Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems

Improving and extending recent results of Matuura and Matsui, and less recent results of Feige and Goemans, we obtain improved approximation algorithms for the MAX 2-SAT and MAX DI-CUT problems. These approximation algorithms start by solving semidefinite programming relaxations of these problems. They then rotate the solution obtained, as suggested by Feige and Goemans. Finally, they round the rotated vectors using random hyperplanes chosen according to skewed distributions. The performance ratio obtained by the MAX 2-SAT algorithm is at least 0.940, while that obtained by the MAX DI-CUT algorithm is at least 0.874. We show that these are essentially the best performance ratios that can be achieved using any combination of prerounding rotations and skewed distributions of hyperplanes, and even using more general families of rounding procedures. The performance ratio obtained for the MAX 2-SAT problem is fairly close to the inapproximability bound of about 0.954 obtained by Hastad. The performance ratio obtained for the MAX DI-CUT problem is very close to the performance ratio of about 0.878 obtained by Goemans and Williamson for the MAX CUT problem.