Approximate Counting via Correlation Decay on Planar Graphs Yitong - PowerPoint PPT Presentation

Approximate Counting via Correlation Decay on Planar Graphs Yitong Yin Nanjing University Chihao Zhang Shanghai Jiaotong University

Holant Problems (Valiant 2004) instance: Ω = ( G ( V, E ) , { f v } v ∈ V ) graph G =( V , E ) edges: variables (domain [ q ] ) f v vertices: constraints (arity=degree) f v : [ q ] ��� ( v ) → C symmetric configuration ( solution , coloring , ...) : σ ∈ [ q ] E holant ( counting ) : � � � � ������ ( Ω ) = σ | E ( v ) f v v ∈ V σ ∈ [ q ] E #matchings: σ ∈ { 0 , 1 } E f v ≡ At-Most-One q =2

Holant problem: ������ ( G , F ) graph family function family � G ∈ G input : with Ω = ( G ( V, E ) , { f v } v ∈ V ) f v ∈ F output : � � � � ������ ( Ω ) = σ | E ( v ) f v v ∈ V σ ∈ [ q ] E spin system / graph homomorphism ( G.H. ): F = { f : [ q ] d → C , d ≤ 2 } ∪ { = } f f • #IS, #VC = f = • #q-colorings, # H -colorings f = f • hardcore / Ising / Potts models, MRF = f G =( V , E ) V E spin model holant

Holant Problems Holant problem: ������ ( G , F ) graph family function family characterize the tractability of by and F ������ ( G , F ) G Bad news: for general/planar , almost all nontrivial : #P-hard G F (Dyer-Greenhill’00, Bulatov-Grohe’05, Dyer-Goldberg’07, Bulatov’08, Goldberg-Grohe-Jerrum’10, Cai-Chen’10, Cai-Chen-Lu’10, Cai-Lu-Xia’10, Dyer-Richerby’10, Dyer-Richerby’11, Cai-Chen’12, Cai-Lu-Xia ’13 ) Good news: tractable if is tree , is Spin or Matching G F (arity ≤ 2 and =) (At-Most-One) Our result: is planar (locally like a tree) G � is regular FPTAS (like spin/matching) F (local info is enough) correlation decay

Gibbs Measure f v : [ q ] ��� ( v ) → R ≥ 0 Ω = ( G ( V, E ) , { f v } v ∈ V ) � � � � ������ ( Ω ) = σ | E ( v ) f v v ∈ V σ ∈ [ q ] E � v ∈ V f v ( σ | E ( v ) ) Gibbs measure: �� ( σ ) = ������ marginal probability: σ A ∈ [ q ] A A ⊂ E �� ( σ ( e ) = c | σ A ) self- compute reduction FPTAS for �� ( σ ( e ) = c | τ A ) ± 1 n ������ ( Ω ) in time ���� ( n )

Correlation Decay strong spatial mixing (SSM): ∀ σ B ∈ [ q ] B � � � �� ( σ ( e ) = c | σ A ) − �� ( σ ( e ) = c | σ A , σ B ) � ≤ ���� ( | V | ) ��� ( − Ω ( t )) SSM : sufficiency of local information G B for approx. of �� ( σ ( e ) = c | σ A ) ? efficiency of (FPTAS) t local computation e A such implication was known for: Spin (Weitz’06) � q =2 , is F Matching (Bayati-Gamarnik-Katz-Nair-Tetali’08)

Regularity Pinning: symmetric f : [ q ] d → C τ ∈ [ q ] k where g : [ q ] d − k → C ��� τ ( f ) = g ∀ σ ∈ [ q ] d − k , g ( σ ) = f ( σ 1 , . . . , σ d − k , τ 1 , . . . , τ k ) when write symmetric q =2 f = [ f 0 , f 1 , . . . , f d ] f : [ q ] d → C is C - regular if where that � σ � 1 = i f i = f ( σ ) � ≤ C � ��� τ ( f ) | ∀ τ ∈ [ q ] k �� �� ∀ 0 ≤ k ≤ d, a family of symmetric functions is regular if F ∃ a finite C s.t. ∀ f ∈ F , f is C -regular examples: bounded-arity [ f 0 , f 1 , f 2 , . . . , f i , . . . , f d − 1 , f d ] equality [1,0,...,0,1] � �� � d − k +1 at-most-one [1,1,0,...,0] counterexample: [0 , . . . , 0 , 1 , 0 , . . . , 0 ] � �� � � �� � cyclic [ a , b , c , a , b , c ,...] d d 2 2

Holant can be computed F is Spin ( junction-tree BP ) � ���� ( n ) · 2 �� in time if has bounded-arity F ( tensor network , Markov-Shi’09) Theorem I is regular, If then ������ ( G, { f v } v ∈ V ⊂ F ) F ���� ( | V | ) · 2 O ( ��������� ( G )) can be computed in time Theorem II If G is planar (apex-minor-free), is regular, then F FPTAS for SSM ������ ( G , F )

Theorem I is regular, If then ������ ( G, { f v } v ∈ V ⊂ F ) F ���� ( | V | ) · 2 O ( ��������� ( G )) can be computed in time SSM : � � ≤ ���� ( | V | ) ��� ( − t ) � � �� ( σ ( e ) = c | σ A ) − �� ( σ ( e ) = c | σ A , σ B ) compute FPTAS B �� ( σ ( e ) = c | τ A ) ± 1 G n for Holant in time ���� ( n ) t Theorem (Demaine-Hajiaghayi’04) For apex-minor-free graphs, e A treewidth of t -ball is O( t ) . Theorem II If G is planar (apex-minor-free), is regular, then F FPTAS for SSM ������ ( G , F )

Theorem I is regular, If then ������ ( G, { f v } v ∈ V ⊂ F ) F ���� ( | V | ) · 2 O ( ��������� ( G )) can be computed in time tree-decomposition: a tree of “bags” of vertices: 1.Every vertex is in some bag. 2.Every edge is in some bag. 3.If two bags have a same vertex, then all bags in the path between them have that vertex. width: max bag size -1 treewidth: width of optimal tree decomposition

Separator-Decomposition of G ( V, E ) : T G each node i ∈ T G corresponds to ( V i , S i ) V ���� = V and V ���� = ∅ V ���� = V � such that is a vertex separator S i � = � of in G [ V i ] V j , V k ⊂ V i ∂ V i is vertex boundary of in G [ V i ] V i V i width: ��� i ∈ T G {| S i | , | ∂ V i |} V j V k S i ∂ V i separator-width sw ( G ) : width of optimal T G V j V k Theorem: and sw ( G ) = Θ ( tw ( G )) can be constructed T G ∅ ∅ in time ���� ( n ) · 2 O ( tw ( G ))

Theorem I is regular, If then ������ ( G, { f v } v ∈ V ⊂ F ) F ���� ( | V | ) · 2 O ( ��������� ( G )) can be computed in time conditional independence: V i V j V k S i ∂ V i L R S V j V k and �� ( σ L | σ S ) �� ( σ R | σ S ) are independent for fixed σ S : vertex separator S i S : edge separator : vertex boundary ∂ V i

τ ∈ [ q ] k Peering: given f : [ q ] d → C ���� τ ( f ) : [ q ] k → { 0 , 1 } defined as � ��� σ ( f ) = ��� τ ( f ) 1 ∀ σ ∈ [ q ] k , ���� τ ( f )( σ ) = ���� 0 ���� τ ( f ) = { σ ∈ [ q ] k | ��� σ ( f ) = ��� τ ( f ) } peering classifies configurations around a vertex into equivalent classes f v states of a vertex: peer classes V i σ τ ρ f v ( στρ ) depends only on V j V k S i peer classes of ∂ V i σ , τ , ρ Holant value can be figured out by V j keeping track of only peer classes V k for regular f , # of peer classes is always finite

Algorithmic Implications applying the SSM obtained by a “decay-only” technique recursive coupling (Goldberg-Martin-Paterson’05), we have FPTAS for: • # q -coloring of triangle-free planar graphs of max- degree Δ for q >1.76322 Δ - 0.47031 • ferromagnetic Ising model with temperature β and field B on planar graphs of max-degree Δ , when � 2 � e 2 β B + e − 2 β B ∆ < 1 e β B + e − β B 4 • ferromagnetic Potts model with temperature β on planar graphs of max-degree Δ for β = O ( 1 ∆ ) (conjectured by Gamarnik-Katz’06)

Conclusions and Open Problems for Holant problems defined by regular constraints: • a poly( n ) · 2 treewidth time algorithm for exact computation; • SSM implies FPTAS on planar graphs. open problems: • in terms of reliance on treewidth, tightness of 2 tw for regular Holant and n tw for all symmetric Holant (under some assumption); • using SSM for FPTAS on general graphs.