the complexity of counting problems
play

The Complexity of Counting Problems Tyson Williams (University of - PowerPoint PPT Presentation

Nov 05, 2022 •214 likes •1.35k views

The Complexity of Counting Problems Tyson Williams (University of Wisconsin-Madison) Joint with: Jin-Yi Cai and Heng Guo (University of Wisconsin-Madison) Michael Kowalczyk (Northern Michigan University) Tyson Williams (UW-M) Counting

  1. One theorem to rule them all Tyson Williams (UW-M) Counting Complexity Google Madison 2013 9 / 32

  2. One theorem to rule them all Theorem (Schaefer’s dichotomy theorem ’78) For any set of constraint functions F , the problem CSP ( F ) is NP-Complete unless one of the following conditions holds, in which case the problem is in P: Tyson Williams (UW-M) Counting Complexity Google Madison 2013 9 / 32

  3. One theorem to rule them all Theorem (Schaefer’s dichotomy theorem ’78) For any set of constraint functions F , the problem CSP ( F ) is NP-Complete unless one of the following conditions holds, in which case the problem is in P: All constraints in F ... that are not constantly false are true when all its arguments are true; 1 that are not constantly false are true when all its arguments are false; 2 are equivalent to a conjunction of binary clauses; 3 are equivalent to a conjunction of Horn clauses; 4 are equivalent to a conjunction of dual-Horn clauses; 5 are equivalent to a conjunction of affine formula. 6 Tyson Williams (UW-M) Counting Complexity Google Madison 2013 9 / 32

  4. One theorem to rule them all Theorem (Schaefer’s dichotomy theorem ’78) For any set of constraint functions F , the problem CSP ( F ) is NP-Complete unless one of the following conditions holds, in which case the problem is in P: All constraints in F ... that are not constantly false are true when all its arguments are true; 1 that are not constantly false are true when all its arguments are false; 2 are equivalent to a conjunction of binary clauses; 3 are equivalent to a conjunction of Horn clauses; 4 are equivalent to a conjunction of dual-Horn clauses; 5 are equivalent to a conjunction of affine formula. 6 Tyson Williams (UW-M) Counting Complexity Google Madison 2013 9 / 32

  5. One theorem to rule them all Theorem (Schaefer’s dichotomy theorem ’78) For any set of constraint functions F , the problem CSP ( F ) is NP-Complete unless one of the following conditions holds, in which case the problem is in P: All constraints in F ... that are not constantly false are true when all its arguments are true; 1 that are not constantly false are true when all its arguments are false; 2 are equivalent to a conjunction of binary clauses; 3 are equivalent to a conjunction of Horn clauses; 4 are equivalent to a conjunction of dual-Horn clauses; 5 are equivalent to a conjunction of affine formula. 6 Tyson Williams (UW-M) Counting Complexity Google Madison 2013 9 / 32

  6. Three takeaways from Schaefer’s dichotomy theorem Tyson Williams (UW-M) Counting Complexity Google Madison 2013 10 / 32

  7. Three takeaways from Schaefer’s dichotomy theorem Theorist: describes the line between easy problems and hard problems Tyson Williams (UW-M) Counting Complexity Google Madison 2013 10 / 32

  8. Three takeaways from Schaefer’s dichotomy theorem Theorist: describes the line between easy problems and hard problems Practitioner (i.e. employee of Google): a complexity dictionary Tyson Williams (UW-M) Counting Complexity Google Madison 2013 10 / 32

  9. Three takeaways from Schaefer’s dichotomy theorem Theorist: describes the line between easy problems and hard problems Practitioner (i.e. employee of Google): a complexity dictionary Observation: no problems of intermediate complexity Tyson Williams (UW-M) Counting Complexity Google Madison 2013 10 / 32

  10. Three takeaways from Schaefer’s dichotomy theorem Theorist: describes the line between easy problems and hard problems Practitioner (i.e. employee of Google): a complexity dictionary Observation: no problems of intermediate complexity Theorem (Ladner’s theorem ’75) If P � = NP , then there exists problems in NP of intermediate complexity. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 10 / 32

  11. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 11 / 32

  12. http://commons.wikimedia.org/wiki/File:P_np_np-complete_np-hard.svg Tyson Williams (UW-M) Counting Complexity Google Madison 2013 11 / 32

  13. A Motivation for Counting Tyson Williams (UW-M) Counting Complexity Google Madison 2013 12 / 32

  14. Independent Set Want large independent set. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 13 / 32

  15. Independent Set Want large independent set. http://commons.wikimedia.org/wiki/File:Independent_set_graph.svg Tyson Williams (UW-M) Counting Complexity Google Madison 2013 13 / 32

  16. The Facts Problem: IndependentSet Input: A graph G and k ∈ N . Output: “ Yes ” if G contains an independent set of size at least k “ No ” otherwise. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 14 / 32

  17. The Facts Problem: IndependentSet Input: A graph G and k ∈ N . Output: “ Yes ” if G contains an independent set of size at least k “ No ” otherwise. IndependentSet ∈ NP-Complete Tyson Williams (UW-M) Counting Complexity Google Madison 2013 14 / 32

  18. The Facts Problem: IndependentSet Input: A graph G and k ∈ N . Output: “ Yes ” if G contains an independent set of size at least k “ No ” otherwise. IndependentSet ∈ NP-Complete Next question: how close to optimal can we get? Tyson Williams (UW-M) Counting Complexity Google Madison 2013 14 / 32

  19. A Different Approach Let I ( G ) be the set of independent sets in G . Tyson Williams (UW-M) Counting Complexity Google Madison 2013 15 / 32

  20. A Different Approach Let I ( G ) be the set of independent sets in G . Want to randomly sample I from I ( G ) such that Pr( I ) ∝ w ( I ) . Tyson Williams (UW-M) Counting Complexity Google Madison 2013 15 / 32

  21. A Different Approach Let I ( G ) be the set of independent sets in G . Want to randomly sample I from I ( G ) such that Pr( I ) ∝ w ( I ) . Then must have Pr( I ) = w ( I ) Z ( G ) , where � Z ( G ) = w ( I ) . I ∈I ( G ) Tyson Williams (UW-M) Counting Complexity Google Madison 2013 15 / 32

  22. A Different Approach Let I ( G ) be the set of independent sets in G . Want to randomly sample I from I ( G ) such that Pr( I ) ∝ w ( I ) . Then must have Pr( I ) = w ( I ) Z ( G ) , where � Z ( G ) = w ( I ) . I ∈I ( G ) Know as the partition function in statistical physics. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 15 / 32

  23. Weight functions Pr( I ) = w ( I ) � where Z ( G ) = w ( I ) . Z ( G ) I ∈I ( G ) Tyson Williams (UW-M) Counting Complexity Google Madison 2013 16 / 32

  24. Weight functions Pr( I ) = w ( I ) � where Z ( G ) = w ( I ) . Z ( G ) I ∈I ( G ) If w ( I ) = 1, then Z ( G ) = |I ( G ) | is the number of independent sets. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 16 / 32

  25. Weight functions Pr( I ) = w ( I ) � where Z ( G ) = w ( I ) . Z ( G ) I ∈I ( G ) If w ( I ) = 1, then Z ( G ) = |I ( G ) | is the number of independent sets. Statistical physicists considered w ( I ) = λ | I | from some nonnegative λ ∈ R . Tyson Williams (UW-M) Counting Complexity Google Madison 2013 16 / 32

  26. Weight functions Pr( I ) = w ( I ) � where Z ( G ) = w ( I ) . Z ( G ) I ∈I ( G ) If w ( I ) = 1, then Z ( G ) = |I ( G ) | is the number of independent sets. Statistical physicists considered w ( I ) = λ | I | from some nonnegative λ ∈ R . If λ = 1, then Z ( G ) = |I ( G ) | again. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 16 / 32

  27. Weight functions Pr( I ) = w ( I ) � where Z ( G ) = w ( I ) . Z ( G ) I ∈I ( G ) If w ( I ) = 1, then Z ( G ) = |I ( G ) | is the number of independent sets. Statistical physicists considered w ( I ) = λ | I | from some nonnegative λ ∈ R . If λ = 1, then Z ( G ) = |I ( G ) | again. If λ = 0, then Z ( G ) = 1. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 16 / 32

  28. Weight functions Pr( I ) = w ( I ) � where Z ( G ) = w ( I ) . Z ( G ) I ∈I ( G ) If w ( I ) = 1, then Z ( G ) = |I ( G ) | is the number of independent sets. Statistical physicists considered w ( I ) = λ | I | from some nonnegative λ ∈ R . If λ = 1, then Z ( G ) = |I ( G ) | again. If λ = 0, then Z ( G ) = 1. Theorem (Sly,Sun ’12) For d ≥ 3 and λ > λ c ( d ) = ( d − 1) d − 1 ( d − 2) d , unless NP = RP there is no approximation algorithm for the partition function with w ( I ) = λ | I | on d-regular graphs. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 16 / 32

  29. Local Constraints Tyson Williams (UW-M) Counting Complexity Google Madison 2013 17 / 32

  30. #VertexCover Definition A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 18 / 32

  31. #VertexCover Definition A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 18 / 32

  32. #VertexCover Definition A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 18 / 32

  33. #VertexCover Definition A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set. X � � � Tyson Williams (UW-M) Counting Complexity Google Madison 2013 18 / 32

  34. Systematic Approach to #VertexCover G = ( V , E ) � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 19 / 32

  35. Systematic Approach to #VertexCover G = ( V , E ) � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 19 / 32

  36. Systematic Approach to #VertexCover G = ( V , E ) 0 σ : V → { 0 , 1 } 1 1 1 � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 19 / 32

  37. Systematic Approach to #VertexCover G = ( V , E ) 0 σ : V → { 0 , 1 } OR OR OR 1 OR OR 1 1 OR � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 19 / 32

  38. Systematic Approach to #VertexCover G = ( V , E ) 0 σ : V → { 0 , 1 } OR OR OR 1 OR OR 1 1 OR � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 19 / 32

  39. Systematic Approach to #VertexCover G = ( V , E ) 0 σ : V → { 0 , 1 } OR OR OR 1 OR OR 1 0 OR � OR( σ ( u ) , σ ( v )) = 1 · 1 · 0 · 1 · 1 · 1 = 0 ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 19 / 32

  40. Systematic Approach to #VertexCover G = ( V , E ) σ : V → { 0 , 1 } � � # VertexCover ( G ) = OR( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 19 / 32

  41. Generalize � � OR ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 20 / 32

  42. Generalize � � OR ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E Input Output p q OR( p , q ) 0 0 0 0 1 1 1 0 1 1 1 1 Tyson Williams (UW-M) Counting Complexity Google Madison 2013 20 / 32

  43. Generalize � � f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E Input Output Input Output p q OR( p , q ) p q f ( p , q ) 0 0 0 0 0 w 0 1 1 0 1 x 1 0 1 1 0 y 1 1 1 1 1 z where w , x , y , z ∈ C Tyson Williams (UW-M) Counting Complexity Google Madison 2013 20 / 32

  44. Generalize Partition Function: Z ( · ) � � Z ( G ) = f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E Input Output Input Output p q OR( p , q ) p q f ( p , q ) 0 0 0 0 0 w 0 1 1 0 1 x 1 0 1 1 0 y 1 1 1 1 1 z where w , x , y , z ∈ C Tyson Williams (UW-M) Counting Complexity Google Madison 2013 20 / 32

  45. A Counting Dichotomy Theorem (Cai, Kowalczyk, W ’12) Over 3-regular graphs G, the exact counting problem for any (binary) complex-weighted function f � � Z ( G ) = f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E is either computable in polynomial time or # P -hard. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 21 / 32

  46. Nonlocal Examples Problem: HamiltonianCycle Input: A graph G . Output: “ Yes ” if G contains an Hamiltonian cycle “ No ” otherwise. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 22 / 32

  47. Nonlocal Examples Problem: HamiltonianCycle Input: A graph G . Output: “ Yes ” if G contains an Hamiltonian cycle “ No ” otherwise. Problem: Connected Input: A graph G . Output: “ Yes ” if G is connected “ No ” otherwise. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 22 / 32

  48. Nonlocal Examples Problem: HamiltonianCycle Input: A graph G . Output: “ Yes ” if G contains an Hamiltonian cycle “ No ” otherwise. Problem: Connected Input: A graph G . Output: “ Yes ” if G is connected “ No ” otherwise. Confessions of a theorists: Tyson Williams (UW-M) Counting Complexity Google Madison 2013 22 / 32

  49. Nonlocal Examples Problem: HamiltonianCycle Input: A graph G . Output: “ Yes ” if G contains an Hamiltonian cycle “ No ” otherwise. Problem: Connected Input: A graph G . Output: “ Yes ” if G is connected “ No ” otherwise. Confessions of a theorists: Some proofs of this depending on definition of “local”. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 22 / 32

  50. Nonlocal Examples Problem: HamiltonianCycle Input: A graph G . Output: “ Yes ” if G contains an Hamiltonian cycle “ No ” otherwise. Problem: Connected Input: A graph G . Output: “ Yes ” if G is connected “ No ” otherwise. Confessions of a theorists: Some proofs of this depending on definition of “local”. Formally, just think of these as conjectures. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 22 / 32

  51. Symmetric Function Definition A function is symmetric if invariant under any permutation of its inputs. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 23 / 32

  52. Symmetric Function Definition A function is symmetric if invariant under any permutation of its inputs. Examples: OR 2 = [0 , 1 , 1] AND 3 = [0 , 0 , 0 , 1] EVEN-PARITY 4 = [1 , 0 , 1 , 0 , 1] MAJORITY 5 = [0 , 0 , 0 , 1 , 1 , 1] (= 6 ) = EQUALITY 6 = [1 , 0 , 0 , 0 , 0 , 0 , 1] Tyson Williams (UW-M) Counting Complexity Google Madison 2013 23 / 32

  53. Constraint Graph for #CSP( F ) Instance F = { EVEN-PARITY 3 , MAJORITY 3 , OR 3 } Tyson Williams (UW-M) Counting Complexity Google Madison 2013 24 / 32

  54. Constraint Graph for #CSP( F ) Instance F = { EVEN-PARITY 3 , MAJORITY 3 , OR 3 } EVEN-PARITY 3 ( x , y , z ) ∧ MAJORITY 3 ( x , y , z ) ∧ OR 3 ( x , y , z ) Tyson Williams (UW-M) Counting Complexity Google Madison 2013 24 / 32

  55. Constraint Graph for #CSP( F ) Instance F = { EVEN-PARITY 3 , MAJORITY 3 , OR 3 } EVEN-PARITY 3 ( x , y , z ) ∧ MAJORITY 3 ( x , y , z ) ∧ OR 3 ( x , y , z ) x EVEN-PARITY 3 y MAJORITY 3 z OR 3 Tyson Williams (UW-M) Counting Complexity Google Madison 2013 24 / 32

  56. Constraint Graph for #CSP( F ) Instance F = { EVEN-PARITY 3 , MAJORITY 3 , OR 3 } EVEN-PARITY 3 ( x , y , z ) ∧ MAJORITY 3 ( x , y , z ) ∧ OR 3 ( x , y , z ) x EVEN-PARITY 3 y MAJORITY 3 z OR 3 NOT planar, so NOT an instance of Pl-#CSP( { EVEN-PARITY 3 , MAJORITY 3 , OR 3 } ) Tyson Williams (UW-M) Counting Complexity Google Madison 2013 24 / 32

  57. Constraint Graph for #CSP( F ) Instance F = { EVEN-PARITY 3 , MAJORITY 3 , OR 3 } EVEN-PARITY 3 ( x , y , z ) ∧ MAJORITY 3 ( x , y , z ) ∧ OR 3 ( x , y , z ) x EVEN-PARITY 3 y MAJORITY 3 z OR 3 NOT planar, so NOT an instance of Pl-#CSP( { EVEN-PARITY 3 , MAJORITY 3 , OR 3 } ) Tyson Williams (UW-M) Counting Complexity Google Madison 2013 24 / 32

  58. Constraint Graph for #CSP( F ) Instance F = { EVEN-PARITY 3 , MAJORITY 3 , OR 2 } EVEN-PARITY 3 ( x , y , z ) ∧ MAJORITY 3 ( x , y , z ) ∧ OR 2 ( x , y ) , z x EVEN-PARITY 3 y MAJORITY 3 z OR 2 Tyson Williams (UW-M) Counting Complexity Google Madison 2013 24 / 32

  59. Constraint Graph for #CSP( F ) Instance F = { EVEN-PARITY 3 , MAJORITY 3 , OR 2 } EVEN-PARITY 3 ( x , y , z ) ∧ MAJORITY 3 ( x , y , z ) ∧ OR 2 ( x , y ) , z x x EVEN-PARITY 3 EVEN-PARITY 3 y z MAJORITY 3 OR 2 y z OR 2 MAJORITY 3 VALID instance of Pl-#CSP( { EVEN-PARITY 3 , MAJORITY 3 , OR 2 } ) Tyson Williams (UW-M) Counting Complexity Google Madison 2013 24 / 32

  60. More Counting Dichotomies Theorem (Cai, Lu, Xia ’09) Let F be any set of complex-valued constraints in Boolean variables. Then #CSP( F ) is either # P -hard or computable in polynomial time. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 25 / 32

  61. More Counting Dichotomies Theorem (Cai, Lu, Xia ’09) Let F be any set of complex-valued constraints in Boolean variables. Then #CSP( F ) is either # P -hard or computable in polynomial time. Theorem (Cai, Xia ’12) Let F be any set of complex-valued constraints. Then #CSP( F ) is either # P -hard or computable in polynomial time. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 25 / 32

  62. More Counting Dichotomies Theorem (Cai, Lu, Xia ’09) Let F be any set of complex-valued constraints in Boolean variables. Then #CSP( F ) is either # P -hard or computable in polynomial time. Theorem (Cai, Xia ’12) Let F be any set of complex-valued constraints. Then #CSP( F ) is either # P -hard or computable in polynomial time. Theorem (Guo, W ’13) Let F be any set of symmetric, complex-valued constraints in Boolean variables. Then Pl-#CSP( F ) is either # P -hard or computable in polynomial time. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 25 / 32

  63. Definition of Holant Function Partition Function f f f f f f � � f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 26 / 32

  64. Definition of Holant Function Partition Function Assignments to vertices Functions on edges f f f f f f � � f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 26 / 32

  65. Definition of Holant Function Partition Function Holant Function Assignments to vertices Assignment to edges Functions on edges Functions on vertices f f f f f f � � f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 26 / 32

  66. Definition of Holant Function Partition Function Holant Function Assignments to vertices Assignment to edges Functions on edges Functions on vertices = 3 f f f f f f = 3 f f f f = 3 = 3 f f � � � � � � g v σ | E ( v ) f ( σ ( u ) , σ ( v )) v ∈ V σ : E →{ 0 , 1 } σ : V →{ 0 , 1 } ( u , v ) ∈ E Tyson Williams (UW-M) Counting Complexity Google Madison 2013 26 / 32

  67. Definition of Holant Function Holant Function Holant( { f } | { = 3 } ) is a counting problem defined Assignment to edges Functions on vertices over (2,3)-regular bipartite graphs. = 3 f f f = 3 f f = 3 = 3 f � � � � g v σ | E ( v ) v ∈ V σ : E →{ 0 , 1 } Tyson Williams (UW-M) Counting Complexity Google Madison 2013 26 / 32

  68. Definition of Holant Function Holant Function Holant( { f } | { = 3 } ) is a counting problem defined Assignment to edges Functions on vertices over (2,3)-regular bipartite graphs. = 3 Degree 2 vertices take f . Degree 3 vertices take = 3 . f f f = 3 f f = 3 = 3 f � � � � g v σ | E ( v ) v ∈ V σ : E →{ 0 , 1 } Tyson Williams (UW-M) Counting Complexity Google Madison 2013 26 / 32

  69. Example Holant Problems Holant( { OR 2 } | { = 3 } ) is # VertexCover on 3-regular graphs. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 27 / 32

  70. Example Holant Problems Holant( { OR 2 } | { = 3 } ) is # VertexCover on 3-regular graphs. Holant( { NAND 2 } | { = 3 } ) is # IndependentSet on 3-regular graphs. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 27 / 32

  71. Example Holant Problems Holant( { OR 2 } | { = 3 } ) is # VertexCover on 3-regular graphs. Holant( { NAND 2 } | { = 3 } ) is # IndependentSet on 3-regular graphs. Holant( { = 2 } | { AT-MOST-ONE } ) � is # Matching . Holant(AT-MOST-ONE) Tyson Williams (UW-M) Counting Complexity Google Madison 2013 27 / 32

  72. Example Holant Problems Holant( { OR 2 } | { = 3 } ) is # VertexCover on 3-regular graphs. Holant( { NAND 2 } | { = 3 } ) is # IndependentSet on 3-regular graphs. Holant( { = 2 } | { AT-MOST-ONE } ) � is # Matching . Holant(AT-MOST-ONE) Holant( { = 2 } | { EXACTLY-ONE } ) � is # PerfectMatching . Holant(EXACTLY-ONE) Tyson Williams (UW-M) Counting Complexity Google Madison 2013 27 / 32

  73. Final Dichotomy Theorem (Cai, Guo, W ’13) Let F be any set of symmetric, complex-valued constraints in Boolean variables. Then Holant( F ) is either # P -hard or computable in polynomial time. Tyson Williams (UW-M) Counting Complexity Google Madison 2013 28 / 32

  74. A Proof Technique: Polynomial Interpolation Tyson Williams (UW-M) Counting Complexity Google Madison 2013 29 / 32

  75. Polynomial Interpolation 2 (distinct) points defines a (unique) line Tyson Williams (UW-M) Counting Complexity Google Madison 2013 30 / 32

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

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
The parameterised complexity of subgraph counting problems Kitty Meeks University of Glasgow

The parameterised complexity of subgraph counting problems Kitty Meeks University of Glasgow

The parameterised complexity of subgraph counting problems Kitty Meeks University of Glasgow ACiD, 4th November 2014 Joint work with Mark Jerrum (QMUL) What is a counting problem? Decision problems Given a graph G , does G contain a Hamilton

750 views • 60 slides
The parameterised complexity of subgraph counting problems Kitty Meeks Queen Mary, University of

The parameterised complexity of subgraph counting problems Kitty Meeks Queen Mary, University of

The parameterised complexity of subgraph counting problems Kitty Meeks Queen Mary, University of London Joint work with Mark Jerrum (QMUL) What is a counting problem? Decision problems Given a graph G , does G contain a Hamilton cycle? Given

784 views • 63 slides
Complexity of Counting Lecture 21 #P: Toda s Theorem 1 Last Time 2 Last Time #P:

Complexity of Counting Lecture 21 #P: Toda s Theorem 1 Last Time 2 Last Time #P:

Complexity of Counting Lecture 21 #P: Toda s Theorem 1 Last Time 2 Last Time #P: counting problems of the form #R(x) = |{w: R(x,w)=1}|, where R is a polynomial time relation 2 Last Time #P: counting problems of the form #R(x) = |{w:

1.57k views • 131 slides
The Complexity of Approximate Counting Leslie Ann Goldberg, University of Oxford 8 th

The Complexity of Approximate Counting Leslie Ann Goldberg, University of Oxford 8 th

The Complexity of Approximate Counting Leslie Ann Goldberg, University of Oxford 8 th International Conference on Language and Automata Theory and Applications (LATA 2014) Madrid 1014 March 2014 Computational Counting Computational Problems

409 views • 26 slides
The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems

The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems

The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems Tyson Williams (University of Wisconsin-Madison) Joint with: Jin-Yi Cai and Heng Guo (University of Wisconsin-Madison) 1 / 14 Edge Coloring

637 views • 39 slides
Complexity of Counting (Computer Science) Can we efficiently count, e.g.: # colorings? #

Complexity of Counting (Computer Science) Can we efficiently count, e.g.: # colorings? #

I NAPPROXIMABILITY FOR A NTIFERROMAGNETIC S PIN S YSTEMS IN THE T REE N ON -U NIQUENESS R EGION Andreas Galanis (Oxford) Daniel tefankovi c (Rochester) Eric Vigoda (Georgia Tech) Cargese 14 , August 29 O VERVIEW Complexity of Counting

569 views • 27 slides
Frequency Counting Many problems can be solved by counting the number of times each character

Frequency Counting Many problems can be solved by counting the number of times each character

CPSC 3200 Practical Problem Solving University of Lethbridge Frequency Counting Many problems can be solved by counting the number of times each character appears in a stringthe order does not matter. e.g. Anagram

500 views • 19 slides
Counting CMPS/MATH 2170: Discrete Mathematics 1 Counting problems How many different subsets

Counting CMPS/MATH 2170: Discrete Mathematics 1 Counting problems How many different subsets

Counting CMPS/MATH 2170: Discrete Mathematics 1 Counting problems How many different subsets that a finite set ! has? How many different North American telephones numbers are possible? How many different sequences of bases in the DNA

460 views • 26 slides
Some counting problems related to A relational structure M consists of a set X and a family of

Some counting problems related to A relational structure M consists of a set X and a family of

Three counting problems: 1 Some counting problems related to A relational structure M consists of a set X and a family of relations on X . permutation groups The age of M is the class of finite relational Peter J. Cameron

214 views • 6 slides
Trapdoor Problems Basing the solution on the complexity of problems, which are easy to solve for

Trapdoor Problems Basing the solution on the complexity of problems, which are easy to solve for

Trapdoor Problems Basing the solution on the complexity of problems, which are easy to solve for the legal users, but are very difficult to the eavesdroppers. Public Key Cryptography 1 Such problems are called trapdoor problems . They allow to

379 views • 5 slides
Trapdoor Problems Basing the solution on the complexity of problems, which are easy to solve for

Trapdoor Problems Basing the solution on the complexity of problems, which are easy to solve for

Trapdoor Problems Basing the solution on the complexity of problems, which are easy to solve for the legal users, but are very difficult to the eavesdroppers. Public Key Cryptography 1 Such problems are called trapdoor problems .

345 views • 3 slides
Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory Marc Roth

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory Marc Roth

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory Marc Roth (Merton College, Oxford University, United Kingdom) , Philip Wellnitz (MPII, SIC, Saarbrcken, Germany) SODA 2020 Basic Definitions and General

955 views • 56 slides
Abstract: Computational Complexity theory deals with the classification of problems into classes

Abstract: Computational Complexity theory deals with the classification of problems into classes

Hierarchies of complexity classes 1 Abstract: Computational Complexity theory deals with the classification of problems into classes of hardness called complexity classes. In Abstract Complexity (in contrast to Concrete Complexity) we define

247 views • 12 slides
The Average-Case Complexity of Counting Cliques in Erd os-R enyi Hypergraphs Enric

The Average-Case Complexity of Counting Cliques in Erd os-R enyi Hypergraphs Enric

The Average-Case Complexity of Counting Cliques in Erd os-R enyi Hypergraphs Enric Boix-Adser` a, Matthew Brennan and Guy Bresler MIT June 29, 2020 Boix-Adser` a, Brennan and Bresler (MIT) Avg-Case Complexity of Counting Cliques June

699 views • 45 slides
An introduction to the theory of Borel complexity of classification problems J. Melleray

An introduction to the theory of Borel complexity of classification problems J. Melleray

An introduction to the theory of Borel complexity of classification problems J. Melleray Institut Camille Jordan (Universit e Lyon 1) Lausanne, May 29 2017 J. Melleray Borel complexity of classification problems I. Some context. J.

948 views • 78 slides
Conquering Generals: an NP-Hard Proof-of-Useful-Work Angelique Faye Loe Elizabeth A. Quaglia

Conquering Generals: an NP-Hard Proof-of-Useful-Work Angelique Faye Loe Elizabeth A. Quaglia

Conquering Generals: an NP-Hard Proof-of-Useful-Work Angelique Faye Loe Elizabeth A. Quaglia Information Security Group Information Security Group angelique.loe.2016@rhul.ac.uk elizabeth.quaglia@rhul.ac.uk Introduction Information Security

348 views • 13 slides
CSCI 3210: Computational Game Theory Computational Complexity Ref: Algorithm Design (Ch 8) on

CSCI 3210: Computational Game Theory Computational Complexity Ref: Algorithm Design (Ch 8) on

10/26/20 CSCI 3210: Computational Game Theory Computational Complexity Ref: Algorithm Design (Ch 8) on Blackboard Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Famous complexity classes u NP (non-deterministic

340 views • 19 slides
Prolog programming: a do-it-yourself course for beginners Day 4 Kristina Striegnitz Department

Prolog programming: a do-it-yourself course for beginners Day 4 Kristina Striegnitz Department

Prolog programming: a do-it-yourself course for beginners Day 4 Kristina Striegnitz Department of Computational Linguistics Saarland University, Saarbr ucken, Germany kris@coli.uni-sb.de http://www.coli.uni-sb.de/kris Day 4: Definite

373 views • 16 slides
NP-Hardness Textbook Reading Chapter 34 Overview Computational (in)tractability Decision

NP-Hardness Textbook Reading Chapter 34 Overview Computational (in)tractability Decision

NP-Hardness Textbook Reading Chapter 34 Overview Computational (in)tractability Decision problems and optimization problems Decision problems and formal languages The class P Decision and verification The class NP NP

2.01k views • 188 slides
CS 301 Lecture 25 NP -complete Stephen Checkoway April 29, 2018 1 / 30 Polynomial-time

CS 301 Lecture 25 NP -complete Stephen Checkoway April 29, 2018 1 / 30 Polynomial-time

CS 301 Lecture 25 NP -complete Stephen Checkoway April 29, 2018 1 / 30 Polynomial-time reducibility A function f is a polynomial-time computable function if some poly-time TM M exists that halts with just f ( w ) on its

1.36k views • 99 slides
Autumn 2019 Ling 5201 Syntax I 2: Syntax as deduction?aria7dne Robert Levine Ohio State

Autumn 2019 Ling 5201 Syntax I 2: Syntax as deduction?aria7dne Robert Levine Ohio State

Autumn 2019 Ling 5201 Syntax I 2: Syntax as deduction?aria7dne Robert Levine Ohio State University levine.1@osu.edu Robert Levine 2019 5201 1 / 9 Where we left off. . . Since we can treat VP as characterizing a string of words looking for

358 views • 9 slides
In Its Usual Formulation, Fuzzy Computation Is, In General, NP-Hard, But a More Realistic

In Its Usual Formulation, Fuzzy Computation Is, In General, NP-Hard, But a More Realistic

In Its Usual Formulation, Fuzzy Computation Is, In General, NP-Hard, But a More Realistic Formulation Can Make It Feasible Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, and Luc Longpr e University of Texas at El Paso El Paso TX

290 views • 16 slides
Probabilistic Graphical Models David Sontag New York University Lecture 4, February 21, 2013

Probabilistic Graphical Models David Sontag New York University Lecture 4, February 21, 2013

Probabilistic Graphical Models David Sontag New York University Lecture 4, February 21, 2013 David Sontag (NYU) Graphical Models Lecture 4, February 21, 2013 1 / 29 Conditional random fields (CRFs) A CRF is a Markov network on variables X

428 views • 29 slides

More recommend