Locations of Zeros and Approximate Counting Chihao Zhang Shanghai - - PowerPoint PPT Presentation

Chihao Zhang

Locations of Zeros and Approximate Counting

Shanghai Jiao Tong University

• Apr. 11 2019

Generating Function

FIS(x) =

∞

∑

k=0

akxk = 1 + 4x + 2x2

FIS(1) =

n

∑

k=1

ak = # of independent sets #P-hard to evaluate (not easier than NP)

∀ε > 0, 1 − ε < ̂ F FIS(1) < 1 + ε

Approximation

Technique for Approximate Counting

Simulate a Markov chain on all independent sets

Markov Chain Monte Carlo “Sample from Gibbs Measure” Approximately compute the marginals of Gibbs measure using graph recursions Correlation Decay

Barvinok’s Method

Alexander Barvinok Brook Taylor

A bit of math…

Is TmG(1) a good approximation of G? (for sufficiently small m)

G = log FIS TmG(z) = First m terms of the Maclaurin series of G

( = G(0) + G′(0)z + G(2)(0) 2 z2 + G(3)(0) 3! z3 + … + G(m−1)(0) (m − 1)! zm−1 )

|TmG(t) − G(t)| ≤ poly(|G|) ⋅ (1 − γ)m Assume F(x) has roots α1, …, αn ∈ ℂ A simple calculation yields if t < min

i

|αi| 1 − γ = t mini |αi|

Question Remains…

How to evaluate TmG(t)?

For IS, trivially in |G|O(m)

up to m = O(log|G|)

• A. Barvinok

Mathematician: I don’t care

(nO(log n) is already a good notion for efficient computation)

Viresh Patel Guus Regts

Computer Scientists: We do care about poly-time

Patel & Regts (2017) : Can be done in |G| ⋅ 2O(m) time

The Message

(Many) generating functions are approximable in zero-free regions It is challenging to identify the location of zeros For partition functions of physical model, lots of work has been done…

Zero-free Region

Partition Function (Physics) A.Sokal… Generating Function (Math) D.Wagner, A.Barvinok… Graph Polynomial (CS)

C.N. Yang T.D. Lee

Lee-Yang Theorem (1952): The partition function of the Ising model has all of its complex roots on the 1-circle

Zero-free Region of IS

Cropped from “Note on the zero-free region of the hard-core model” by F . Bencs and P . Csikvári. (http://arxiv.org/abs/1807.08963)

“cardiac zero-free region” on trees

[Bezakova, Galanis, Goldberg, Štefankovič 2018]

It is NP-hard to approximate IS polynomial out of the zero-free region

Matchings

Matching Polynomial : FM(x) =

m

∑

k=0

mkxk

Holant Problem

FM(1) = #ℳ = ∑

σ∈{0,1}E ∏ v∈V

fv(σ), fv(σ) is the indicator function of the event σ chooses at most one of v′s incident edges

1 2 4 3

f2 f1 f3 f4

Holant : fv can be arbitrary {0,1}E(v) → ℂ This talk : fv is symmetric Holant(ℱ) : fv ∈ ℱ

Dichotomy Theorems

#P-complete PTIME

[Cai, Guo, Williams 2013]

Holant(ℱ) is either in P or #P-complete

Most problems are hard, unless it enjoys some good algebraic property

Approximate Counting

#ind. sets #matchings #edge covers

• We know a few tractable islands
• We know a few intractable islands

Approximation: FPTAS/FPRAS for the partition function

A dichotomy for approximating Holant(ℱ)?

Barvinok’s Method

Theorem (Barvinok 16; Patel & Regts 18 )

There exists an FPTAS to approximate FH(1) if the polynomial FH(x) is zero-free in a δ-strip containing [0,1] for some δ > 0 True for Matching (Heilmann-Lieb)

General Holant?

1

δ

ℂ

Asano Contraction

In a proof of the Lee-Yang theorem…

Holant Polynomial

Recall that FH(x) =

|E|

∑

i=0

bi ⋅ xi

We obtain the polynomial by gluing small polynomials

x(1)

1

x(2)

1

x(1)

2

x(2)

2

x(2)

3

x(1)

3

x1 x2 x3

F(x(1)

1 , x(2) 1 , x(1) 2 , x(2) 2 , x(1) 3 , x(2) 3 )

F(x1, x2, x3)

Asano’s contraction

Ruelle’s Bound

x1 x2

F(x1, x2)

x

F′(x)

Ruelle’s Theorem (Ruelle 71)

F(x1, x2) is zero-free in A F′(x) is zero-free in − ¯ A ⋅ ¯ A

Ruelle’s Theorem

F(x1, x2) is zero-free in A F′(x) is zero-free in − ¯ A ⋅ ¯ A

x(1)

1

x(2)

1

x(1)

2

x(2)

2

x(2)

3

x(1)

3

x1 x2 x3

Corollary

pf(x(1)

1 , x(2) 2 ) is zero-free in Hε

F(x1, x2, x3) is zero-free in δ-strip of [0,1]

f f f

F(x1, x2, x3)

Hε := {z ∈ ℂ : ℜz > − ε}

Local Polynomial

f = [f0, f1, …, fd] pf(z) =

d

∑

k=0 (

d k) ⋅ fk ⋅ zk Constraint Function Local Polynomial Identify the family of f such that pf is Hε zero-free

Second-Order Recurrence

f = [f0, f1, …, fd] afk + bfk+1 + cfk+2 = 0 ∀0 ≤ k ≤ d − 2 fk ≥ 0 & for some (a, b, c) ≠ (0,0,0)

• matchings and perfect matchings
• parity
• edge covers
• Fibonacci gates
• all ternary symmetric functions

Local Polynomial of Second-Order Recurrence

The local polynomial is of the form pf(z) = (p + qz)d + (r + sz)d Good if it is Hε zero-free Not much information when it is not

Holographic Reduction

A tweak of the local polynomial without changing the global value

pf p′

f

FH F′

H

FH(1) = F′

H(1)

Leslie Valiant

Holographic Reduction

• A many-to-many reduction between counting problems
• Central tool in establishing dichotomy results for exact counting

pf(z) = (p + qz)d + (r + sz)d ((pw + q) + (p − wq)z)

d + ((rw + s) + (r − ws)z) d

a free w

Results for SO Holant

In a joint work with C. Liao, H.Guo and P . Lu, we obtain approximation algorithm for SO Holant except

The later case is equivalent to approximating #PM

#PM: central open problem in approximate counting

[0,λ sin π d , λ2 sin 2π d , …, λi sin iπ d , …,0] [0,1,0,λ,0,…,0,λ

d − 2 2 ,0] if d even

[0,1,0,λ,0,…,0,λ

d − 1 2 ] if d odd

Approximating Holant(f ) on cubic graph is either in poly-time or equivalent to aprroximating #PM Ternary functions satisfy SO recurrence [0,1,0,λ] or [λ,0,1,0] #PM

• therwise

FPTAS/FPRAS A “Dichotomy”

Conclusion

Barvinok’s method is powerful and seems to be universal

Conjecture Uniqueness of Gibbs measure (Spatial mixing) implies absence of complex zero

We are more familiar with Gibbs measure… No idea how to prove absence of zero in general

Conclusion

independent sets q-coloring more Holant

Recent progress

• IS: tree is not the worst case (arxiv 1903.05462)
• Potts (q-coloring): matching the best uniqueness bound (Last week)
• some progress on asymmetric Holant