Lee-Yang Theorems and the Complexity of Computing Averages Piyush Srivastava Caltech Joint work with Alistair Sinclair Western States Mathematical Physics Meeting February 17, 2015 Journal ref. : Communications in Mathematical Physics, 329 (3), 827–858, 2014.
Outline Background: Ising model and computational complexity Zeros of polynomials: From phase transitions to complexity Extended Lee-Yang type theorems Beyond the Ising model (if time permits. . . ) 1
Background: The Ising model 2
The Ising model [Ising, 1925] Classical statistical physics model of magnetism in bulk Graph G = ( V, E ) . Configuration σ assigns { + , −} spin to each vertex in V Edge activity 0 < J < 1 models local interactions Vertex activity λ > 0 models magnetic field ( λ > 1 favors + spins) + w ( σ ) = λ 2 J 2 J + − J w ( σ ) = λ #(+) J #(+ , − ) Weight of σ : π ( σ ) = 1 Gibbs distribution: Z w ( σ ) 3
The Ising model [Ising, 1925] Classical statistical physics model of magnetism in bulk Graph G = ( V, E ) . Configuration σ assigns { + , −} spin to each vertex in V Edge activity 0 < J < 1 models local interactions Vertex activity λ > 0 models magnetic field ( λ > 1 favors + spins) + w ( σ ) = λ 2 J 2 J + − J w ( σ ) = λ #(+) J #(+ , − ) Weight of σ : π ( σ ) = 1 Gibbs distribution: Z w ( σ ) Partition function: Z ( J, λ ) = � σ w ( σ ) Magnetization: µ ( J, λ ) is the average number of ‘+’ spins 3
The Ising model: Computational problems Question 1 Can we get efficient algorithms for computing averages like the magnetization? Failing that. . . Question 2 Can we prove that these problems are computationally hard? 4
Proving hardness: Computational complexity 5
Computational complexity: Complexity classes Complexity theory: classifies problems based on algorithmic “hardness” Consider a given logic circuit C C ( x ) x Efficiency Efficient algorithm ≡ Runs in time polynomial in input size 6
Computational complexity: Complexity classes Complexity theory: classifies problems based on algorithmic “hardness” Consider a given logic circuit C C ( x ) x Circuit-Eval Given C and x , is C ( x ) = True ? Efficient algorithm Complexity class P Efficiency Efficient algorithm ≡ Runs in time polynomial in input size 6
Computational complexity: Complexity classes Complexity theory: classifies problems based on algorithmic “hardness” Consider a given logic circuit C C ( x ) x Circuit-Eval Circuit-SAT Given C and x , is Given C , is there an x such that C ( x ) = True ? C ( x ) = True ? ⊆ Efficient checking Efficient algorithm No efficient search! Complexity class P Complexity class NP Efficiency Efficient algorithm ≡ Runs in time polynomial in input size 6
Computational complexity: Complexity classes Complexity theory: classifies problems based on algorithmic “hardness” Consider a given logic circuit C C ( x ) x Circuit-Eval Circuit-SAT #Circuit-SAT Given C and x , is Given C , is there an Given C , how many x x such that are there such that C ( x ) = True ? C ( x ) = True ? C ( x ) = True ? ⊆ ⊆ Efficient checking Efficient checking Efficient algorithm No efficient counting! No efficient search! Complexity class P Complexity class NP Complexity class #P Efficiency Efficient algorithm ≡ Runs in time polynomial in input size 6
Computational complexity: Hard problems Definition A problem A is hard for a class C if an efficient algorithm for A implies efficient algorithms for all problems in C . 7
Computational complexity: Hard problems Definition A problem A is hard for a class C if an efficient algorithm for A implies efficient algorithms for all problems in C . Theorem [Cook 1971; Levin 1973; Valiant 1979] Circuit-SAT is hard for NP. #Circuit-SAT is hard for #P. Many natural optimization and counting problems have been proved hard [Karp, 1972; Valiant, 1979, . . . ] ◮ . . . Traveling salesperson problem, finding satisfying assignments of a Boolean formula, finding maximum cuts in a graph, counting dimer coverings, etc. P � = NP conjecture Efficient algorithms do not exist for these hard problems One of the most important open problems in mathematics and computer science 7
Computational complexity: Proving hardness How do we prove a problem A is hard for a class (say #P)? Reductions: Start with a problem H known to be hard for # P Hypothetical algorithm α Solves A efficiently 8
Computational complexity: Proving hardness How do we prove a problem A is hard for a class (say #P)? Reductions: Start with a problem H known to be hard for # P Hypothetical algorithm α Subroutine Construct algorithm β Solves A efficiently Solves H efficiently calls Efficient algorithm α for A = ⇒ Efficient algorithm β for H ⇓ H is hard = ⇒ A is hard 8
Computational complexity and spin systems Partition Functions: Exact Computation Ising model: #P-hard for any fixed λ > 0 and 0 < J < 1 Extensive theory on the classification of partition functions of various spin systems based on complexity [e.g., Cai et al., 2010] Averages: Exact Computation Unlike the rich theory for the case of partition functions, not much was known ◮ Ising model: Computing magnetization is trivial for λ = 1 (spins are symmetric, so magnetization = n/ 2 ). Other λ ? ◮ Other spin systems: ? 9
Complexity of averages: Our results Ising model Theorem For any fixed λ � = 1 and 0 < J < 1 computing the magnetization of the Ising model is #P-hard. This is true even for bounded degree graphs (with degree ≥ 4 ). Comm. Math. Phys. (2014) 10
Complexity of averages: Our results Ising model Theorem For any fixed λ � = 1 and 0 < J < 1 computing the magnetization of the Ising model is #P-hard. This is true even for bounded degree graphs (with degree ≥ 4 ). Monomer-dimer model Theorem For any fixed λ > 0 , computing the average monomer number in the monomer-dimer model with edge weights from the set { 1 , 2 , 3 } is #P-hard. This is true even for bounded degree graphs (with degree ≥ 5 ). Comm. Math. Phys. (2014) 10
Complexity of averages: Our results Ising model Theorem For any fixed λ � = 1 and 0 < J < 1 computing the magnetization of the Ising model is #P-hard. This is true even for bounded degree graphs (with degree ≥ 4 ). Monomer-dimer model Theorem For any fixed λ > 0 , computing the average monomer number in the monomer-dimer model with edge weights from the set { 1 , 2 , 3 } is #P-hard. This is true even for bounded degree graphs (with degree ≥ 5 ). Comm. Math. Phys. (2014) 10
Complexity of averages and Zeros of polynomials 11
Proving #P-hardness: Partition functions Recall that � J #(+ , − ) λ #(+) Z ( J, λ ) = σ ∈{ + , −} V Interpolation k =1 α k λ k as a polynomial in λ (here n = | V | ) View Z ( J, λ ) = � n Show that coefficients α k encode the solution to a #P-hard problem (e.g. # Max-Cut ) Find the coefficients α k using polynomial interpolation 12
Proving #P-hardness: Partition functions Recall that � J #(+ , − ) λ #(+) Z ( J, λ ) = σ ∈{ + , −} V Interpolation k =1 α k λ k as a polynomial in λ (here n = | V | ) View Z ( J, λ ) = � n Show that coefficients α k encode the solution to a #P-hard problem (e.g. # Max-Cut ) Find the coefficients α k using polynomial interpolation Shows that computing Z ( J, λ ) is hard—at least when λ is part of the input 12
Proving #P-hardness: Magnetization The magnetization µ ( J, λ ) can be written as σ # (+) λ #(+) J # of cut edges = λZ ′ � σ #(+) w ( σ ) � µ ( J, λ ) = = Z , Z ( J, λ ) Z ( J, λ ) where Z ′ = ∂ ∂λ Z ( J, λ ) 13
Proving #P-hardness: Magnetization The magnetization µ ( J, λ ) can be written as σ # (+) λ #(+) J # of cut edges = λZ ′ � σ #(+) w ( σ ) � µ ( J, λ ) = = Z , Z ( J, λ ) Z ( J, λ ) where Z ′ = ∂ ∂λ Z ( J, λ ) Interpolation View µ ( J, λ ) as a rational function in λ The coefficients encode the solution to a #P-hard problem Find the coefficients of Z (and Z ′ ) using rational interpolation 13
Proving #P-hardness: Magnetization The magnetization µ ( J, λ ) can be written as σ # (+) λ #(+) J # of cut edges = λZ ′ � σ #(+) w ( σ ) � µ ( J, λ ) = = Z , Z ( J, λ ) Z ( J, λ ) where Z ′ = ∂ ∂λ Z ( J, λ ) Interpolation � View µ ( J, λ ) as a rational function in λ The coefficients encode the solution to a #P-hard problem Find the coefficients of Z (and Z ′ ) using rational interpolation 13
Proving #P-hardness: Magnetization The magnetization µ ( J, λ ) can be written as σ # (+) λ #(+) J # of cut edges = λZ ′ � σ #(+) w ( σ ) � µ ( J, λ ) = = Z , Z ( J, λ ) Z ( J, λ ) where Z ′ = ∂ ∂λ Z ( J, λ ) Interpolation � View µ ( J, λ ) as a rational function in λ � The coefficients encode the solution to a #P-hard problem Find the coefficients of Z (and Z ′ ) using rational interpolation 13
Proving #P-hardness: Magnetization The magnetization µ ( J, λ ) can be written as σ # (+) λ #(+) J # of cut edges = λZ ′ � σ #(+) w ( σ ) � µ ( J, λ ) = = Z , Z ( J, λ ) Z ( J, λ ) where Z ′ = ∂ ∂λ Z ( J, λ ) Interpolation � View µ ( J, λ ) as a rational function in λ � The coefficients encode the solution to a #P-hard problem ? Find the coefficients of Z (and Z ′ ) using rational interpolation 13
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