Vanishing Signatures A Complete Dichotomy Rises from the Capture of - PowerPoint PPT Presentation

. .. .. . . .. . . . . . .. . . .. . . .. .. (joint work with Jin-Yi Cai and Tyson Williams) STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) June 3rd 2013 Palo Alto University of Wisconsia-Madison Heng Guo . Vanishing Signatures A Complete Dichotomy Rises from the Capture of . . . .. . . . .. . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . . . .. . .. . . .. . . 1 / 29

. .. .. . . .. . . . . . .. . . .. . . .. .. . STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) . . . . . . Contents Counting Problems . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . . .. 2 / 29 . . .. . . .. . . 1 Counting Problems 2 Dichotomy 3 Vanishing Signatures

. . . .. . . .. . .. .. . . .. . . .. . . . Approximate an integral by a weighted sum; STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) Ising model, Potts model, Hard-core gas model, … Partition functions. Classical simulation of quantum circuits; e expectation of any random variable; . be computed is usually expressed as a sum of products. learning, quantum computation, information theory, and so on. e quantity to Computational Counting problems appear often in statistical physics, machine Counting problems Counting Problems . .. . .. .. .. .. . . .. . . . . . .. . . .. . . . .. . . . .. . . .. . .. . . . .. . . .. . 3 / 29

. . . .. . . .. . .. .. . . .. . . .. . . . Approximate an integral by a weighted sum; STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) Ising model, Potts model, Hard-core gas model, … Partition functions. Classical simulation of quantum circuits; e expectation of any random variable; . be computed is usually expressed as a sum of products. learning, quantum computation, information theory, and so on. e quantity to Computational Counting problems appear often in statistical physics, machine Counting problems Counting Problems . .. . .. .. .. .. . . .. . . . . . .. . . .. . . . .. . . . .. . . .. . .. . . . .. . . .. . 3 / 29

. . . .. . . .. . .. .. . . .. . . .. . . . Approximate an integral by a weighted sum; STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) Ising model, Potts model, Hard-core gas model, … Partition functions. Classical simulation of quantum circuits; e expectation of any random variable; . be computed is usually expressed as a sum of products. learning, quantum computation, information theory, and so on. e quantity to Computational Counting problems appear often in statistical physics, machine Counting problems Counting Problems . .. . .. .. .. .. . . .. . . . . . .. . . .. . . . .. . . . .. . . .. . .. . . . .. . . .. . 3 / 29

where f I 0 0 f I 1 1 f I 0 1 f I 1 0 . . . .. . . .. . . .. . . . .. . . .. Let us take a closer look at the partition functions. Counting Problems i STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) 1 . , j f I Partition functions E i j 0 1 V We can rewrite it in the following form: Ising model (without an external �eld): . .. .. . . . . .. . . .. . .. . . . .. . . .. . . .. .. . . . . .. . . .. . . .. . .. .. . . .. . . 4 / 29 ∑ β n ( σ ) , σ : V �→{ + , −} where n ( σ ) is the number of (+ , +) and ( − , − ) neighbours in the graph given σ .

. . .. . . .. . . .. . . .. . . .. . .. . Partition functions STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) We can rewrite it in the following form: Ising model (without an external �eld): Let us take a closer look at the partition functions. Counting Problems . . .. . . .. . . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . . .. 4 / 29 . .. . .. . . .. . ∑ β n ( σ ) , σ : V �→{ + , −} where n ( σ ) is the number of (+ , +) and ( − , − ) neighbours in the graph given σ . ∑ ∏ f I ( σ ( i ) , σ ( j )) , σ : V �→{ 0 , 1 } ( i , j ) ∈ E where f I ( 0 , 0 ) = f I ( 1 , 1 ) = β , f I ( 0 , 1 ) = f I ( 1 , 0 ) = 1 .

where f  0 0 f  1 0 f  0 1 f  1 1 g  0 1 and g  1 . Counting Problems . .. . . .. . . .. Hard-core gas model: . . .. . . .. Partition functions i j We can rewrite it in the following form: i STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) . 0 , 1 , g  V i V j i f  E . 0 1 .. . .. . . .. . . .. . . .. . .. .. . . .. . . .. . . . . . . . .. . . .. . . .. . .. . . . .. . . .. 5 / 29 ∑ λ | V ′ | 1 { V ′ is an independent set } V ′ ⊆ V

. . .. . . .. .. . .. . . .. . . .. . .. . . . .. . . .. . Counting Problems Partition functions Hard-core gas model: We can rewrite it in the following form: Heng Guo (CS, UW-Madison) Complex Holant STOC 2013 . . .. .. . . .. . . .. . . . . . .. . . .. . . . .. . . .. . . .. . .. . . 5 / 29 . .. ∑ λ | V ′ | 1 { V ′ is an independent set } V ′ ⊆ V ∑ ∏ ∏ f  ( σ ( i ) , σ ( j )) g  ( σ ( i )) , i ∈ V σ : V �→{ 0 , 1 } ( i , j ) ∈ E where f  ( 0 , 0 ) = f  ( 1 , 0 ) = f  ( 0 , 1 ) = 1 , f  ( 1 , 1 ) = 0 , g  ( 0 ) = 1 and g  ( 1 ) = λ .

E v is the assignment and f  is the E-O function. . . . . .. . . .. .. . . . . .. . . .. .. . f  STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) restricted to the set E v of incident edges of v , where E v v V . 0 1 E We can rewrite it in the following form: #Perfect-Matching: Perfect matchings Counting Problems . .. .. . .. .. . . .. . . . .. . .. . . .. . . . . .. .. . . .. . . .. . . . . . .. . . .. . 6 / 29 ∑ 1 { E ′ is a perfect matching } E ′ ⊆ E

. . .. . .. .. . . .. . . .. . . .. . .. . . . .. . . .. . Counting Problems Perfect matchings #Perfect-Matching: We can rewrite it in the following form: Heng Guo (CS, UW-Madison) Complex Holant STOC 2013 . . .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . . .. 6 / 29 . . . . .. .. . .. ∑ 1 { E ′ is a perfect matching } E ′ ⊆ E ∑ ∏ f  ( σ | E ( v ) ) , v ∈ V σ : E �→{ 0 , 1 } where σ | E ( v ) is the assignment σ restricted to the set E ( v ) of incident edges of v , and f  is the E-O function.

. .. . . .. . . .. . . .. . . .. . . . . Instance is a graph. STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) Quantity to compute is an exponential sum over all possible assignments. Functions take assignments on adjacent edges/vertices as inputs. Vertices and edges are associated with some functions. Common features . Counting Problems . .. . . .. .. . .. . . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . .. .. . . .. . . 7 / 29

. .. . . .. . . .. . . .. . . .. . . . . Instance is a graph. STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) Quantity to compute is an exponential sum over all possible assignments. Functions take assignments on adjacent edges/vertices as inputs. Vertices and edges are associated with some functions. Common features . Counting Problems . .. . . .. .. . .. . . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . .. .. . . .. . . 7 / 29

. .. . . .. . . .. . . .. . . .. . . . . Instance is a graph. STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) Quantity to compute is an exponential sum over all possible assignments. Functions take assignments on adjacent edges/vertices as inputs. Vertices and edges are associated with some functions. Common features . Counting Problems . .. . . .. .. . .. . . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . .. .. . . .. . . 7 / 29

. . . . .. . . .. . . .. . . .. . . .. .. . . . STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) e expressive power is increasing in order. . . . .. . . specify where to put the functions and to sum over what assignments. Counting problems are often parameterized by constraint functions. Frameworks Frameworks Counting Problems . .. . .. .. .. . . .. . . . . . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. . 8 / 29 1 Graph Homomorphisms 2 Constraint Satisfaction Problems (#CSP) 3 Holant Problems

. . Counting Problems . .. . . .. . .. . . . .. . . .. . . Instance - signature grid . .. . STOC 2013 Complex Holant Heng Guo (CS, UW-Madison) Figure: A signature grid f 2 . f 4 f 3 f 1 . . f 1 . f 3 . f 1 . f 2 .. . . . .. . . .. . . .. . .. . . . .. . . .. . . . 9 / 29 .. .. . . .. . . .. . . . . .. . . .. . . .. A signature grid Ω = ( G , F , π ) consists of a graph G = ( V , E ) , where each vertex is labeled by a function f v ∈ F , and π : V → F is the labelling.