Max - cut Problem

Max-cut problem is one of many NP-hard graph theory problems which attracted many researchers over the years. Though there is almost no hope in finding a polynomialtime algorithm for max-cut problem, various heuristics, or combination of optimization and heuristic methods have been developed to solve this problem. Among them is the efficient algorithm of Goemans and Williamson. Their algorithm combines Semidefinite programming and a rounding procedure to produce an approximate solution to the max-cut problem. This approximate solution, however, turns out to be quite good and its error is bounded by 0.878. This report is organized as follows. We first define the max-cut problem, and review the notation in 1.1 and 1.2. In section 2, we formulate the max-cut problem. Section 3 is dedicated to the Goemans-Williamson algorithm. Finally in section 4, we talk about "Dual-scaling algorithm". This algorithm is proposed for solving large-scale sparse SDP’s by Benson et al. in [2]. We explain the algorithm, and we show how it can be used to solve the SDP relaxation of max-cut problem.