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

I NAPPROXIMABILITY FOR A NTIFERROMAGNETIC S PIN S YSTEMS IN THE T REE N ON -U NIQUENESS R EGION Andreas Galanis (Oxford) Daniel Štefankoviˇ c (Rochester) Eric Vigoda (Georgia Tech) Cargese ’14 , August 29

O VERVIEW 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 on graphs of maximum degree ∆ .

M OTIVATING E XAMPLE : T HE H ARD -C ORE M ODEL 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 := Z G ( λ ) = λ | I | I ∈I ( G ) Gibbs Distribution: for I ∈ I ( G ) , µ ( I ) = λ | I | Z When λ = 1 : Z = |I ( G ) | = number of independent sets in G .

C OMPUTING THE P ARTITION F UNCTION � Partition function: Z := Z G ( λ ) = λ | I | independent set I #H ARD -C ORE ( 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 #I ND S ETS is #P-complete, even for graphs of max degree ∆ = 3 . Can we approximate Z on graphs of max degree ∆ ? As λ ↑ , #H ARD -C ORE ( λ ) becomes computationally harder (more weight on the max independent set)

A PPROXIMATING THE P ARTITION F UNCTION 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

C OMPUTATIONAL T RANSITION - H ARD C ORE M ODEL FPTAS for λ c ( T ∆ ) constant ∆ Hard Weitz ’06 Sly ’10 Activity λ

C OMPUTATIONAL T RANSITION - H ARD C ORE M ODEL G., Štefankoviˇ c, FPTAS for λ c ( T ∆ ) Vigoda ’12 constant ∆ Hard Weitz ’06 Sly ’10 Sly, Sun ’12 Activity λ λ < λ c ( T ∆ ) : FPTAS for all graphs with constant max degree ∆ λ > λ c ( T ∆ ) : No FPRAS on graphs with max degree ∆ λ c ( 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?

U NIQUENESS P HASE T RANSITION ON THE I NFINITE T REE (H ARD -C ORE ) For ∆ -regular tree of height ℓ : p even = Pr ( root is occupied in Gibbs dist. | leaves are occupied ) ℓ p odd = Pr ( root is occupied in Gibbs dist. | leaves are unoccupied ) ℓ ℓ →∞ p even ℓ →∞ p odd Does ? lim = lim 2 ℓ 2 ℓ Uniqueness ( λ ≤ λ c ( T ∆ )) : No boundary affects root. Non-Uniqueness ( λ > λ c ( T ∆ )) : Exist boundaries affect root. [Kelly ’91]: λ c ( T ∆ ) = (∆ − 1 ) ∆ − 1 ≈ e / (∆ − 2 ) . (∆ − 2 ) ∆

U NIQUENESS P HASE T RANSITION ON THE I NFINITE T REE (H ARD -C ORE ) For ∆ -regular tree of height ℓ : p even = Pr ( root is occupied in Gibbs dist. | leaves are occupied ) ℓ p odd = Pr ( root is occupied in Gibbs dist. | leaves are unoccupied ) ℓ ℓ →∞ p even ℓ →∞ p odd Does ? lim = lim 2 ℓ 2 ℓ Uniqueness ( λ ≤ λ c ( T ∆ )) : No boundary affects root. Non-Uniqueness ( λ > λ c ( T ∆ )) : Exist boundaries affect root. [Kelly ’91]: λ c ( T ∆ ) = (∆ − 1 ) ∆ − 1 ≈ e / (∆ − 2 ) . (∆ − 2 ) ∆ Key: Unique vs. Multiple fixed points of 2-level tree recursions: λ ( 1 − p 2 ℓ + 1 ) ∆ − 1 λ ( 1 − p 2 ℓ ) ∆ − 1 p 2 ℓ + 2 = 1 + λ ( 1 − p 2 ℓ + 1 ) ∆ − 1 , p 2 ℓ + 1 = 1 + λ ( 1 − p 2 ℓ ) ∆ − 1

T HIS T ALK 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 q -state Potts model Spins: { 1 , . . . , q } Spins: { 1 , . . . , q } , Parameter: B > 0 Configurations: proper q - Config.: assignments σ : V → [ q ] � colorings of G . B monochromatic edges under σ Z = Z = # of proper colorings σ : V → [ q ] of G Ferromagnetic vs Antiferromagnetic ( B ≥ 1 ) ( B < 1 )

M AIN R ESULTS - C OLORINGS [Vigoda ’99]: FPRAS for q ≥ 11 ∆ 6 . [Jonasson ’02]: Uniqueness for colorings iff q ≥ ∆ + 1 . T HEOREM 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.

M AIN R ESULTS - A NTIFERRO P OTTS T HEOREM 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 B c ( T ∆ ) = (∆ − q ) / ∆ . Also: general inapprox theorem for antiferro models with #spins ≥ 2 in the tree non-uniqueness region.

S LY ’ S G ADGET Sly uses in his reduction random bipartite ∆ -regular graphs. [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 > λ c : a typical ind set from Gibbs distribution is unbalanced ( α � = β ). Used to establish slow (“torpid") mixing of Glauber dynamics α = 2 β = 4 Phase of an indep set I : side with more vertices in I .

S LY ’ S R EDUCTION H : Input to M AX -C UT 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 M AX -C UT input to H ARD -C ORE (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 = M AX -C UT (H).

E STABLISHING THE B IMODALITY (H ARD -C ORE ) Z α,β ( λ ) → contribution of sets with α n vertices occupied on the left and β n vertices on the right. � Z( λ ) = Z α,β ( λ ) α,β Mossel-Weitz-Wormald: Second moment analysis to establish that both peaks appear.

E STABLISHING THE S ECOND M OMENT [Mossel-Weitz-Wormald ’09]: λ c ( T ∆ ) < λ < λ c ( T ∆ ) + ǫ ∆ [Sly ’10]: λ = 1 , ∆ = 6 [G.-Ge-Štefankoviˇ c-Vigoda-Yang ’11]: λ > λ c ( 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 on random ∆ -regular bipartite graphs. Key technique: connection of moments to induced matrix norms.

M ULTI -S PIN S YSTEMS General q -spin system Graph G = ( V , E ) : Spins: { 1 , . . . , q } Interaction: specified by q × q symmetric matrix B = ( B ij ) i , j ∈ [ q ] , B ij ≥ 0 Configurations: assignments σ : V → [ q ] � Weight of a configuration: w ( σ ) = B σ ( u ) σ ( v ) ( u , v ) ∈ E � Partition function: Z = w ( σ ) σ : V → [ q ]     . . . . . . B 1 1 0 1 1     1 B . . . 1 1 0 . . . 1     Potts: B =  , Colorings: B =  . . . . . . .  ...   ...  . . . . . .  . . .  . . . . . . . . . 1 1 B 1 1 0

T REE N ON -U NIQUENESS FOR G ENERAL S PIN S YSTEMS Uniqueness threshold hard to capture for general q -spin systems. � � � ∆ − 1 � � � ∆ − 1 Tree recursions: � � . R i ∝ B ij C j , C j ∝ B ij R j j ∈ [ q ] i ∈ [ q ] Fixed points: vectors ( r , c ) with r = ( R 1 , . . . , R q ) , c = ( C 1 , . . . , C q ) and � � R i ∝ R i , C j ∝ C j . 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 ) / ∆ .

M ULTIMODALITY IN S PIN S YSTEMS For q -dimensional probability vectors α = ( α 1 , . . . , α q ) , β = ( β 1 , . . . , β q ) : Σ α , β = configurations where α i n vertices in V 1 and β i n vertices in V 2 have spin i . G ... ... n α 1 vertices spin 1 n α i vertices spin i n α q vertices spin q Graph G ... ... n β q vertices spin q n β 1 vertices spin 1 n β j vertices spin j � Z α , β w ( σ ) . := G σ ∈ Σ α , β G α , β ln E G [ Z α , β Dominant phase: pair ( α , β ) which achieves the maximum max ] . 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, � q � Antiferro Potts/Colorings (even q) → dominant phases. q / 2