New bounds on the unconstrained quadratic integer programming problem

We consider the maximization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \max\{x^{T}\!Ax : x\in \{-1, 1\}^n\}$$\end{document} for a given symmetric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\in\mathcal{R}^{n\times n}$$\end{document}. 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 = VVT where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in\mathcal{R}^{n\times m}$$\end{document} with m < n, then there exists a polynomial time algorithm (polynomial in n for a given m) to solve the problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max\{x^TV V^Tx : x\in \{-1, 1\}^n\}$$\end{document}. 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.