Approximation Bounds for Max-Cut Problem with Semidefinite Programming Relaxation

In this paper, we consider the max-cut problem as studied by Goemans and Williamson [8]. Since the problem is NP-hard in general, following Goemans and Williamson, we apply the approximation method based on the semidefinite programming (SDP) relaxation. 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 show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at least α + δd if every weight is strictly positive, where δd > 0 is a constant depending on the problem dimension and data. Alon, Sudakov and Zwick [2] show that the local analyses of the Goemans and Williamson algorithm and its extension by Zwick [25] are tight even if arbitrary collections of valid linear constraints are added to the SDP relaxation. Hence the improvement is in this sense best possible for the Goemans and Williamson type approach to the max-cut problem.