Improved FPTAS for Multi - spin Systems Pinyan Lu Yitong Yin - PowerPoint PPT Presentation

Improved FPTAS for Multi - spin Systems Pinyan Lu Yitong Yin Microsoft Research Asia Nanjing University presented by: Sangxia Huang KTH

Colorings instance : undirected G(V,E) with max-degree ≤Δ q colors: goal : counting the number of proper q -colorings for G • exact counting is #P-hard • when q < Δ , decision of existence is NP-hard

Colorings instance : undirected G(V,E) with max-degree ≤Δ q colors: goal : counting the number of proper q -colorings for G • exact counting is #P-hard • when q < Δ , decision of existence is NP-hard approximately counting the number of proper q -colorings for G when q ≥αΔ + β equivalent to sampling an almost uniform random q -coloring

Spin System (pairwise Markov random field ) F u F v A a instance : A b • undirected G(V,E); A f A d F w • q states : [ q ]; A c A e F y F x • each edge e ∈ E associated with an activity : a symmetric nonnegative q × q matrix A e : [ q ] × [ q ] → R ≥ 0 • each vertex v ∈ V associated with an external field : a nonnegative q -vector F v : [ q ] → R ≥ 0 goal : computing the partition function: X Y Y Z = A e ( x u , x v ) F v ( x v ) e = uv ∈ E x ∈ [ q ] V v ∈ V

Spin System F u F v A a A b A e : [ q ] × [ q ] → R ≥ 0 A f A d F w F v : [ q ] → R ≥ 0 A c A e F y F x partition function: count the # of solutions to an CSP X Y Y Z = A e ( x u , x v ) F v ( x v ) e = uv ∈ E x ∈ [ q ] V v ∈ V binary constraints unary constraints enumerate all configurations

Spin System F u F v A a A b A e : [ q ] × [ q ] → R ≥ 0 A f A d F w F v : [ q ] → R ≥ 0 A c A e F y F x partition function: count the # of solutions to an CSP X Y Y Z = A e ( x u , x v ) F v ( x v ) e = uv ∈ E x ∈ [ q ] V v ∈ V binary constraints unary constraints enumerate all configurations     1 0 1 1 0   coloring:   F = A =     . ... . 1     .     1 0

Examples of Spin systems • 2-spin: q =2 • hardcore model (independent set), Ising model, etc. • multi-spin: general q     1 0 • coloring: 1 1 0     F = A =     . ... .    1  .     0 1

Examples of Spin systems • 2-spin: q =2 • hardcore model (independent set), Ising model, etc. • multi-spin: general q     1 0 • coloring: 1 1 0     F = A =     . ... .    1  .     0 1 • Potts model: inverse temperature β e β   1 e β   arbitrary F A =   1 ...     e β when β =- ∞ and F =(1,1,... ,1) , it is coloring

Results sufficient conditions for FPTAS for classes of spin systems • coloring: q ≥αΔ + β • randomized algorithms: by simulating a random walk (the Glauber dynamics) over colorings • α =11/6 (Jerrum’95 ⇢ Bubley-Dyer’97 ⇢ Vigoda’99) • deterministic algorithms: by exploiting the correlation decay (spatial mixing) property • α≈ 2.8432 (Gamarnik-Katz’07) • just correlation decay (no FPTAS): α≈ 1.763 (Goldberg-Martin-Paterson’05, Gamarnik-Katz-Misra’ 12 ) this paper : deterministic FPTAS for α≈ 2.58071

Results sufficient conditions for FPTAS for classes of spin systems • general multi-spin system: A e ( x, y ) in terms of c = max A e ( w, z ) e ∈ E w,x,y,z ∈ [ q ] • Gamarnik-Katz’07: ( c ∆ − c − ∆ ) ∆ q ∆ < 1 an exponential 3 ∆ ( c ∆ − 1) ≤ 1 this paper : improvement! • on Potts model (with inverse temperature β ) : it implies : 3 ∆ (e | β | − 1) ≤ 1

Results sufficient conditions for FPTAS for classes of spin systems • general multi-spin system: A e ( x, y ) in terms of c = max A e ( w, z ) e ∈ E w,x,y,z ∈ [ q ] • Gamarnik-Katz’07: ( c ∆ − c − ∆ ) ∆ q ∆ < 1 an exponential 3 ∆ ( c ∆ − 1) ≤ 1 this paper : improvement! • on Potts model (with inverse temperature β ) : it implies : 3 ∆ (e | β | − 1) ≤ 1 • confirming the conjecture of in [GK’07] � 1 � | β | = O ∆ • asymptotically matching the inaproximability e β < 1 − q ∆ bound for in [Galanis-Stefankovic-Vigoda’ 13] β < 0

The standard first step: reducing to the computing of marginal probability X Y Y Z = A e ( x u , x v ) F v ( x v ) e = uv ∈ E x ∈ [ q ] V v ∈ V for any configuration x ∈ [ q ] V Gibbs measure: Q e = uv ∈ E A e ( x u , x v ) Q v ∈ V F v ( x v ) P [ X = x ] = Z marginal probability: P [ X v = x v ]

The standard first step: reducing to the computing of marginal probability X Y Y Z = A e ( x u , x v ) F v ( x v ) e = uv ∈ E x ∈ [ q ] V v ∈ V for any configuration x ∈ [ q ] V Gibbs measure: Q e = uv ∈ E A e ( x u , x v ) Q v ∈ V F v ( x v ) P [ X = x ] = Z marginal probability: P [ X v = x v ] Jerrum-Valiant-Vazirani’86 for self-reducible class of spin-systems: efficient approximation FPTAS for Z of marginal probability (with additive error)

The standard first step: • self-reducible: general spin systems, Potts models • not self-reducible: coloring • self-reducible superclass of coloring: list-coloring instance : undirected G(V,E) each vertex v associated { } { } with a list L v of colors { } allowed to use on v coloring ⊂ list-coloring ⊂ Potts ⊂ multi-spin { } { } | {z } self-reducible

The standard first step: • self-reducible: general spin systems, Potts models • not self-reducible: coloring • self-reducible superclass of coloring: list-coloring instance : undirected G(V,E) each vertex v associated { } { } with a list L v of colors { } allowed to use on v coloring ⊂ list-coloring ⊂ Potts ⊂ multi-spin { } { } | {z } self-reducible new goal: multi-spin system ( ) approximate the marginal for Potts model (with additive error) P [ X v = x ] list-coloring

The standard first step: • self-reducible: general spin systems, Potts models • not self-reducible: coloring • self-reducible superclass of coloring: list-coloring instance : undirected G(V,E) each vertex v associated { } { } with a list L v of colors { } allowed to use on v coloring ⊂ list-coloring ⊂ Potts ⊂ multi-spin { } { } | {z } self-reducible new goal: multi-spin system ( ) approximate the marginal for Potts model (with additive error) P [ X v = x ] list-coloring new way: correlation decay classic way: random walk

Recursion for List-Coloring list-coloring instance Ω v ’s neighbors: v 1 , v 2 , . . . , v d : delete v Ω i color: x v,x ∀ j < i , delete x v v from list L v j v 1 v d v 1 v d v i − 1 n v i G G L v j − { x } Gamarnik-Katz’07: Q d v,x ( X v i 6 = x ) i =1 P Ω i P Ω ( X v = x ) = Q d P v,x ( X v i 6 = y ) i =1 P Ω i y ∈ L v ⇣ ⌘ Q d 1 � P Ω i v,x ( X v i = x ) i =1 = ⇣ ⌘ Q d P 1 � P Ω i v,x ( X v i = y ) y ∈ L v i =1

Recursion for List-Coloring list-coloring instance Ω v ’s neighbors: v 1 , v 2 , . . . , v d : delete v Ω i color: x v,x ∀ j < i , delete x v v from list L v j v 1 v d v 1 v d v i − 1 n v i G G L v j − { x } Gamarnik-Katz’07: Q d telescopic v,x ( X v i 6 = x ) i =1 P Ω i P Ω ( X v = x ) = products Q d P v,x ( X v i 6 = y ) i =1 P Ω i y ∈ L v ⇣ ⌘ Q d 1 � P Ω i v,x ( X v i = x ) i =1 = ⇣ ⌘ Q d P 1 � P Ω i v,x ( X v i = y ) y ∈ L v i =1

Recursion for general multi-spin system a natural generalization of list-coloring: : delete v Ω i multi-spin system Ω v,x ∀ j < i , new external field v ’s neighbors: v 1 , v 2 , . . . , v d state: x F 0 v j ( y ) = A vv j ( x, y ) F v j ( y ) v v augmented by edge v 1 v d v 1 v d v i − 1 n v i G G activity F 0 v j ✓ ◆ z 6 = x ( A vvi ( x,x ) − A vvi ( x,z ) ) P Ω i F v ( x ) Q d A vvi ( x,x ) − P v,x ( X vi = z ) i =1 P Ω ( X v = x ) = ✓ ◆ z 6 = y ( A vvi ( y,y ) − A vvi ( y,z ) ) P Ω i y 2 [ q ] F v ( y ) Q d P A vvi ( y,y ) − P v,y ( X vi = z ) i =1

Recursion for general multi-spin system a natural generalization of list-coloring: : delete v Ω i multi-spin system Ω v,x ∀ j < i , new external field v ’s neighbors: v 1 , v 2 , . . . , v d state: x F 0 v j ( y ) = A vv j ( x, y ) F v j ( y ) v v augmented by edge v 1 v d v 1 v d v i − 1 n v i G G activity F 0 v j ✓ ◆ z 6 = x ( A vvi ( x,x ) − A vvi ( x,z ) ) P Ω i F v ( x ) Q d A vvi ( x,x ) − P v,x ( X vi = z ) i =1 P Ω ( X v = x ) = ✓ ◆ z 6 = y ( A vvi ( y,y ) − A vvi ( y,z ) ) P Ω i y 2 [ q ] F v ( y ) Q d P A vvi ( y,y ) − P v,y ( X vi = z ) i =1 for list-coloring: special case ⇣ ⌘ Q d 1 − P Ω i v,x ( X v i = x ) i =1 P Ω ( X v = x ) = ⇣ ⌘ Q d P 1 − P Ω i v,x ( X v i = y ) y ∈ L v i =1

Correlation Decay vector p = ( p i,y,z ) 1  i  d ; y,z 2 [ q ]; y 6 = z where p i,y,z = P Ω i v,y ( X v i = z ) i =1 ( A vvi ( x,x ) − P z 6 = x ( A vvi ( x,x ) − A vvi ( x,z ) ) p i,x,z ) F v ( x ) Q d f ( p ) = i =1 ( A vvi ( y,y ) − P z 6 = y ( A vvi ( y,y ) − A vvi ( y,z ) ) p i,y,z ) y 2 [ q ] F v ( y ) Q d P • an exponential-time exact P Ω [ X v = x ] = f ( p ) algorithm p i,y,z recursion tree