Approximation Bounds for Quadratic Maximization with Semidefinite Programming Relaxation

In this paper, we consider a class of quadratic maximization problems. One important instance in that class is the famous quadratic maximization formulation of the max-cut problem studied by Goemans and Williamson [6]. Since the problem is NP-hard in general, following Goemans and Williamson, we apply the approximation method based on the semidefinite programming (SDP) relaxation. For a subclass of the problems, including the ones studied by Helmberg [9] and Zhang [23], we show that the SDP relaxation approach yields an approximation solution with the worst-case performance ratio at least α = 0.87856 · · ·. This is a generalization of the results obtained in [6, 9, 23]. In fact, the estimated worst-case performance ratio is dependent on the data of the problem with α being a uniform lower bound. In light of this new bound, we study the original max-cut problem and show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at least α + δd, where δd > 0 is a constant depending on the problem dimension and data. Karloff [10] showed that for any positive ǫ > 0 there is an instance of the max-cut problem such that the SDP relaxation (with triangle inequalities) bound is worse than α+ ǫ. Hence the improvement is in this sense best possible for the Goemans and Williamson type approach to the max-cut problem.