Approximate Counting fix A= partition function: zA(G)=∑Π Aa(u),o(v) o∈{0,1}V(u,w)∈E is a well-define computational problem poly-time computable if By=1 or (B,Y)=(0,0) #P-hard if otherwise Approximation!
Approximate Counting ZA(G) = {0,1}V (u,v)E A(u),(v) partition function: A = A0,0 A0,1 A1,0 A1,1 = 1 1 fix is a well-define computational problem poly-time computable if = 1 or (, ) = (0, 0) #P-hard if otherwise Approximation!
US93] Jerrum-Sinclair'93 [GJP03]Goldberg-Jerrum-Paterson'03 Y 3 ferromagnetic 2-state spin 2.5 87=1 FPRAS [GJP03] ferromagnetic Ising Model 1.5 anti- .FPRAS [JS93] ferromagnetic 1.11017 1 0≤B,Y≤1 0.5 no FPRAS unless NPCRP FPRAS [GJP03] [GJP03] 、heat-bath 8 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0< , <1 = 1 uniqueness threshold threshold achieved by heatbath random walk 1.11017 0 , 1 ferromagnetic Ising Model ferromagnetic 2-state spin FPRAS [JS93] FPRAS [GJP03] FPRAS [GJP03] heat-bath no FPRAS unless NP⊆RP [GJP03] antiferromagnetic Goldberg-Jerrum-Paterson’03 Jerrum-Sinclair’93 [GJP03] [JS93]
Uniqueness Threshold 1.11017 1 0四 @(+ d 产0四 1.11017 元=f(金) 四 B Y f'()川<1 W 0 0 W 0 for all d B 0≤B<1<Y
1.11017 Uniqueness Threshold f(x) = x + 1 x + d x ˆ = f(ˆx) |f (ˆx)| < 1 0 < 1 < for all d
Our Result Y 3 2.5 ←—3y=1 2 1.5 ------- 1.11017 0≤B,Y≤1 0.5 FPTAS 8 0.5 1 1.5 2 2.5 3 B
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0< , <1 = 1 uniqueness threshold threshold achieved by heatbath random walk 1.11017 0 , 1 Our Result FPTAS
Marginal Distribution weight:w()=A(u).() (u,v)∈E w(o) Gibbs measure: L()=ZA(G) Z4(G)=∑ ΠA(w,o) o∈{0,1V(u,w)∈E marginal distributions at vertex v: p=Pr [o(v)=0] ΛCVoA∈{0,1}A fixed v∈A free vA Pr [o(v)=01(A)=OA]
Marginal Distribution weight: w() = (u,v)E A(u),(v) Gibbs measure: µ() = w() ZA(G) ZA(G) = {0,1}V (u,v)E A(u),(v) pv = 8Z µ [(v) = 0] p v = 8Z µ [(v) = 0 | () = ] {0, 1} marginal distributions at vertex v: V fixed v free v