Approximability of the Ground State Problem for Certain Ising Spin Glasses

We consider polynomial time algorithms for finding approximate solutions to the groundstateproblem for the following three-dimensional case of an Isingspinglass: 2nspins are arranged on a two-level grid with at mostn\u3b3vertical interactions (0 64 \u3b3 64 1). The main results are:1. Let 64 \u3b3 0 such that every approximate polynomial time algorithm has absolute error greater than \u3b2n\u3b3infinitely often, unlessP=NP.2. Let \u3b3 = 1. There is an approximate polynomial time algorithm with absolute error less thann/lgn; there exists a numberk> 1 such that every approximate polynomial time algorithm has absolute error greater thann/(lgn)kinfinitely often iffNP 88 29\u3b5 > 0DTIME(2n\u3b5)