Rank-Two Relaxation Heuristics for MAX-CUT and Other Binary Quadratic Programs

The Goemans-Williamson randomized algorithm guarantees a high-quality approximation to the Max-Cut problem, but the cost associated with such an approximation can be excessively high for large-scale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, rank-two relaxation and develop a specialized version of the Goemans-Williamson technique. The proposed approach leads to continuous optimization heuristics applicable to Max-Cut as well as other binary quadratic programs, for example the Max-Bisection problem. A computer code based on the rank-two relaxation heuristics is compared with two state-of-the-art semidefinite programming codes that implement the Goemans-Williamson randomized algorithm, as well as with a purely heuristic code for effectively solving a particular Max-Cut problem arising in physics. Computational results show that the proposed approach is fast and scalable and, more importantly, attains a higher approximation quality in practice than that of the Goemans-Williamson randomized algorithm. An extension to Max-Bisection is also discussed as well as an important difference between the proposed approach and the Goemans-Williamson algorithm, namely that the new approach does not guarantee an upper bound on the Max-Cut optimal value.