New bounds on the unconstrained quadratic integer programming problem

We consider the maximization = max{xTAx : x {1, 1}n} for a given symmetric A Rnn. It was shown recently, using properties of zonotopes and hyperplane arrangements, that in the special case that A has a small rank, so that A = VV T where V Rnm with m < n, then there exists a polynomial time algorithm (polynomial in n for a given m) to solve the problem max{xT VV T x : x {1, 1}n}. In this paper, we use this result, as well as a spectral decomposition of A to obtain a sequence of non-increasing upper bounds on with no constraints on the rank of A. We also give easily computable necessary and sufcient conditions for the absence of a gap between the solution and its upper bound. Finally, we incorporate the semidenite relaxation upper bound into our scheme and give an illustrative example. [PUBLICATION ABSTRACT]