SPIN SYSTEMS: HARDNESS OF APPROXIMATE COUNTING VIA PHASE TRANSITIONS

Transcription

1 SPIN SYSTEMS: HARDNESS OF APPROXIMATE COUNTING VIA PHASE TRANSITIONS Andreas Galanis Joint work with: Jin-Yi Cai Leslie Ann Goldberg Heng Guo Mark Jerrum Daniel Štefankovič Eric Vigoda The Power of Randomness in Computation, March 16-20

2 OVERVIEW Connections between Phase transitions from statistical physics. Computational complexity of approximate counting/sampling problems.

3 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 := 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.

4 COMPUTING THE PARTITION FUNCTION Partition function: Z := Z G (λ) = independent set I λ I #HARD-CORE(λ): On input G = (V, E), compute Z. Z is typically exponential 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? (FPRAS) As λ increases, #HARD-CORE(λ) becomes harder (more weight on the max independent set)

5 COMPUTATIONAL TRANSITION (HARD-CORE MODEL) FPTAS for constant Weitz 06 λ c (T ) Hard Sly 10 Activity λ [Weitz 06]: FPTAS for all graphs with constant max degree when λ < λ c (T ). [Sly 10]: No FPRAS for graphs with max degree when λ c (T ) < λ < λ c (T ) + ɛ for some ɛ > 0. λ c (T ): Uniqueness Threshold of the infinite -regular tree T.

6 COMPUTATIONAL TRANSITION (HARD-CORE MODEL) FPTAS for constant Weitz 06 λ c (T ) Hard Sly 10 Activity λ G., Štefankovič, Vigoda 12 Sly, Sun 12 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 12, Sly-Sun 12]: general antiferro 2-spin models. What happens for spin models with more than 2 spins?

7 UNIQUENESS PHASE TRANSITION ON THE INFINITE TREE (HARD-CORE) For -regular tree of height l: p even l p odd l Does = Pr ( root is occupied in Gibbs dist. leaves are occupied) = Pr ( root is occupied in Gibbs dist. leaves are unoccupied) lim peven 2l = lim podd 2l? l 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)

8 UNIQUENESS PHASE TRANSITION ON THE INFINITE TREE (HARD-CORE) For -regular tree of height l: p even l p odd l Does = Pr ( root is occupied in Gibbs dist. leaves are occupied) = Pr ( root is occupied in Gibbs dist. leaves are unoccupied) lim peven 2l = lim podd 2l? l 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: p 2l+2 = λ(1 p 2l+1) λ(1 p 2l+1 ) 1, p 2l+1 = λ(1 p 2l) λ(1 p 2l ) 1

9 MULTI-SPIN MODELS 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 of G q-state Potts model Spins: {1,..., q}, Parameter: B > 0 Config.: assignments σ : V [q] Z = monochromatic edges under σ B σ:v [q] Ferromagnetic vs Antiferromagnetic (B 1) (B < 1)

10 MAIN RESULTS - COLORINGS [Vigoda 99]: FPRAS for q [Jonasson 02]: Uniqueness for colorings iff q + 1. THEOREM[G.-ŠTEFANKOVIČ-VIGODA 14] 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.

11 MAIN RESULTS - ANTIFERRO POTTS THEOREM[G.-ŠTEFANKOVIČ-VIGODA 14] 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.

12 SLY S REDUCTION (HARD-CORE) 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. α = 2 β = 4 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 Phase of an indep set I: side with more vertices in I.

13 SLY S REDUCTION MAX-CUT(H) HARD-CORE(G) Gadget: random -regular bipartite graph with a few degree 1 vertices (yellow below). Key idea: replace each vertex of H by a gadget. Phases for gadget correspond to cut (S, S). For each edge of H: add parallel edges between their gadgets using 1 deg. vertices. Graph H input to MAX-CUT Main Idea: input to HARD-CORE(λ) Adjacent gadgets prefer opposite phases. Dominant Configuration: {0, 1} assignment to vertices of H with fewest monochromatic edges = MAX-CUT(H).

14 BIMODALITY ON RANDOM BIPARTITE GRAPHS (HARD-CORE) Z α,β (λ) contribution of sets with αn vertices occupied on the left and βn vertices on the right. Z(λ) = α,β Z α,β (λ) Mossel-Weitz-Wormald: Second moment analysis to establish that both peaks appear.

15 ESTABLISHING THE BIMODALITY [Mossel-Weitz-Wormald 09]: λ c (T ) < λ < λ c (T ) + ɛ [Sly 10]: λ = 1, = 6 [G.-Ge-Štefankovič-Vigoda-Yang 11]: λ > λ c (T ) for 4, 5 [G.-Štefankovič-Vigoda 12]: Remaining cases = 4, 5 [G.-Štefankovič-Vigoda 14]: Simple analysis for general spin systems on random -regular bipartite graphs. Key technique: connection of moments to induced matrix norms.

16 MULTI-SPIN SYSTEMS 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) = Partition function: Z = σ:v [q] w(σ) (u,v) E B σ(u)σ(v) Potts: B = B B B, Colorings: B =

17 UNIQUENESS PHASE TRANSITION FOR GENERAL SPIN SYSTEMS? Examples: (Brightwell-Winkler 02) Multiple fixed points for colorings iff q <, (G.-Štefankovič-Vigoda 14) Multiple fixed points for antiferro Potts iff B < ( q)/. Uniqueness threshold hard to capture for general q-spin systems. ( ) 1, ( ) 1. Tree recursions: R i B ij C j Ĉj 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, Ĉ j C j. Interested in: Unique vs Multiple fixed points (r, c).

18 MULTIMODALITY IN SPIN SYSTEMS: PART I For q-dimensional probability vectors α = (α 1,..., α q ), β = (β 1,..., β q ): Σ α,β G = configurations where α i n vertices in V 1 and β i n vertices in V 2 have spin i. nα 1 vertices spin 1 Graph G... nα i vertices spin i... nα q vertices spin q nβ 1 vertices spin 1... nβ j vertices spin j... nβ q vertices spin q Z α,β G := σ Σ α,β G w(σ). Dominant phase: pair (α, β) which achieves the maximum max α,β log E G[Z α,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, q Antiferro Potts/Colorings (even q) dominant phases. q/2

19 ESTABLISHING THE MULTIMODALITY WHP (SECOND MOMENT) Define Ψ B 1 1 (α, β) := lim n n log E G[Z α,β G ], 1 ΨB 2 (α, β) := lim n n log E G[(Z α,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 (E G [(Z α,β G )2 ] C (E G [Z α,β G ])2 ). 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 q 2 -dimensional probability vectors.

20 THE FIRST MOMENT AS AN INDUCED MATRIX NORM ( Recall for p, q > 0, x p := x i p) 1/p Bx q, B p q := max. x i p >0 x p We will show max α,β Ψ 1(α, β) = log B p, where p = /( 1). Lemma. max α,β Ψ 1(α, β) = max r,c Φ(r, c), where Φ(r, c) = log r T Bc r p c p. Proof (main idea): One-to-one correspondence between (i) critical points of Ψ 1 (α, β), (ii) critical points of Φ(r, c), and (iii) fixed points of tree recursions. Moreover, Ψ 1 (α, β) = Φ(r, c) at their corresponding critical points.

21 THE FIRST MOMENT AS AN INDUCED MATRIX NORM ( Recall for p, q > 0, x p := x i p) 1/p Bx q, B p q := max. x i p >0 x p We will show max α,β Ψ 1(α, β) = log B p, where p = /( 1). Lemma. max α,β Ψ 1(α, β) = max r,c Φ(r, c), where Φ(r, c) = ln r T Bc r p c p. Corollary. max Φ(r, c) = log B r,c p, where p = /( 1). Proof: max r,c r T Bc = max r p c c p max r r T Bc Bc = max = B r p c c p c p. p

22 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 B p. max α,β ΨB 2 (α, β) = max γ,δ ΨB B 1 (γ, δ) = ln B B p. Key Fact (Bennett 77): for matrix norms p q with p q it holds that C D p q = C p q D p q.

23 REMARKS For random -regular bipartite graph with V = V 1 V 2 : ( ) q For even q, dominant phases in non-uniqueness regime: q/2 0 < a < b < 1, for V 1 : half colors S have marginal a, other half [q] \ S have b, for V 2 : [q] \ S have marginal a and S have b. For odd q, which of the two types dominate: 1 (1, q/2, q/2 ) (conjecture: q = 3, < 10), 2 ( q/2, q/2 ) (conj.: q = 3, 10, all odd q > 3) Challenge: compute B B B 1 for 0 B < 1 and 3.

24 RESTRICTING TO BIPARTITE INPUTS Second part: what happens when the input graph is bipartite? MAX-CUT reduction no longer works. [ B 1 ] Antiferro Ising model, i.e., B := with B < 1: 1 B On bipartite graphs, equivalent to ferro Ising model with parameter 1/B. [Jerrum-Sinclair 93]: FPRAS for ferro Ising model (B > 1) on general graphs

25 COUNTING INDEPENDENT SETS ON BIPARTITE GRAPHS #BIS: counting independent sets in bipartite graphs [Dyer-Goldberg-Greenhill-Jerrum 03]: Various approximate counting problems are at least as hard as approximating #BIS. E.g., #STABLE-MATCHINGS,#BIPARTITE-q-COL, etc. [Goldberg-Jerrum 10]: #FERROPOTTS AP #BIS We study #BIS in a bounded-degree setting More generally, antiferro 2-spin systems on bipartite graphs

26 MAIN RESULT CAI-G.-GOLDBERG-GUO-JERRUM-ŠTEFANKOVIČ-VIGODA 14 For every antiferro 2-spin system (other than the antiferro Ising model), in the tree non-uniqueness region: #BIS-hard to approx the partition function on bipartite graphs of max degree. [Liu-Lu-Zhang 14]: Used the result to show #BIS-hardness for ferromagnetic models with β γ and sufficiently large external field.

27 CONSEQUENCES FOR #BIS COROLLARY #BIS on bipartite graphs with degree at most 6 is as hard as #BIS on bipartite graphs without degree bound. [Weitz 06]: FPTAS for graphs with degree at most 5. [Liu-Lu 14]: FPTAS for bipartite graphs where one side has degree at most 5.

28 PROOF OVERVIEW Main Lemma: When a 2-spin system supports Nearly-independent phase-correlated spins" Symmetry breaking" on bipartite graphs of max degree, it is #BIS-hard to approx the partition function on bipartite graphs. Holds also for =, i.e., on graphs with no degree bound.

29 NEARLY-INDEPENDENT PHASE-CORRELATED SPINS (HARD-CORE) Gadget: random -regular bipartite graph with a few degree 1 vertices (yellow below). + - : degree vertices : degree 1 vertices Phase of an independent set I: Y(I) = +, if side +" has more vertices in I. Otherwise, Y(I) =. Bimodality (when λ > λ c (T )): Y(I)=+ w.p. roughly 1/2. roughly 1/2, Y(I)=- w.p. Key fact (Sly 10): Conditioned on the phase, the spins of yellow vertices are approx independent. Distribution is Q + if Y(σ) = +, Q if Y(σ) = (correlated with the phase). Uses [Martinelli-Sinclair-Weitz 04] to make error of approx polynomially small.

30 A FIRST ATTEMPT Let λ > λ c(t ). Let H = (V, E) be a bipartite graph. (H can have arbitrarily large degree) Gadget: random -regular bipartite graph G, few degree 1 vertices. Red(H): bipartite graph max degree, input to HARD-CORE(λ) Key idea: replace each vertex of H by a gadget. Phases for gadgets correspond to a cut (S, S ). Graph H Red(H) (input to HARD-CORE(λ)) Phase interaction: An edge of H contributes multipl factor B 1 if phases on its gadgets agree, B 2 otherwise (B 1 < B 2). With B = B 1/B 2 < 1: Z Ising;B (H) = (1 ± ɛ) Z λ(red(h)) (Z λ (G)/2) V

31 #BIS-HARDNESS (REDUCTION SKETCH) Use instead the following intermediate problem. NON-UNIFORM-ISING(B, λ) Input. Bipartite graph G = (V, E), subset U V. Compute Z = B m(σ) λ {u U:σu=1}. σ:v {0,1} We prove that for any 0 < B < 1 and λ 1: Remaining argument: #BIS AP NON-UNIFORM-ISING(B, λ) Simulate an external field λ 1 using a bipartite graph H with degree at most. A spin system supports symmetry breaking" on bipartite graphs of max degree when such a graph H exists.

32 CONCLUSIONS Phase transitions on infinite -regular tree (random -regular graphs) = Hardness of approximate counting on graphs of max degree Open questions: Matching positive results for multi-spin systems (e.g. colorings)? Classify antiferro multi-spin systems on bipartite graphs of max degree (e.g. bipartite q colorings)?

33 Thank You!