Approximability of the ground state problem for certain Ising spin glasses

Abstract

We consider polynomial time algorithms for finding approximate solutions to the ground state problem for the following three-dimensional case of an Ising spin glass: 2n spins are arranged on a two-level grid with at most nγ vertical interactions (0 ≤ γ ≤ 1). The main results are: 1. Let 1/2 ≤ γ < 1. There is an approximate polynomial time algorithm with absolute error less than nγ for all n; there exists a constant β > 0 such that every approximate polynomial time algorithm has absolute error greater than βnγ infinitely often, unless P = NP. 2. Let γ = 1. There is an approximate polynomial time algorithm with absolute error less than n/lg n; there exists a number k > 1 such that every approximate polynomial time algorithm has absolute error greater than n/(1g n)k infinitely often iff NP ⊈ ∩ε>0 DTIME(2nε). © 1997 Academic Press.