Complexity of Counting (Computer Science) Can we efficiently count, - - PowerPoint PPT Presentation

INAPPROXIMABILITY FOR ANTIFERROMAGNETIC SPIN SYSTEMS

IN THE

TREE NON-UNIQUENESS REGION

Andreas Galanis (Oxford) Daniel Štefankoviˇ c (Rochester) Eric Vigoda (Georgia Tech) Cargese ’14, August 29

OVERVIEW

Complexity of Counting

(Computer Science) Can we efficiently count, e.g.: # colorings? # independent sets? Connections between: Phase transitions in the infinite ∆-regular tree T∆ and Computational complexity of approximating the partition function

• n graphs of maximum degree ∆.

MOTIVATING EXAMPLE: THE HARD-CORE MODEL Hard-core model: Lattice gas model For a graph G = (V, E), let I(G) = set of independent sets of G For activity λ > 0, I ∈ I(G) has weight λ|I| Partition function: Z := ZG(λ) =

• I∈I(G)

λ|I| Gibbs Distribution: for I ∈ I(G), µ(I) = λ|I| Z When λ = 1 : Z = |I(G)| = number of independent sets in G.

COMPUTING THE PARTITION FUNCTION Partition function: Z := ZG(λ) =

• independent set I

λ|I| #HARD-CORE(λ): For input G = (V, E), compute Z. Z is typically exponential in |V|. Goal: compute Z in time polynomial in |V|. [Valiant ’79, Greenhill’00]: Exact computation of #INDSETS is #P-complete, even for graphs of max degree ∆ = 3. Can we approximate Z on graphs of max degree ∆? As λ ↑, #HARD-CORE(λ) becomes computationally harder (more weight on the max independent set)

APPROXIMATING THE PARTITION FUNCTION Approximating Z: FPTAS = fully-polynomial (deterministic) approximation scheme FPRAS = fully-polynomial randomized approximation scheme FPTAS: Given input graph G = (V, E) and parameter ǫ > 0, the algorithm outputs OUT which satisfies (1 − ǫ)OUT ≤ |Z| ≤ (1 + ǫ)OUT and runs in time polynomial in |V|, 1/ǫ. FPRAS: Same as FPTAS, only OUT satisfies Pr((1 − ǫ)OUT ≤ |Z| ≤ (1 + ǫ)OUT) ≥ 3/4

COMPUTATIONAL TRANSITION - HARD CORE MODEL Weitz ’06 FPTAS for constant ∆ Sly ’10 Hard Activity λ λc(T∆)

COMPUTATIONAL TRANSITION - HARD CORE MODEL Weitz ’06 FPTAS for constant ∆ Sly ’10 Hard Activity λ λc(T∆)

G., Štefankoviˇ c, Vigoda ’12

Sly, Sun ’12 λ < λc(T∆): FPTAS for all graphs with constant max degree ∆ λ > λc(T∆): No FPRAS on graphs with max degree ∆ λc(T∆): Uniqueness Threshold on the infinite ∆-regular tree T∆. [Li-Lu-Yin ’13, Sly-Sun ’12]: general antiferro 2-spin models. What happens for spin models with more than 2 spins?

UNIQUENESS PHASE TRANSITION ON THE INFINITE TREE (HARD-CORE)

For ∆-regular tree of height ℓ: peven

ℓ

= Pr ( root is occupied in Gibbs dist. | leaves are occupied) podd

ℓ

= Pr ( root is occupied in Gibbs dist. | leaves are unoccupied) Does lim

ℓ→∞ peven 2ℓ

= lim

ℓ→∞ podd 2ℓ

? Uniqueness (λ ≤ λc(T∆)): No boundary affects root. Non-Uniqueness (λ > λc(T∆)): Exist boundaries affect root. [Kelly ’91]: λc(T∆) = (∆ − 1)∆−1 (∆ − 2)∆ ≈ e/(∆ − 2).

UNIQUENESS PHASE TRANSITION ON THE INFINITE TREE (HARD-CORE)

For ∆-regular tree of height ℓ: peven

ℓ

= Pr ( root is occupied in Gibbs dist. | leaves are occupied) podd

ℓ

= Pr ( root is occupied in Gibbs dist. | leaves are unoccupied) Does lim

ℓ→∞ peven 2ℓ

= lim

ℓ→∞ podd 2ℓ

? Uniqueness (λ ≤ λc(T∆)): No boundary affects root. Non-Uniqueness (λ > λc(T∆)): Exist boundaries affect root. [Kelly ’91]: λc(T∆) = (∆ − 1)∆−1 (∆ − 2)∆ ≈ e/(∆ − 2).

Key: Unique vs. Multiple fixed points of 2-level tree recursions: p2ℓ+2 = λ(1 − p2ℓ+1)∆−1 1 + λ(1 − p2ℓ+1)∆−1 , p2ℓ+1 = λ(1 − p2ℓ)∆−1 1 + λ(1 − p2ℓ)∆−1

THIS TALK What happens for spin models with more than 2 spins? Canonical examples of multi-spin systems: for a graph G = (V, E),

q-colorings problem

Spins: {1, . . . , q} Configurations: proper q- colorings of G. Z = # of proper colorings

• f G

q-state Potts model

Spins: {1, . . . , q}, Parameter: B > 0 Config.: assignments σ : V → [q] Z =

• σ:V→[q]

Bmonochromatic edges under σ Ferromagnetic vs Antiferromagnetic (B ≥ 1) (B < 1)

MAIN RESULTS - COLORINGS [Vigoda ’99]: FPRAS for q ≥ 11∆ 6 . [Jonasson ’02]: Uniqueness for colorings iff q ≥ ∆ + 1. THEOREM For all q, ∆ ≥ 3 with q even, whenever q < ∆, there is no FPRAS to approximate the number of colorings on ∆-regular graphs (even within an exponential factor, even for triangle-free graphs). [Johansson ’96]: Triangle-free graphs are colorable with O(∆/ log ∆) colors.

MAIN RESULTS - ANTIFERRO POTTS THEOREM For all q, ∆ ≥ 3 with q even, whenever 0 < B < (∆ − q)/∆, there is no FPRAS to approximate the partition function for the antiferro q-state Potts model at parameter B on ∆-regular graphs (even within an exponential factor). Uniqueness threshold for antiferromagnetic Potts model not known, conjectured to be at Bc(T∆) = (∆ − q)/∆. Also: general inapprox theorem for antiferro models with #spins≥ 2 in the tree non-uniqueness region.

SLY’S GADGET Sly uses in his reduction random bipartite ∆-regular graphs. α = 2 β = 4 [Mossel-Weitz-Wormald ’09] For an indep set I: α: # vertices in I on the left side. β: # vertices in I on the right side. Bimodality for λ > λc: a typical ind set from Gibbs distribution is unbalanced (α = β). Used to establish slow (“torpid") mixing of Glauber dynamics Phase of an indep set I: side with more vertices in I.

SLY’S REDUCTION

H: Input to MAX-CUT Gadget: random ∆-regular bipartite graph with a few degree ∆ − 1 vertices (yellow below). Key idea: replace each vertex of H by a gadget. Phases of gadgets correspond to cut (S, S). For each edge of H: add edges between their gadgets using ∆ − 1 deg. vertices.

Graph H input to MAX-CUT

→

input to HARD-CORE (max degree ∆)

Main Idea: Adjacent gadgets prefer opposite phases (antiferro interaction). Dominant Phase Configuration: {0, 1} assignment to vertices of H with fewest monochromatic edges = MAX-CUT(H).

ESTABLISHING THE BIMODALITY (HARD-CORE) Zα,β(λ) → contribution of sets with αn vertices

• ccupied on the left and βn

vertices on the right.

Z(λ) =

• α,β

Zα,β(λ)

Mossel-Weitz-Wormald: Second moment analysis to establish that both peaks appear.

ESTABLISHING THE SECOND MOMENT [Mossel-Weitz-Wormald ’09]: λc(T∆) < λ < λc(T∆) + ǫ∆ [Sly ’10]: λ = 1, ∆ = 6 [G.-Ge-Štefankoviˇ c-Vigoda-Yang ’11]: λ > λc(T∆) for ∆ = 4, 5 [G.-Štefankoviˇ c-Vigoda ’12]: Remaining cases ∆ = 4, 5, also antiferro Ising model with no external field Current approach: Simple analysis for general spin systems

• n random ∆-regular bipartite graphs.

Key technique: connection of moments to induced matrix norms.

MULTI-SPIN SYSTEMS

General q-spin system

Graph G = (V, E): Spins: {1, . . . , q} Interaction: specified by q×q symmetric matrix B = (Bij)i,j∈[q], Bij ≥ 0 Configurations: assignments σ : V → [q] Weight of a configuration: w(σ) =

• (u,v)∈E

Bσ(u)σ(v) Partition function: Z =

• σ:V→[q]

w(σ) Potts: B =      B 1 . . . 1 1 B . . . 1 . . . . . . ... . . . 1 1 . . . B     , Colorings: B =      1 . . . 1 1 . . . 1 . . . . . . ... . . . 1 1 . . .     .

TREE NON-UNIQUENESS FOR GENERAL SPIN SYSTEMS Uniqueness threshold hard to capture for general q-spin systems. Tree recursions: Ri ∝

j∈[q]

BijCj ∆−1 ,

• Cj ∝

i∈[q]

BijRj ∆−1 . Fixed points: vectors (r, c) with r = (R1, . . . , Rq), c = (C1, . . . , Cq) and

• Ri ∝ Ri,
• Cj ∝ Cj.

Interested in: Unique vs Multiple fixed points (r, c). Examples: (Brightwell-Winkler ’02) Multiple fixed points for colorings iff q < ∆, (this paper) Multiple fixed points for antiferro Potts iff B < (∆ − q)/∆.

MULTIMODALITY IN SPIN SYSTEMS For q-dimensional probability vectors α = (α1, . . . , αq), β = (β1, . . . , βq):

Σα,β

G

= configurations where αin vertices in V1 and βin vertices in V2 have spin i.

nαi vertices spin i nα1 vertices spin 1 nαq vertices spin q

... ...

nβ1 vertices spin 1 nβj vertices spin j nβq vertices spin q

... ...

Graph G Zα,β

G

:=

• σ∈Σα,β

G

w(σ). Dominant phase: pair (α, β) which achieves the maximum max

α,β ln EG[Zα,β G

]. In “uniqueness", unique dominant phase which satisfies α = β. In “non-uniqueness", multiple dominant phases which satisfy α = β. Examples (in non-uniqueness): Hard-core model → 2 dominant phases, Antiferro Potts/Colorings (even q)→ q q/2

• dominant phases.

THE FIRST MOMENT

nαi vertices spin i nα1 vertices spin 1 nαq vertices spin q

... ...

nβ1 vertices spin 1 nβj vertices spin j nβq vertices spin q

... ...

Graph G

EG

• Zα,β

G

• =
• n

α1n, . . . , αqn

• n

β1n, . . . , βqn

• 



x

• i
• αin

xi1n,...,xiqn j

• βjn

x1jn,...,xqjn

• n

x11n,...,xiqn

• ij

Bxijn

ij

 

∆

where the sum is over x satisfying

• j xij = αi
• i xij = βj

DOMINANT PHASES FOR COLORINGS/ANTIFERRO POTTS For random ∆-regular bipartite graph with V = V1 ∪ V2: For even q, q q/2

• dominant phases in non-uniqueness regime:

∃ 0 < a < b < 1, for V1: half colors S have marginal a, other half [q] \ S have b, for V2: [q] \ S have marginal a and S have b. For odd q, which of the two types dominate:

1

(1, ⌊q/2⌋, ⌊q/2⌋),

2

(⌊q/2⌋, ⌈q/2⌉)

ESTABLISHING THE MULTIMODALITY WHP (SECOND MOMENT) Define ΨB

1 (α, β) := lim n→∞

1 n ln EG[Zα,β

G

], ΨB

2 (α, β) := lim n→∞

1 n ln EG[(Zα,β

G

)2]. Main Task: Show that max

α,β ΨB 2 (α, β) = 2 max α,β ΨB 1 (α, β).

Equality captures that the contribution to the second moment comes from uncorrelated configurations (EG[(Zα,β

G

)2] ≤ C · (EG[Zα,β

G

])2). (Simple) Observation: The second moment can be formulated as the first moment of a paired-spin system with interaction B ⊗ B. Immediate consequence: max

α,β ΨB 2 (α, β) = max γ,δ ΨB⊗B 1

(γ, δ), where γ, δ are q2-dimensional probability vectors.

THE FIRST MOMENT AS AN INDUCED MATRIX NORM Recall for p, q′ > 0, xp :=

i

|xi|p1/p , Bp→q′ := max

xp>0

Bxq′ xp . We will show max

α,β Ψ1(α, β) = ∆ ln Bp→∆ , where p = ∆/(∆ − 1).

• Lemma. max

α,β Ψ1(α, β) = max r,c Φ(r, c), where Φ(r, c) = ∆ ln

rTBc rp cp . Proof (main idea): One-to-one correspondence between (i) critical points

• f Ψ1(α, β), (ii) critical points of Φ(r, c), and (iii) fixed points of tree

recursions. Moreover, Ψ1(α, β) = Φ(r, c) at their corresponding critical points.

THE FIRST MOMENT AS AN INDUCED MATRIX NORM Recall for p, q′ > 0, xp :=

i

|xi|p1/p , Bp→q′ := max

xp>0

Bxq′ xp . We will show max

α,β Ψ1(α, β) = ∆ ln Bp→∆ , where p = ∆/(∆ − 1).

• Lemma. max

α,β Ψ1(α, β) = max r,c Φ(r, c), where Φ(r, c) = ∆ ln

rTBc rp cp .

• Corollary. max

r,c Φ(r, c) = ∆ ln Bp→∆, where p = ∆/(∆ − 1).

Proof: max

r,c

rTBc rp cp = max

c

max

r

rTBc rp cp = max

c

Bc∆ cp = Bp→∆ .

PROOF OF SECOND MOMENT Recall, we need to show that max

α,β ΨB 2 (α, β) = 2 max α,β ΨB 1 (α, β).

Proof: with p = ∆/(∆ − 1), max

α,β ΨB 1 (α, β) = ∆ ln Bp→∆ .

max

α,β ΨB 2 (α, β) = max γ,δ ΨB⊗B 1

(γ, δ) = ∆ ln B ⊗ Bp→∆ . Key Fact (Bennett ’77): for matrix norms ·p→q′ with p ≤ q′ it holds that C ⊗ Dp→q′ = Cp→q′ Dp→q′ .

CONCLUSIONS Phase transitions on infinite ∆-regular tree (random ∆-regular bipartite graphs) = ⇒ Hardness of approximate counting on graphs of max degree ∆ Open question: Matching positive results, say, for colorings? Current bound: q ≥ 11∆/6 (Vigoda ’99) Conjecture (?): q ≥ ∆ + 2

Thank You!