New bounds on the unconstrained quadratic integer programming problem

We consider the maximization γ = max{x^TAx : x∈{-1, 1}^n} for a given symmetric A∈ℛ^n× n . 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∈ℛ^n× m with m < n , then there exists a polynomial time algorithm (polynomial in n for a given m ) to solve the problem max{x^TV V^Tx : 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 γ with no constraints on the rank of A . We also give easily computable necessary and sufficient conditions for the absence of a gap between the solution and its upper bound. Finally, we incorporate the semidefinite relaxation upper bound into our scheme and give an illustrative example.