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 Ω(1/ logn). 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 Ω (1/logn) 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.