems via A · · · · · · · · · · · · · · · · · · · · · · · · b x c b c y b , c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b , c b , c · · · · · · · · · · · · · · · · · · b x a b , x , c b , c , y · · · · · · · · · · · · · · · · · · b , x , a 6 / 22

  12. Tree Decompositions Exploiting Tree Decompositions (TDs) Dynamic Programming DP A via local algorithm A x b c · · · · · · c y a · · · · · · · · · · · · · · · · · · b b c b c 1. Decompose graph · · · · · · · · · · · · · · · · · · · · · · · · 2. Solve problems via A · · · · · · · · · · · · 3. Combine solutions · · · · · · · · · · · · b x c b c y b , c · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b , c b , c · · · · · · · · · · · · · · · · · · b x a b , x , c b , c , y · · · · · · · · · · · · · · · · · · b , x , a 6 / 22

  13. PMC and Tree Decompositions? Outline Introduction Tree Decompositions PMC and Tree Decompositions? Towards Solving in Practice Summary & Future Work 7 / 22

  14. PMC and Tree Decompositions? How would we solve PMC by means of Tree Decompositions? 7 / 22

  15. PMC and Tree Decompositions? How would we solve PMC by means of Tree Decompositions? Relax! Let us go one step back to SAT! 7 / 22

  16. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) 7 / 22

  17. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) Mod ( ϕ ) = { { b } , { a , b } , { b , c } , { a , b , c } , { b , c , x } , { a , b , c , x } , { b , y } , { a , b , y }} 7 / 22

  18. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x c y a b 1. Decompose graph b , c b , c b , c b , x , c b , c , y b , x , a 7 / 22

  19. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y 1 0 a a 1 1 b c b c b 1 0 0 0 1. Decompose graph 1 1 1 0 1 1 b x c 2. Solve problems via S 1 0 0 b c y 1 0 1 0 0 0 1 1 1 0 0 1 b , c 1 0 0 b x a 1 0 1 1 0 0 1 1 0 b , c b , c 1 0 1 1 1 0 b , x , c b , c , y 1 1 1 0 1 1 b , x , a b , x , a 7 / 22

  20. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y 1 0 a 1 1 b c b c b 1 0 0 0 1. Decompose graph 1 1 1 0 1 1 b x c 2. Solve problems via S 1 0 0 b c y 1 0 1 0 0 0 1 1 1 0 0 1 b , c 1 0 0 b x a 1 0 1 1 0 0 1 1 0 b , c b , c 1 0 1 1 1 0 b , x , c b , x , c b , c , y 1 1 1 0 1 1 b , x , a 7 / 22

  21. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y 1 0 a 1 1 b c b c b 1 0 0 0 1. Decompose graph 1 1 1 0 1 1 b x c 2. Solve problems via S 1 0 0 b c y 1 0 1 0 0 0 1 1 1 0 0 1 b , c 1 0 0 b x a 1 0 1 1 0 0 1 1 0 b , c b , c b , c 1 0 1 1 1 0 b , x , c b , c , y 1 1 1 0 1 1 b , x , a 7 / 22

  22. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y 1 0 a 1 1 b c b c b 1 0 0 0 1. Decompose graph 1 1 1 0 1 1 b x c 2. Solve problems via S 1 0 0 b c y 1 0 1 0 0 0 1 1 1 0 0 1 b , c 1 0 0 b x a 1 0 1 1 0 0 1 1 0 b , c b , c 1 0 1 1 1 0 b , x , c b , c , y b , c , y 1 1 1 0 1 1 b , x , a 7 / 22

  23. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y 1 0 a 1 1 b c b c b 1 0 0 0 1. Decompose graph 1 1 1 0 1 1 b x c 2. Solve problems via S 1 0 0 b c y 1 0 1 0 0 0 1 1 1 0 0 1 b , c 1 0 0 b x a 1 0 1 1 0 0 1 1 0 b , c b , c b , c b , c 1 0 1 1 1 0 b , x , c b , c , y 1 1 1 0 1 1 b , x , a 7 / 22

  24. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y 1 0 a 1 1 b c b c b 1 0 0 0 1. Decompose graph 1 1 1 0 1 1 b x c 2. Solve problems via S 1 0 0 b c y 1 0 1 0 0 0 1 1 1 0 0 1 b , c b , c 1 0 0 b x a 1 0 1 1 0 0 1 1 0 b , c b , c 1 0 1 1 1 0 b , x , c b , c , y 1 1 1 0 1 1 b , x , a 7 / 22

  25. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y 1 0 a 1 1 b c b c b 1 0 0 0 1. Decompose graph 1 1 1 0 1 1 b x c 2. Solve problems via S 1 0 0 b c y 1 0 1 0 0 0 1 1 1 0 0 1 b , c 1 0 0 b x a 1 0 1 1 0 0 1 1 0 b , c b , c 1 0 1 1 1 0 b , x , c b , c , y 1 1 1 0 1 1 b , x , a 7 / 22

  26. PMC and Tree Decompositions? Local algorithm S for SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y 1 0 a 1 1 b c b c b 1 0 0 0 1. Decompose graph 1 1 1 0 1 1 b x c 2. Solve problems via S 1 0 0 b c y 3. Combine solutions 1 0 1 0 0 0 1 1 1 0 0 1 b , c 1 0 0 b x a 1 0 1 1 0 0 1 1 0 b , c b , c 1 0 1 1 1 0 b , x , c b , c , y 1 1 1 0 1 1 b , x , a 7 / 22

  27. PMC and Tree Decompositions? Local algorithm S # for #SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c # c y 1 0 4 a 1 1 4 b c # b c # b 1 0 0 0 2 2 1. Decompose graph 1 1 4 1 0 2 1 1 1 b x c # 2. Solve problems via S 1 0 0 b c y # 2 3. Combine solutions 1 0 1 2 0 0 0 1 1 1 1 0 0 1 2 1 b , c 1 0 0 1 b x a # 1 0 1 1 1 0 0 1 1 1 0 1 b , c b , c 1 0 1 1 1 1 0 1 b , x , c b , c , y 1 1 1 1 0 1 1 1 b , x , a 7 / 22

  28. PMC and Tree Decompositions? Local algorithm S # for #SAT [SamerS10] ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x # b c c 1 0 y 4 a 1 1 4 b c # b c # b 1 0 2 0 0 2 1. Decompose graph 1 1 4 1 0 2 1 1 1 b x c # 2. Solve problems via S 1 0 0 2 b c y # 3. Combine solutions 1 0 1 0 0 0 2 1 1 1 1 2 0 0 1 1 1 0 0 b , c 1 b x a # 1 0 1 1 1 0 0 1 1 1 0 1 b , c b , c 1 0 1 1 1 1 0 1 b , x , c b , c , y Runtime: 2 O ( tw ) · poly ( | ϕ | ) 1 1 1 1 0 1 1 1 b , x , a 7 / 22

  29. PMC and Tree Decompositions? Solving PMC using S ? As already shown, algorithm S can be extended to solve #SAT! 8 / 22

  30. PMC and Tree Decompositions? Solving PMC using S ? As already shown, algorithm S can be extended to solve #SAT! Can we trivially extend S to solve PMC? 8 / 22

  31. PMC and Tree Decompositions? Solving PMC using S ? As already shown, algorithm S can be extended to solve #SAT! Can we trivially extend S to solve PMC? Short answer: NO! Theorem Unless ETH fails, there is no algorithm for PMC running in time 2 2 o ( tw ) · poly ( | ϕ | ) . 8 / 22

  32. PMC and Tree Decompositions? Solving PMC using S ? As already shown, algorithm S can be extended to solve #SAT! Can we trivially extend S to solve PMC? Short answer: NO! Theorem Unless ETH fails, there is no algorithm for PMC running in time 2 2 o ( tw ) · poly ( | ϕ | ) . Proof. ◮ Solve closed QBF ∀ X . ∃ Y .ϕ by checking whether 2 | X | solves PMC of ( ϕ, P = X ) 8 / 22

  33. PMC and Tree Decompositions? Solving PMC using S ? As already shown, algorithm S can be extended to solve #SAT! Can we trivially extend S to solve PMC? Short answer: NO! Theorem Unless ETH fails, there is no algorithm for PMC running in time 2 2 o ( tw ) · poly ( | ϕ | ) . Proof. ◮ Solve closed QBF ∀ X . ∃ Y .ϕ by checking whether 2 | X | solves PMC of ( ϕ, P = X ) ◮ Unless ETH fails, 2-QSAT can not be solved [LampisM17] in time 2 2 o ( tw ) · 2 o ( | ϕ | ) 8 / 22

  34. PMC and Tree Decompositions? Side Question: Does it get simpler eventually? 9 / 22

  35. PMC and Tree Decompositions? Side Question: Does it get simpler eventually? #Σ ℓ : “PMC” for QBFs ◮ Given: QBF ψ = ∃ X 1 . ∀ X 2 . . . . QX ℓ .ϕ , where set P is the set of free variables of ψ ◮ Task: Compute PMCQ P ( ψ ) := |{ I | I ⊆ P , ψ [ I ] valid }| 9 / 22

  36. PMC and Tree Decompositions? Side Question: Does it get simpler eventually? #Σ ℓ : “PMC” for QBFs ◮ Given: QBF ψ = ∃ X 1 . ∀ X 2 . . . . QX ℓ .ϕ , where set P is the set of free variables of ψ ◮ Task: Compute PMCQ P ( ψ ) := |{ I | I ⊆ P , ψ [ I ] valid }| Theorem ([FHP19]) o ( tw ) 2 2 ... 2 · 2 o ( | ϕ | ) . Unless ETH fails, there is no algorithm for #Σ ℓ running in time ���� height ℓ + 1 9 / 22

  37. PMC and Tree Decompositions? Side Question: Does it get simpler eventually? #Σ ℓ : “PMC” for QBFs ◮ Given: QBF ψ = ∃ X 1 . ∀ X 2 . . . . QX ℓ .ϕ , where set P is the set of free variables of ψ ◮ Task: Compute PMCQ P ( ψ ) := |{ I | I ⊆ P , ψ [ I ] valid }| Theorem ([FHP19]) o ( tw ) 2 2 ... 2 · 2 o ( | ϕ | ) . Unless ETH fails, there is no algorithm for #Σ ℓ running in time ���� height ℓ + 1 Proof. ◮ Argument similar to before, boils down to: 9 / 22

  38. PMC and Tree Decompositions? Side Question: Does it get simpler eventually? #Σ ℓ : “PMC” for QBFs ◮ Given: QBF ψ = ∃ X 1 . ∀ X 2 . . . . QX ℓ .ϕ , where set P is the set of free variables of ψ ◮ Task: Compute PMCQ P ( ψ ) := |{ I | I ⊆ P , ψ [ I ] valid }| Theorem ([FHP19]) o ( tw ) 2 2 ... 2 · 2 o ( | ϕ | ) . Unless ETH fails, there is no algorithm for #Σ ℓ running in time ���� height ℓ + 1 Proof. ◮ Argument similar to before, boils down to: o ( tw ) 2 2 ... 2 · 2 o ( | ϕ | ) ◮ Unless ETH fails, ( ℓ + 1 ) -QSAT can not be solved [FHP19] in time ���� height ℓ + 1 9 / 22

  39. PMC and Tree Decompositions? Solving PMC using S ? Wanted ◮ PMC P ( ϕ ) := |{ I ∩ P | I ⊆ var ( ϕ ) , I | = ϕ }| 10 / 22

  40. PMC and Tree Decompositions? Solving PMC using S ? Wanted ◮ PMC P ( ϕ ) := |{ I ∩ P | I ⊆ var ( ϕ ) , I | = ϕ }| � “Directly” computing PMC P : too expensive 10 / 22

  41. PMC and Tree Decompositions? Solving PMC using S ? Wanted ◮ PMC P ( ϕ ) := |{ I ∩ P | I ⊆ var ( ϕ ) , I | = ϕ }| � “Directly” computing PMC P : too expensive Approach: Compute numbers “locally” � We will still be using S 10 / 22

  42. PMC and Tree Decompositions? Solving PMC using S ? Wanted ◮ PMC P ( ϕ ) := |{ I ∩ P | I ⊆ var ( ϕ ) , I | = ϕ }| � “Directly” computing PMC P : too expensive Approach: Compute numbers “locally” � We will still be using S PMC in three steps 1. Run algorithm DP S 10 / 22

  43. PMC and Tree Decompositions? Solving PMC using S ? Wanted ◮ PMC P ( ϕ ) := |{ I ∩ P | I ⊆ var ( ϕ ) , I | = ϕ }| � “Directly” computing PMC P : too expensive Approach: Compute numbers “locally” � We will still be using S PMC in three steps 1. Run algorithm DP S 2. Purge non-solutions 10 / 22

  44. PMC and Tree Decompositions? Solving PMC using S ? Wanted ◮ PMC P ( ϕ ) := |{ I ∩ P | I ⊆ var ( ϕ ) , I | = ϕ }| � “Directly” computing PMC P : too expensive Approach: Compute numbers “locally” � We will still be using S PMC in three steps 1. Run algorithm DP S 2. Purge non-solutions 3. Projected counting via DP P Step 3 requires the following: � Assign counter to a set of rows 10 / 22

  45. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P 11 / 22

  46. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P ◮ Map subsets of resulting “row classes” to a counter ◮ Maintain counters using the principle of inclusion and exclusion 11 / 22

  47. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P ◮ Map subsets of resulting “row classes” to a counter ◮ Maintain counters using the principle of inclusion and exclusion Example: Computing pmc t given P = { x , y , z } . . . x . . . . . . 0 . . . . . . t : 0 . . . . . . 1 . . . . . . x counter . . . . . . . . . 0 1 . . . . . . . . . 1 0 3 1 . . . . . . . . . 2 0 2 . . . . . . . . . 1 3 . . . . . . . . . . . . 11 / 22

  48. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P ◮ Map subsets of resulting “row classes” to a counter ◮ Maintain counters using the principle of inclusion and exclusion Example: Computing pmc t given P = { x , y , z } . . . x . . . . . . 0 . . . . . . t : 0 . . . . . . 1 . . . . . . x counter . . . . . . . . . 0 1 . . . . . . . . . 1 0 3 1 . . . . . . . . . 2 0 2 . . . . . . . . . 1 3 . . . . . . . . . . . . 11 / 22

  49. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P ◮ Map subsets of resulting “row classes” to a counter ◮ Maintain counters using the principle of inclusion and exclusion Example: Computing pmc t given P = { x , y , z } . . . x . . . . . . 0 . . . . . . t : 0 . . . . . . 1 . . . . . . x counter . . . . . . . . . 0 1 . . . . . . . . . 1 0 3 1 . . . . . . . . . 2 0 2 . . . . . . . . . 1 3 . . . . . . . . . . . . 11 / 22

  50. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P ◮ Map subsets of resulting “row classes” to a counter ◮ Maintain counters using the principle of inclusion and exclusion Example: Computing pmc t given P = { x , y , z } . . . x . . . . . . 0 . . . . . . t : 0 . . . . . . 1 . . . . . . x counter . . . . . . . . . 0 1 . . . . . . . . . 1 0 3 1 . . . . . . . . . 2 0 2 . . . . . . . . . 1 3 . . . . . . . . . . . . 11 / 22

  51. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P ◮ Map subsets of resulting “row classes” to a counter ◮ Maintain counters using the principle of inclusion and exclusion Example: Computing pmc t given P = { x , y , z } . . . x . . . . . . 0 . . . . . . pmc t = 6 t : 0 . . . . . . 1 . . . . . . x counter . . . . . . . . . 0 1 . . . . . . . . . 1 0 3 1 . . . . . . . . . 2 0 2 . . . . . . . . . 1 3 . . . . . . . . . . . . 11 / 22

  52. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P ◮ Map subsets of resulting “row classes” to a counter ◮ Maintain counters using the principle of inclusion and exclusion Example: Computing pmc t given P = { x , y , z } . . . x . . . . . . 0 . . . . . . pmc t = 6 − 4 ∗ t : 0 . . . . . . 1 . . . . . . x counter . . . . . . . . . 0 1 . . . . . . . . . 1 *: Counter for 1 st and 3 rd row: 1 0 3 1 . . . . . . . . . 2 0 2 . . . . . . . . . 1 3 . . . . . . . . . . . . 11 / 22

  53. PMC and Tree Decompositions? Towards projection via local algorithm P Ingredients for given PMC instance ( ϕ, P ) ◮ Equivalence classes (partitioning) of table rows For rows u , v of a table: u ≡ P v ⇐ ⇒ I ( u ) ∩ P = I ( v ) ∩ P ◮ Map subsets of resulting “row classes” to a counter ◮ Maintain counters using the principle of inclusion and exclusion Example: Computing pmc t given P = { x , y , z } . . . x . . . . . . 0 . . . . . . pmc t = 6 − 4 ∗ + 1 = 3 t : 0 . . . . . . 1 . . . . . . x counter . . . . . . . . . 0 1 . . . . . . . . . 1 *: Counter for 1 st and 3 rd row: 1 0 3 1 . . . . . . . . . 2 0 2 . . . . . . . . . 1 3 . . . . . . . . . . . . 11 / 22

  54. PMC and Tree Decompositions? Towards projection via local algorithm P Principle of Inclusion-Exclusion (PIE) for family X of finite sets X i � = | X 1 | + | X 2 | + . . . − | X 1 ∩ X 2 | + . . . + ( − 1 ) |X|− 1 · | | � X i | X i | X i ∈X X i ∈X 12 / 22

  55. PMC and Tree Decompositions? Towards projection via local algorithm P Principle of Inclusion-Exclusion (PIE) for family X of finite sets X i � = | X 1 | + | X 2 | + . . . − | X 1 ∩ X 2 | + . . . + ( − 1 ) |X|− 1 · | | � X i | X i | X i ∈X X i ∈X � � = Σ I � { 1 ,..., |X|} ( − 1 ) | I |− 1 · | + ( − 1 ) |X|− 1 · | X i | X i | i ∈ I X i ∈X 12 / 22

  56. PMC and Tree Decompositions? Towards projection via local algorithm P Principle of Inclusion-Exclusion (PIE) for family X of finite sets X i � � = | X 1 | + | X 2 | + . . . − | X 1 ∩ X 2 | + . . . + ( − 1 ) |X|− 1 · | | X i | X i | X i ∈X X i ∈X � �� � pmc t � � = Σ I � { 1 ,..., |X|} ( − 1 ) | I |− 1 · | + ( − 1 ) |X|− 1 · | X i | X i | i ∈ I X i ∈X 12 / 22

  57. PMC and Tree Decompositions? Towards projection via local algorithm P Principle of Inclusion-Exclusion (PIE) for family X of finite sets X i � � = | X 1 | + | X 2 | + . . . − | X 1 ∩ X 2 | + . . . + ( − 1 ) |X|− 1 · | | X i | X i | X i ∈X X i ∈X � �� � pmc t � � = Σ I � { 1 ,..., |X|} ( − 1 ) | I |− 1 · | + ( − 1 ) |X|− 1 · | X i | X i | i ∈ I X i ∈X � �� � ipmc t 12 / 22

  58. PMC and Tree Decompositions? Towards projection via local algorithm P Principle of Inclusion-Exclusion (PIE) for family X of finite sets X i � � = | X 1 | + | X 2 | + . . . − | X 1 ∩ X 2 | + . . . + ( − 1 ) |X|− 1 · | | X i | X i | X i ∈X X i ∈X � �� � pmc t � � = Σ I � { 1 ,..., |X|} ( − 1 ) | I |− 1 · | + ( − 1 ) |X|− 1 · | X i | X i | i ∈ I X i ∈X � �� � ipmc t 12 / 22

  59. PMC and Tree Decompositions? Towards projection via local algorithm P Principle of Inclusion-Exclusion (PIE) for family X of finite sets X i � � = | X 1 | + | X 2 | + . . . − | X 1 ∩ X 2 | + . . . + ( − 1 ) |X|− 1 · | | X i | X i | X i ∈X X i ∈X � �� � pmc t � � = Σ I � { 1 ,..., |X|} ( − 1 ) | I |− 1 · | + ( − 1 ) |X|− 1 · | X i | X i | i ∈ I X i ∈X � �� � ipmc t Local algorithm P in more details: ◮ Only store (“counter”) ipmc t -values ◮ pmc t -values are used intermediately ◮ Generalization of PIE that – instead of cardinalities – relies on stored ipmc t -values 12 / 22

  60. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC 13 / 22

  61. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) Mod ( ϕ ) = { { b } , { a , b } , { b , c } , { a , b , c } , { b , c , x } , { a , b , c , x } , { b , y } , { a , b , y }} PMC P ( ϕ ) = |{∅ , { x } , { y }}| = 3 13 / 22

  62. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c 1 0 y a 1 1 b c b c b 1 0 0 0 1 1 1. Local algorithm S 1 0 1 1 b x c 1 0 0 b c y 1 0 1 0 0 0 1 1 1 1 0 0 b , c 1 1 0 b x a 0 0 1 1 0 0 b , c b , c 1 0 1 1 0 1 1 1 0 b , x , c b , c , y 0 1 1 1 1 1 b , x , a 13 / 22

  63. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) x b c c y a 1 0 1 1 b c b c b 1 0 1 0 1. Local algorithm S 1 1 1 1 2. Purge non-solutions b x c b c y 1 0 0 1 0 0 1 0 1 1 1 0 1 1 1 1 0 1 b , c b x a b , c b , c 1 0 0 1 0 1 b , x , c b , c , y 1 1 0 1 1 1 b , x , a 13 / 22

  64. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a b , x , a 13 / 22

  65. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a b , x , a 13 / 22

  66. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a b , x , a 13 / 22

  67. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

  68. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

  69. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

  70. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

  71. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

  72. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

  73. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b x a ipmc b , c b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

  74. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

  75. PMC and Tree Decompositions? Dynamic Programming via local algorithm P for PMC P = { x , y } , ϕ =( ¬ a ∨ b ∨ x ) ∧ ( a ∨ b ) ∧ ( c ∨ ¬ x ) ∧ ( b ∨ ¬ c ) ∧ ( ¬ b ∨ ¬ c ∨ ¬ y ) b c ipmc 1 0 2 1 1 1 2 b c ipmc b c ipmc 1 0 1 0 1 2 1. Local algorithm S 1 1 1 1 2 1 1 1 2. Purge non-solutions b x c ipmc b c y ipmc 1 0 0 1 1 0 0 1 3. Solve PMC via P 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 b , c b , c b x a ipmc b , c b , c 1 0 0 1 1 1 0 1 1 b , x , c b , c , y 1 1 0 1 1 1 1 1 1 b , x , a 13 / 22

Recommend

A Faster Algorithm for Propositional Model Counting Parameterized by Incidence Treewidth

A Faster Algorithm for Propositional Model Counting Parameterized by Incidence Treewidth

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) SAT Does have a satisfying

1.31k views • 102 slides
On the Sparsity of XORs in Approximate Model Counting Durgesh Agrawal 1 Bhavishya 1 Kuldeep S.

On the Sparsity of XORs in Approximate Model Counting Durgesh Agrawal 1 Bhavishya 1 Kuldeep S.

On the Sparsity of XORs in Approximate Model Counting Durgesh Agrawal 1 Bhavishya 1 Kuldeep S. Meel 2 1 Indian Institute of Technology, Kanpur 2 School of Computing, National University of Singapore SAT 2020 1/18 Model Counting Given

684 views • 57 slides
Part I. Introduction to treewidth SCHOOL ON PARAMETERIZED ALGORITHMS AND COMPLEXITY 17-22

Part I. Introduction to treewidth SCHOOL ON PARAMETERIZED ALGORITHMS AND COMPLEXITY 17-22

FEDOR V. FOMIN Part I. Introduction to treewidth SCHOOL ON PARAMETERIZED ALGORITHMS AND COMPLEXITY 17-22 August 2014 B dlewo, Poland Why treewidth? Very general idea in science: large structures can be understood by breaking them into

1.99k views • 135 slides
BIRD: Engineering an Efficient CNF-XOR SAT Solver and its Applications to Approximate Model

BIRD: Engineering an Efficient CNF-XOR SAT Solver and its Applications to Approximate Model

BIRD: Engineering an Efficient CNF-XOR SAT Solver and its Applications to Approximate Model Counting Mate Soos Kuldeep S. Meel National University of Singapore AAAI 2019 1/15 Model Counting Given Boolean variables X 1 , X 2 ,

741 views • 33 slides
Treewidth reduction and algorithmic applications Treewidth reduction and algorithmic applications

Treewidth reduction and algorithmic applications Treewidth reduction and algorithmic applications

Content Treewdith Reduction Theorem The Algorithmic Motivation Proof Idea Concluding remarks Treewidth reduction and algorithmic applications Treewidth reduction and algorithmic applications Content Treewdith Reduction Theorem The

388 views • 34 slides
Computability: Three Millenia and Counting Jon Kleinberg Modeling Computation Goal: Model the

Computability: Three Millenia and Counting Jon Kleinberg Modeling Computation Goal: Model the

Computability: Three Millenia and Counting Jon Kleinberg Modeling Computation Goal: Model the notion of computation mathematically. Analogy to physics: if we build a mathematical model, we can derive laws that guide the practice of computer

292 views • 28 slides
On the variants of treewidth and minor-closedness property O-joung Kwon KAIST in Daejeon, Korea

On the variants of treewidth and minor-closedness property O-joung Kwon KAIST in Daejeon, Korea

On the variants of treewidth On the variants of treewidth and minor-closedness property O-joung Kwon KAIST in Daejeon, Korea GRASTA 2014 Joint work with Hans Bodlaender, Vincent Kreuzen, Stefan Kratsch and Seongmin Ok 1 / 22 On the variants

756 views • 31 slides
Counting algorithms and complexity. A brief overview of the field . Ioannis Nemparis 1 1

Counting algorithms and complexity. A brief overview of the field . Ioannis Nemparis 1 1

Introduction Exploring the counting class Counting in FP Correlation decay Holant problems Above # P Open questions Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting algorithms and complexity. A brief

686 views • 41 slides
Model Counting Aditya A. Shrotri Dept. of Computer Science, Rice University Research Overview:

Model Counting Aditya A. Shrotri Dept. of Computer Science, Rice University Research Overview:

Theory and Practice of Efficient Approximate Model Counting Aditya A. Shrotri Dept. of Computer Science, Rice University Research Overview: Approximate Model Counting Aditya A. Shrotri 1/15/2019 1 Beyond SAT: #SAT SAT: Does there exist

276 views • 10 slides
Exploiting compositionality to explore a large space of model structures R. Grosse, R.

Exploiting compositionality to explore a large space of model structures R. Grosse, R.

Exploiting compositionality to explore a large space of model structures R. Grosse, R. Salakhutdinov, W. Freeman, & J. Tenenbaum Best Student Paper at UAI 2012 Jan Gasthaus Tea talk 31st Aug 2012 1 / 15 Motivation Goal: Given a data set,

248 views • 23 slides
Exploiting compositionality to explore a large space of model structures Roger Grosse Dept. of

Exploiting compositionality to explore a large space of model structures Roger Grosse Dept. of

Exploiting compositionality to explore a large space of model structures Roger Grosse Dept. of Computer Science, University of Toronto Introduction How has the life of a machine learning engineer changed in the past decade? Many tasks that

443 views • 43 slides
Exploiting model structure to encode transition relations and transition rate matrices

Exploiting model structure to encode transition relations and transition rate matrices

Exploiting model structure to encode transition relations and transition rate matrices Gianfranco Ciardo Department of Computer Science and Engineering University of California, Riverside Supported in part by the National Science Foundation

591 views • 44 slides
The role of planarity in connectivity problems parameterized by treewidth Julien Baste and Ignasi

The role of planarity in connectivity problems parameterized by treewidth Julien Baste and Ignasi

Treewidth Some problems parametrized by treewidth Results Further research The role of planarity in connectivity problems parameterized by treewidth Julien Baste and Ignasi Sau 1/22 Julien Baste and Ignasi Sau The role of planarity in

517 views • 28 slides
Exploiting sparsity in model matrices Douglas Bates and Martin Maechler Department of Statistics

Exploiting sparsity in model matrices Douglas Bates and Martin Maechler Department of Statistics

Exploiting sparsity in model matrices Douglas Bates and Martin Maechler Department of Statistics University of Wisconsin Madison U.S.A. Seminar f ur Statistik ETH Zurich Switzerland (bates|maechler)@R-project.org (R-Core) DSC2009,

1.08k views • 51 slides
Di ff erentially-Private Batch Query Answering Exploiting the Workload vs. Exploiting the Data

Di ff erentially-Private Batch Query Answering Exploiting the Workload vs. Exploiting the Data

Di ff erentially-Private Batch Query Answering Exploiting the Workload vs. Exploiting the Data Gerome Miklau University of Massachusetts, Amherst DIMACS Workshop on Recent Work on Di ff erential Privacy across Computer Science October 2012

1.39k views • 136 slides
Binary Counters Integer Representations towards Efficient Counting in the Bit Probe Model

Binary Counters Integer Representations towards Efficient Counting in the Bit Probe Model

Binary Counters Integer Representations towards Efficient Counting in the Bit Probe Model (presented at TAMC 2011) Gerth Stlting Brodal (Aarhus University) Mark Greve (Aarhus University) Vineet Pandey (BITS Pilani, India) S. Srinivasa Rao

486 views • 35 slides
The Erd os-Ko-Rado Theorem and the Treewidth of the Kneser Graph Daniel Harvey School of

The Erd os-Ko-Rado Theorem and the Treewidth of the Kneser Graph Daniel Harvey School of

The Erd os-Ko-Rado Theorem and the Treewidth of the Kneser Graph Daniel Harvey School of Mathematical Sciences, Monash University 7/7/14 1 Erd os-Ko-Rado Theorem 2 Separators 3 The Kneser Graph 4 Treewidth Erd os-Ko-Rado Theorem

761 views • 52 slides
Counting with automorphisms Lectures for CO 430 / 630 March 24 April 2, 2020 1. Counting

Counting with automorphisms Lectures for CO 430 / 630 March 24 April 2, 2020 1. Counting

Counting with automorphisms Lectures for CO 430 / 630 March 24 April 2, 2020 1. Counting orbits 2. Type generating functions 3. Cycle index functions 4. Examples 5. Main theorem 6. Unlabelled trees 1. Counting orbits of

653 views • 43 slides
Probabilistic Inference in BN2T Models by Weighted Model Counting Jirka Vomlel Institute of

Probabilistic Inference in BN2T Models by Weighted Model Counting Jirka Vomlel Institute of

Probabilistic Inference in BN2T Models by Weighted Model Counting Jirka Vomlel Institute of Information Theory and Automation Academy of Sciences of the Czech Republic http://www.utia.cz/vomlel Aalborg, Denmark, November, 21, 2013 A medical

741 views • 41 slides
Side Channel Analysis Using a Model Counting Constraint Solver and Symbolic Execution Tevfik

Side Channel Analysis Using a Model Counting Constraint Solver and Symbolic Execution Tevfik

Side Channel Analysis Using a Model Counting Constraint Solver and Symbolic Execution Tevfik Bultan Computer Science Department University of California, Santa Barbara (UCSB) Joint work with: Abdulbaki Aydin, Lucas Bang, UCSB Corina

1.1k views • 91 slides
Binary Counters Integer Representations towards Efficient Counting in the Bit Probe Model (paper

Binary Counters Integer Representations towards Efficient Counting in the Bit Probe Model (paper

Binary Counters Integer Representations towards Efficient Counting in the Bit Probe Model (paper presented at TAMC 2011) Gerth Stlting Brodal (Aarhus Universitet) Mark Greve (Aarhus Universitet) Vineet Pandey (BITS Pilani, Indien) S.

661 views • 34 slides
OmpSs + OpenACC Multi-target Task-Based Programming Model Exploiting OpenACC GPU Kernel Guray

OmpSs + OpenACC Multi-target Task-Based Programming Model Exploiting OpenACC GPU Kernel Guray

www.bsc.es www.bsc.es OmpSs + OpenACC Multi-target Task-Based Programming Model Exploiting OpenACC GPU Kernel Guray Ozen guray.ozen@bsc.es Exascale in BSC Marenostrum 4 (13.7 Petaflops ) General purpose cluster (3400 nodes) with Intel

464 views • 20 slides
Counting is Hard: Probabilistically Counting Views at Reddit Krishnan Chandra, Data Engineer

Counting is Hard: Probabilistically Counting Views at Reddit Krishnan Chandra, Data Engineer

Counting is Hard: Probabilistically Counting Views at Reddit Krishnan Chandra, Data Engineer What is probabilistic counting? How did probabilistic counting help us scale? Overview What issues did we face along the way? What is

826 views • 38 slides
Counting CS1200, CSE IIT Madras Meghana Nasre April 2, 2020 CS1200, CSE IIT Madras Meghana

Counting CS1200, CSE IIT Madras Meghana Nasre April 2, 2020 CS1200, CSE IIT Madras Meghana

Counting CS1200, CSE IIT Madras Meghana Nasre April 2, 2020 CS1200, CSE IIT Madras Meghana Nasre Counting Counting (without counting) Basic Counting Techniques Pigeon Hole Principle (revisited) Permutations and Combinations

804 views • 13 slides

More recommend