Maximizing Quadratic Programs: Extending Grothendieck's Inequality

This paper considers the following type of quadratic programming problem. Given an arbitrary matrix A, whose diagonal elements are zero, find x {-1, 1}^n such that x^TAx is maximized. Our approximation algorithm for this problem uses the canonical semidefinite relaxation and returns a solution whose ratio to the optimum is in (1/log n). This quadratic programming problem can be seen as an extension to that of maximizing x^TAy (where y's components are also 1). Grothendieck's inequality states that the ratio of the optimum value of the latter problem to the optimum of its canonical semidefinite relaxation is bounded below by a constant. The study of this type of quadratic program arose from a desire to approximate the maximum correlation in correlation clustering. Nothing substantive was known about this problem; we present an (1/log n) approximation, based on our quadratic programming algorithm. We can also guarantee that our quadratic programming algorithm returns a solution to the MAXCUT problem that has a significant advantage over a random assignment.