Lee-Yang Theorems and the Complexity of Computing Averages Piyush - - PowerPoint PPT Presentation

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)

J J + − +

w(σ) = λ2J2

Weight of σ: w(σ) = λ#(+)J#(+,−) Gibbs distribution: π(σ) = 1

Zw(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)

J J + − +

w(σ) = λ2J2

Weight of σ: w(σ) = λ#(+)J#(+,−) Gibbs distribution: π(σ) = 1

Zw(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 x C(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 x C(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 x C(x)

Circuit-Eval Given C and x, is C(x) = True? Efficient algorithm Complexity class P

⊆

Circuit-SAT Given C, is there an x such that C(x) = True? Efficient checking No efficient search! 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 x C(x)

Circuit-Eval Given C and x, is C(x) = True? Efficient algorithm Complexity class P

⊆

Circuit-SAT Given C, is there an x such that C(x) = True? Efficient checking No efficient search! Complexity class NP

⊆

#Circuit-SAT Given C, how many x are there such that C(x) = True? Efficient checking No efficient counting! 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 Solves A efficiently Construct algorithm β Solves H efficiently Subroutine 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 Z(J, λ) =

• σ∈{+,−}V

J#(+,−)λ#(+) Interpolation View Z(J, λ) = n

k=1 αkλk as a polynomial in λ (here n = |V |)

Show that coefficients αk encode the solution to a #P-hard problem (e.g. #Max-Cut) Find the coefficients αk using polynomial interpolation

12

Proving #P-hardness: Partition functions

Recall that Z(J, λ) =

• σ∈{+,−}V

J#(+,−)λ#(+) Interpolation View Z(J, λ) = n

k=1 αkλk as a polynomial in λ (here n = |V |)

Show that coefficients αk encode the solution to a #P-hard problem (e.g. #Max-Cut) Find the coefficients αk 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 µ(J, λ) =

• σ #(+)w(σ)

Z(J, λ) =

• σ # (+) λ#(+)J# of cut edges

Z(J, λ) = λZ′ Z , where Z′ =

∂ ∂λZ(J, λ)

13

Proving #P-hardness: Magnetization

The magnetization µ(J, λ) can be written as µ(J, λ) =

• σ #(+)w(σ)

Z(J, λ) =

• σ # (+) λ#(+)J# of cut edges

Z(J, λ) = λZ′ Z , where Z′ =

∂ ∂λZ(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 µ(J, λ) =

• σ #(+)w(σ)

Z(J, λ) =

• σ # (+) λ#(+)J# of cut edges

Z(J, λ) = λZ′ Z , where Z′ =

∂ ∂λZ(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 µ(J, λ) =

• σ #(+)w(σ)

Z(J, λ) =

• σ # (+) λ#(+)J# of cut edges

Z(J, λ) = λZ′ Z , where Z′ =

∂ ∂λZ(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 µ(J, λ) =

• σ #(+)w(σ)

Z(J, λ) =

• σ # (+) λ#(+)J# of cut edges

Z(J, λ) = λZ′ Z , where Z′ =

∂ ∂λZ(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 µ(J, λ) =

• σ #(+)w(σ)

Z(J, λ) =

• σ # (+) λ#(+)J# of cut edges

Z(J, λ) = λZ′ Z , where Z′ =

∂ ∂λZ(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

• But. . .

Cannot interpolate p(x)

q(x) when p(x) and q(x) share common factors!

13

Rational interpolation and #P-hardness

Rational Interpolation [Macon and Dupree, 1962]

Suppose R(x) = p(x)

q(x) where deg (p(x)) = deg (q(x)) = n.

If gcd(p(x), q(x)) = 1 then p(x) and q(x) can be determined efficiently from 2n + 2 evaluations of R

14

Rational interpolation and #P-hardness

Rational Interpolation [Macon and Dupree, 1962]

Suppose R(x) = p(x)

q(x) where deg (p(x)) = deg (q(x)) = n.

If gcd(p(x), q(x)) = 1 then p(x) and q(x) can be determined efficiently from 2n + 2 evaluations of R We had µ(J, λ) = λZ′ Z Thus, sufficient to prove that gcd (Z′(J, λ), Z(J, λ)) = 1 to show #P-hardness

◮ Or, equivalently, that Z(J, λ) and Z′(J, λ) have no common complex zeros 14

Rational interpolation and #P-hardness (contd. . . )

Conclusion

Z, Z′ have no common zeros

= ⇒

magnetization µ is as hard to compute as Z (and hence #P-hard) The antecedent is not true for disconnected graphs (and for J = 1)—but does suggest a close scrutiny of the zeros of Z

◮ e.g., ZG ˙

∪G = Z2 G, so that ZG ˙ ∪G and Z′ G ˙ ∪G have lots of common zeros

15

Zeros of the partition function: The Lee-Yang theorem

Theorem [Lee and Yang, 1952]

When 0 < J ≤ 1, the zeros of Z(J, z) satisfy |z| = 1.

X Y O

|z| = 1

Lee-Yang theorem: Zeros of Z Gauss-Lucas lemma: Z′(J, z) = 0 implies |z| ≤ 1

◮ . . . but this is not sufficient for showing

that Z and Z′ have no common zeros Original motivation for the theorem was studying phase transitions in the Ising model

16

An extension of the Lee-Yang theorem

Theorem (Sinclair, S.)

For a connected graph with 0 < J < 1, the zeros of Z′(J, z) =

∂ ∂zZ(J, z) satisfy

|z| < 1. In particular, gcd (Z(J, z), Z′(J, z)) = 1.

• Comm. Math. Phys. (2014)

17

An extension of the Lee-Yang theorem

Theorem (Sinclair, S.)

For a connected graph with 0 < J < 1, the zeros of Z′(J, z) =

∂ ∂zZ(J, z) satisfy

|z| < 1. In particular, gcd (Z(J, z), Z′(J, z)) = 1.

X Y O

|z| = 1

Lee-Yang theorem: Zeros of Z

• Comm. Math. Phys. (2014)

17

An extension of the Lee-Yang theorem

Theorem (Sinclair, S.)

For a connected graph with 0 < J < 1, the zeros of Z′(J, z) =

∂ ∂zZ(J, z) satisfy

|z| < 1. In particular, gcd (Z(J, z), Z′(J, z)) = 1.

X Y O

|z| = 1

Lee-Yang theorem: Zeros of Z

X Y O

|z| < 1

Our theorem: Zeros of Z′

• Comm. Math. Phys. (2014)

17

Proving Lee-Yang theorems

Conclusion 18

Multivariate Lee-Yang theorems

Consider the scenario where the vertex activities can vary across vertices: w(σ) = J#(+,−)

• v:σ(v)=+

zv

19

Multivariate Lee-Yang theorems

Consider the scenario where the vertex activities can vary across vertices: w(σ) = J#(+,−)

• v:σ(v)=+

zv We then define the following magnetization operator: D :=

• v

zv ∂ ∂zv so that DZ(J, z1, z2, . . . zn)|z1=z2=...=zn=x = z ∂ ∂z Z(J, z)|z=x The magnetization itself is given by µ(J, z1, z2, . . . zn) = DZ(J, z1, z2, . . . zn) Z(J, z1, z2, . . . zn) Recall that this agrees with the univariate case: µ(J, λ) = λZ′

Z

19

Multivariate Lee-Yang theorems (contd. . . )

Theorem [Lee and Yang, 1952; Asano, 1970]

Suppose 0 < J ≤ 1, and |zi| > 1 for 1 ≤ i ≤ n. Then, Z(J, z1, z2, . . . , zn) = 0. The univariate Lee-Yang theorem follows by setting zi = z for all i

20

Multivariate Lee-Yang theorems (contd. . . )

Theorem [Lee and Yang, 1952; Asano, 1970]

Suppose 0 < J ≤ 1, and |zi| > 1 for 1 ≤ i ≤ n. Then, Z(J, z1, z2, . . . , zn) = 0. The univariate Lee-Yang theorem follows by setting zi = z for all i

Our theorem

On a connected graph, the conditions 0 < J < 1 and |zi| ≥ 1 for all 1 ≤ i ≤ n imply that DZ(J, z1, z2, . . . , zn) = 0. Our univariate theorem follows from the above by setting zi = z for all i

20

Proof Sketch

Theorem

On a connected graph, the conditions 0 < J < 1 and |zi| ≥ 1 for all 1 ≤ i ≤ n imply that DZ(J, z1, z2, . . . , zn) = 0. Say that Z(J, z1, z2, . . . , zn) has property G if it satisfies the conclusion of the above theorem The proof proceeds by induction:

◮ Each step maintains the connectedness of the graph, and the property G

for its partition function

◮ Asano’s proof of the Lee-Yang theorem as a warm-up 21

Asano’s Proof of the Lee-Yang theorem

ZG has property A (denoted Z ∈ A) if 0 < J ≤ 1 and |zi| > 1 for all i = ⇒ ZG(J, z1, z2, . . . , zn) = 0 Note that if G and H are disjoint graphs with ZG, ZH ∈ A, then ZG ˙

∪H = ZGZH ∈ A

22

Asano’s Proof of the Lee-Yang theorem

ZG has property A (denoted Z ∈ A) if 0 < J ≤ 1 and |zi| > 1 for all i = ⇒ ZG(J, z1, z2, . . . , zn) = 0 Note that if G and H are disjoint graphs with ZG, ZH ∈ A, then ZG ˙

∪H = ZGZH ∈ A

Single edge: z1z2 + J(z1 + z2) + 1 ∈ A

Proof

z1z2 + J(z1 + z2) + 1 = 0 = ⇒ |z2| =

• 1 + Jz1

J + z1

• For 0 < J ≤ 1, this is a Möbius transform mapping the exterior of the unit disk to

its interior. Thus, if |z1| > 1 then |z2| ≤ 1

22

Asano’s proof of the Lee-Yang theorem (contd.)

Merging vertices: Az1z2 + Bz1 + Cz2 + D ∈ A = ⇒ Az + D ∈ A

Proof

Let z3, z4, . . . zn be fixed so that |zi| > 1 for i ≥ 3. Az1z2 + Bz1 + Cz2 + D ∈ A = ⇒ Az1z2 + Bz1 + Cz2 + D = 0, for |z1| , |z2| > 1 = ⇒ Az2 + Bz + Cz + D = 0 for |z| > 1 = ⇒

• D

A

• ≤ 1 (Product of zeros)

Thus, Az + D = 0 = ⇒ |z| = |D| / |A| ≤ 1

• 23

Asano’s proof: Putting it together

Repeated use of above operations implies that G ∈ A for all graphs G Example:

24

Asano’s proof: Putting it together

Repeated use of above operations implies that G ∈ A for all graphs G Example: ∈ A Single edges and disjoint products

24

Asano’s proof: Putting it together

Repeated use of above operations implies that G ∈ A for all graphs G Example: ∈ A = ⇒ ∈ A Single edges and disjoint products Merge

24

Asano’s proof: Putting it together

Repeated use of above operations implies that G ∈ A for all graphs G Example: ∈ A = ⇒ ∈ A Single edges and disjoint products Merge

24

Asano’s proof: Putting it together

Repeated use of above operations implies that G ∈ A for all graphs G Example: ∈ A = ⇒ ∈ A Single edges and disjoint products Merge

24

Asano’s proof: Putting it together

Repeated use of above operations implies that G ∈ A for all graphs G Example: ∈ A = ⇒ ∈ A Single edges and disjoint products Merge

24

Asano’s proof: Putting it together

Repeated use of above operations implies that G ∈ A for all graphs G Example: ∈ A = ⇒ ∈ A = ⇒ ∈ A Single edges and disjoint products Merge

24

Proof of our theorem: In Asano’s footsteps

ZG has property G (denoted Z ∈ G) if 0 < J < 1 and |zi| ≥ 1 for all i = ⇒ DZG(J, z1, z2, . . . , zn) = 0 (Recall that D =

v∈V zv ∂ ∂zv )

25

Proof of our theorem: In Asano’s footsteps

ZG has property G (denoted Z ∈ G) if 0 < J < 1 and |zi| ≥ 1 for all i = ⇒ DZG(J, z1, z2, . . . , zn) = 0 (Recall that D =

v∈V zv ∂ ∂zv )

Single edge: z1z2 + J(z1 + z2) + 1 ∈ G

◮ DZ = 2z1z2 + J(z1 + z2) = 0 implies that

1 |z1| + 1 |z2| ≥ 2 J : contradiction

But things become too complicated when we try to merge graphs

25

Proof of our theorem: The two inductive steps

Contracting vertices: Az1z2 + Cz1 + Dz2 + B ∈ G = ⇒ Az2 + B ∈ G

• 26

Proof of our theorem: The two inductive steps

Contracting vertices: Az1z2 + Cz1 + Dz2 + B ∈ G = ⇒ Az2 + B ∈ G Adding a single new edge and a new vertex: Az1 + B ∈ G = ⇒ Az2

1(J + z) + B(1 + Jz) ∈ G

• 26

Proof of our theorem: The two inductive steps

Contracting vertices: Az1z2 + Cz1 + Dz2 + B ∈ G = ⇒ Az2 + B ∈ G Adding a single new edge and a new vertex: Az1 + B ∈ G = ⇒ Az2

1(J + z) + B(1 + Jz) ∈ G

Unlike Asano’s proof, each of the above steps requires a somewhat technical argument relying on a correlation inequality due to Newman [1974] Another technical problem is the change in degree of the activities

• 26

Finishing the proof

Our proof works via an induction on the number of edges, using the above

• perations to construct the graph

◮ See paper for details

. . . or arXiv:1407.5991 for a shorter, more analytic proof of a weaker (but sufficient) version

27

Finishing the proof

Our proof works via an induction on the number of edges, using the above

• perations to construct the graph

◮ See paper for details

. . . or arXiv:1407.5991 for a shorter, more analytic proof of a weaker (but sufficient) version The main theorem then immediately implies the hardness result

Theorem

For any 0 < J < 1 and λ = 1 computing the magnetization of the Ising model is #P-hard. This is true even for bounded degree graphs (with degree ≥ 4)

27

Beyond the Ising model

(If time permits. . . )

It probably won’t Zeros and phase transitions 28

The Monomer-dimer model

Graph G = (V, E). Configurations σ are matchings (“dimer coverings”) of G Monomer Activity λ > 0 models the tendency of vertices to be monomers Dimer activities γe > 0 model the tendency of an edge to be a dimer

γ1 γ2 γ3 × ×

• w(σ) = λγ1

Weight of σ: w(σ) = λ#(•)

e∈σ γe (“Gibbs distribution”)

29

The Monomer-dimer model

Graph G = (V, E). Configurations σ are matchings (“dimer coverings”) of G Monomer Activity λ > 0 models the tendency of vertices to be monomers Dimer activities γe > 0 model the tendency of an edge to be a dimer

γ1 γ2 γ3 × ×

• w(σ) = λγ1

Weight of σ: w(σ) = λ#(•)

e∈σ γe (“Gibbs distribution”)

Partition function: Z(γ, λ) =

σ w(σ)

Average quantity: u(γ, λ) is the average number of monomers

29

Monomer-dimer model

As in the case of the Ising model, the average monomer number u(γ, λ) can be expressed in terms of derivatives u(γ, λ) = λZ′ Z

Conclusion

Again, if we could show that Z does not have repeated zeros, #P-hardness of computing u will follow

30

Zeros of the partition function: the Heilmann-Lieb theorem

Theorem [Heilmann and Lieb, 1972]

For a Hamiltonian graph with any positive collection of edge weights γ, the zeros

• f Z(γ, z) are distinct. Thus, gcd (Z(γ, z), Z′(γ, z)) = 1.

They showed further that connected graphs can have repeated zeros Again, it follows that computing the average monomer number is as hard as computing the monomer-dimer partition function on Hamiltonian graphs

31

Zeros of the partition function: the Heilmann-Lieb theorem

Theorem [Heilmann and Lieb, 1972]

For a Hamiltonian graph with any positive collection of edge weights γ, the zeros

• f Z(γ, z) are distinct. Thus, gcd (Z(γ, z), Z′(γ, z)) = 1.

They showed further that connected graphs can have repeated zeros Again, it follows that computing the average monomer number is as hard as computing the monomer-dimer partition function on Hamiltonian graphs

Theorem

For γe ∈ {1, 2, 3}, computing the monomer-dimer partition function on Hamiltonian graphs of degree at least 5 is #P-hard.

31

Proof Sketch

Reduce from #Monotone 2-SAT, using a variation of Valiant’s reduction to the permanent Valiant’s reduction comprises

◮ (relatively simple) gadgets for clauses and variables in the formula, and ◮ somewhat more elaborate XOR gadget used to connect these

Our first step is to introduce a common variable to help connect these gadgets together φ =

k

• i=1

(yi1 ∨ yi2) − →

k

• i=1

(τ ∨ yi1 ∨ yi2)

32

Proof sketch

The strategy is to find Hamiltonian paths in the individual gadgets and connect them through the variable gadget for τ This (mostly) works except for the XOR gadget, which we need to modify

◮ The −1 weight edges in the XOR gadget need to be replaced by a chain

• f edges, and hence must be forced to be part of the Hamiltonian path

a b c

a c d b

2 −1 3 −1 2

Variable gadget Clause gadget XOR gadget

33

Proof sketch (contd. . . )

With the above technical modifications to Valiant’s reduction, we are able to perform the reduction to ensure that the output graph contains a Hamiltonian path A little more work yields the following theorem

Theorem

For any fixed λ > 0, computing the average monomer number with edge weights from {1, 2, 3} is #P-hard. This is true even for bounded degree graphs of degree ∆ ≥ 5.

34

Comments

(Strong) versions of Lee-Yang theorems immediately yield complexity results for average quantities

◮ But for several spin systems we don’t know any: e.g., for independent

sets, the best known result is only for claw free graphs [Chudnovsky and Seymour, 2007] Stability results for the independence polynomial (hard core lattice gas) have very strong connections to the Lovász local lemma [Scott and Sokal, 2005] Other average quantities? Susceptibility (variance of the magnetization)?

35

Thank you! Questions?

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