Before revision:  We address the question of whether one can achieve a simple, elementary O(\alpha)-approximation algorithm  not based on any LP-based methods, either in the centralized setting or in the distributed setting.
After revision:  We address the question of whether one can achieve a simple, elementary O(\alpha)-approximation algorithm that is elementary, i.e., not based on any LP-based methods, either in the centralized setting or in the distributed setting.