Inapproximability After Uniqueness Phase Transition in Two-Spin Systems
Abstract
A two-state spin system is specified by a 2 x 2 matrix A = A0,0 A0,1, A1,0 A1,1 = β 1, 1 γ where β, γ 0. Given an input graph G=(V,E), the partition function ZA(G) of a system is defined as ZA(G) = Σσ: V -> 0,1 Π(u,v) ∈ E Aσ(u), σ(v) We prove inapproximability results for the partition function in the region specified by the non-uniqueness condition from phase transition for the Gibbs measure. More specifically, assuming NP RP, for any fixed β, γ in the unit square, there is no randomized polynomial-time algorithm that approximates ZA(G) for d-regular graphs G with relative error ε = 10-4, if d = ((β,γ)), where (β,γ) > 1/(1-βγ) is the uniqueness threshold. Up to a constant factor, this hardness result confirms the conjecture that the uniqueness phase transition coincides with the transition from computational tractability to intractability for ZA(G). We also show a matching inapproximability result for a region of parameters β, γ outside the unit square, and all our results generalize to partition functions with an external field.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.