Approximating Quadratic Programs with Semidefinite Relaxations

Given an arbitrary matrix A in which all of the diagonal elements are zero, we would like to find x1, x2, . . . , xn ∈ {−1, 1} such that ∑n i=1 ∑n j=1 aijxixj is maximized. This problem has an important application in correlation clustering, and is also related to the well-known inequality of Grothendieck in functional analysis. While solving quadratic programs is NP-hard, we can approximate the solution by using the canonical semidefinite relaxation. We would like to know how good an approximation this provides. In this paper we explore this problem computationally in an attempt to find matrices A that result in a large gap between the semidefinite relaxation and the quadratic program.