Approximability of the ground state problem for certain Ising spin glasses

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 n(gamma) vertical interactions (0 less than or equal to gamma less than or equal to 1). The main results are: 1. Let 1/2 less than or equal to gamma < 1. There is an approximate polynomial time algorithm with absolute error less than n(gamma) for all n; there exists a constant beta > 0 such that every approximate polynomial time algorithm has absolute error greater than beta n(gamma) infinitely often, unless P = NP. 2. Let gamma = 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/(lg n)k infinitely often iff NP not subset of or equal to boolean AND epsilon > 0 DTIME(2(n epsilon)). (C) 1997 Academic Press.