T wo-State Spin System graph G =( V , E ) 2 states {0,1} - - PowerPoint PPT Presentation

Approximate Counting

via Correlation Decay in Spin Systems

Yitong Yin Nanjing University

Joint work with Liang Li (Peking U) and Pinyan Lu (MSRA)

T wo-State Spin System

2 states {0,1} configuration σ : V → {0, 1} graph G=(V,E) contributions of local interactions: weight:

w(σ) =

• e∈E

(e) β γ 1

T wo-State Spin System

A = A0,0 A0,1 A1,0 A1,1

• =

β 1 1 γ

• 2 states {0,1}

configuration σ : V → {0, 1} graph G=(V,E) contributions of local interactions: weight:

w(σ) = β γ 1

• (u,v)∈E

Aσ(u),σ(v)

T wo-State Spin System

A = A0,0 A0,1 A1,0 A1,1

• =

β 1 1 γ

• 2 states {0,1}

graph G=(V,E) weight:

w(σ) =

• (u,v)∈E

Aσ(u),σ(v)

Gibbs measure:

µ(σ) = w(σ) ZA(G) ZA(G) =

• σ∈{0,1}V

w(σ)

partition function: configuration σ : V → {0, 1}

Partition Function

A = A0,0 A0,1 A1,0 A1,1

• =

β 1 1 γ

• 2 states {0,1}

graph G=(V,E)

ZA(G) =

• σ∈{0,1}V
• (u,v)∈E

Aσ(u),σ(v)

partition function:

β = 0, γ = 1 # independent set # vertex cover

configuration σ : V → {0, 1} weighted Boolean CSP with one symmetric relation

Approximate Counting

ZA(G) =

• σ∈{0,1}V
• (u,v)∈E

Aσ(u),σ(v)

partition function:

A = A0,0 A0,1 A1,0 A1,1

• =

β 1 1 γ

• fix

is a well-define computational problem poly-time computable if

βγ = 1 (β, γ) = (0, 0)

• r

#P-hard if otherwise Approximation!

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

• 0< , <1

= 1

uniqueness threshold

threshold achieved by heatbath random walk

1.11017

0 ≤ β, γ ≤ 1

β γ

ferromagnetic Ising Model ferromagnetic 2-state spin FPRAS [JS93] FPRAS [GJP03]

FPRAS [GJP03]

heat-bath

no FPRAS unless NP⊆RP [GJP03] anti- ferromagnetic

Goldberg-Jerrum-Paterson’03 Jerrum-Sinclair’93 [GJP03] [JS93]

• 1.11017
• Uniqueness Threshold

f(x) = βx + 1 x + γ d ˆ x = f(ˆ x) |f (ˆ x)| < 1 0 ≤ β < 1 < γ

for all d

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

• 0< , <1

= 1

uniqueness threshold

threshold achieved by heatbath random walk

1.11017

0 ≤ β, γ ≤ 1

β γ

Our Result

FPTAS

Marginal Distribution

weight:

w(σ) =

• (u,v)∈E

Aσ(u),σ(v)

Gibbs measure:

µ(σ) = w(σ) ZA(G) ZA(G) =

• σ∈{0,1}V
• (u,v)∈E

Aσ(u),σ(v) pv =

σ∼µ[σ(v) = 0]

pσΛ

v

=

σ∼µ[σ(v) = 0 | σ(Λ) = σΛ]

σΛ ∈ {0, 1}Λ

marginal distributions at vertex v:

Λ ⊂ V v ∈ Λ v Λ fixed free

Self-reduction

σΛ ∈ {0, 1}Λ V = {v1, v2, . . . , vn} σi : v1, v2, . . . , vi 1 =

n

• i=1

(1 − pσi−1

vi

) Λ ⊂ V (σn) (σn) = w(σn) Z(G) = γ|E| Z(G) Z(G) = γ|E| n

i=1(1 − pσi−1 vi

) pσΛ

v

=

σ [σ(v) = 0 | σ(Λ) = σΛ]

=

σ [σ : v1, . . . , vn 1]

=

n

• i=1
• σ [σ(vi) = 1 | σ : v1, . . . , vi−1 1]

(Jerrum-Valiant-Vazirani)

Correlation Decay

“strong spatial mixing” in [Weitz’06] B

∂B

G v

∀σ∂B, τ∂B ∈ {0, 1}∂B Λ

• σ [σ(v) = 0 | σ∂B] ≈

σ [σ(v) = 0 | τ∂B]

• σ [σ(v) = 0 | σ∂B, σΛ] ≈

σ [σ(v) = 0 | τ∂B, σΛ]

t error < exp (-t) exponential correlation decay

pσΛ

v

≈

v T

Recursion for Tree

RσΛ

T

= pσΛ

v

1 − pσΛ

v

RσΛ

T = d

• i=1

βRσΛ

Ti + 1

RσΛ

Ti + γ

w(σT : v 0) w(σT : v 1) = d

i=1 (βw(σTi : vi 0) + w(σTi : vi 1))

d

i=1 (w(σTi : vi 0) + γw(σTi : vi 1))

= σ∼µ|σΛ[σ(v) = 0] σ∼µ|σΛ[σ(v) = 1] v1 v2 vd T1 Td σΛ ∈ {0, 1}Λ Λ ⊂ V

1

Self-Avoiding Walk Tree

due to Weitz (2006)

1 2 3 4 5 6 2 4 4 1 4 4 6 5 5 6 4 3 3 5 6 5 6 4 1

G=(V,E) v

if cycle closing > cycle starting if cycle closing < cycle starting RσΛ

G,v =

pσΛ

v

1 − pσΛ

v

T = T(G, v) RσΛ

G,v = RσΛ T

Weitz (2006) 6 6 6 6 6 σΛ

error= 1

Approximation Algorithm

2 4 4 1 4 4 6 5 5 6 4 3 3 5 6 5 6 4 1 T = T(G, v) 4 4 4 4 6 5

exponential correlation decay:

poly-time on O(1)-degree graphs

preserve degrees

error decreases exponentially in depth ∂B RσΛ∩B

− RσΛ∩B

• = (− ∂B)

Correlation Decay!

T echnique

• amortized analysis of decay:
• the potential method;
• Computationally Efficient Correlation Decay:

dealing with unbounded-degree graphs;

Uniqueness Threshold

• Td infinite (d+1)-regular tree

(Bethe lattice, Cayley tree) f(x) = βx + 1 x + γ d ˆ x = f(ˆ x) |f (ˆ x)| < 1

Uniqueness of Gibbs measure

• 1.11017
• Uniqueness Threshold

f(x) = βx + 1 x + γ d ˆ x = f(ˆ x) |f (ˆ x)| < 1 0 ≤ β < 1 < γ

for all d

1

Correlation Decay

2 4 4 1 4 4 6 5 5 6 4 3 3 5 6 5 6 4 1 T = T(G, v) 4 4 4 4 6 5 6

RσΛ

T

=

d

• i=1

βRσΛ

Ti + 1

RσΛ

Ti + γ

βγ < 1 anti-ferromagnetic monotonically decreasing ∆

= f(RσΛ

T1 , . . . , RσΛ Td )

R ∈ [0, ∞)

upper bound = f(lower bounds) lower bound = f(upper bounds)

δ = R − R

v ∈ Λ fixed to be 0 v ∈ Λ fixed to be 1

lower=upper= ∞ lower=upper= 0

δ = (−Ω( ∆))

Goal:

v T v1 v2 vd T1 Td Rv ≤ RσΛ

T

≤ Rv + δv

[Rv1, Rv1 + δv1] [Rvd, Rvd + δvd]

f(x) = βx + 1 x + γ d

x1 xd

x ∈ [Rv, Rv + δv]

Cheating:

f(x1, . . . , xd) =

d

• i=1

βxi + 1 xi + γ |f (x)| < 1

we do not always have

δ Φ(x) ≤ α · δ Φ(x)

|f (x)|Φ(x) Φ(f(x))

< 1 α = Φ(x) = x

D+1 2D (βx + 1) +1

v T v1 v2 vd T1 Td

[Rv1, Rv1 + δv1] [Rvd, Rvd + δvd]

x1 xd

x ∈ [Rv, Rv + δv]

f(x1, . . . , xd) =

d

• i=1

βxi + 1 xi + γ Φ(x) = x

D+1 2D (βx + 1) +1

Mean Value Thms

α(d, x) = d(1 − βγ)x

D+1 2D (βx + 1) d(D−1) 2D

(x + γ)1+ d(D−1)

2D

• β
• βx+1

x+γ

d + 1

• =

Φ(x) Φ(f(x))|f (x)|

Jensen’s Inequality

v T v1 v2 vd T1 Td

[Rv1, Rv1 + δv1] [Rvd, Rvd + δvd]

x1 xd

x ∈ [Rv, Rv + δv]

Φ(x) = x

D+1 2D (βx + 1) +1

α(d, x) = d(1 − βγ)x

D+1 2D (βx + 1) d(D−1) 2D

(x + γ)1+ d(D−1)

2D

• β
• βx+1

x+γ

d + 1

• if

α(d, x) = d(1 − βγ)x

D+1 2D (βx + 1) d(D−1) 2D

(x + γ)1+ d(D−1)

2D

• β
• βx+1

x+γ

d + 1

v T v1 v2 vd T1 Td Rv ≤ RσΛ

T

≤ Rv + δv

[Rv1, Rv1 + δv1] [Rvd, Rvd + δvd]

x1 xd

x ∈ [Rv, Rv + δv]

1 2 4 4 1 4 4 6 5 5 6 4 3 3 5 6 5 6 4 1 T = T(G, v) 4 4 4 4 6 5 6 R ∈ [0, ∞)

δ = R − R

δ = (−Ω( ∆))

Computationally Efficient Correlation Decay

v T v1 v2 vd T1 Td

[Rv1, Rv1 + δv1] [Rvd, Rvd + δvd]

x1 xd

x ∈ [Rv, Rv + δv]

α(d, x) = d(1 − βγ)x

D+1 2D (βx + 1) d(D−1) 2D

(x + γ)1+ d(D−1)

2D

• β
• βx+1

x+γ

d + 1

• for some

Computationally Efficient Correlation Decay

v T v1 v2 vd T1 Td

[Rv1, Rv1 + δv1] [Rvd, Rvd + δvd]

x1 xd

x ∈ [Rv, Rv + δv]

α(d, x)

for some

for small

• ne-step recursion decays

for large

• ne-step recursion decays

behaves like steps!

Computationally Efficient Correlation Decay

for small

• ne-step recursion decays

for large

• ne-step recursion decays

steps!

v T v1 v2 vd T1 Td v v1 v2 vd “span” d leaves M-ary

• ld metric

new metric exponential decay in behaves like

Computationally Efficient Correlation Decay

v T v1 v2 vd T1 Td

• ld metric

new metric exponential decay in

• nly moderately grows

v v1 v2 vd “span” d leaves M-ary distance = O(log n) 1/poly-precision in poly-time

Conclusion

• FPTAS for 2-state spin system up to

uniqueness threshold (conjectured to be the boundary of approximability).

• Computationally Efficient Correlation

Decay (the first time that Correlation Decay is used to deal with arbitrary graphs).

Thank You!