Before revision:  Applying a covering scheme, we can boost any 2-approximation algorithm for CVP to a (1+\epsilon)-approximation algorithm with the improved run time, either using a straightforward sampling routine or using the deterministic algorithm of Dadush for the construction of an epsilon net.
After revision:  Applying a covering scheme, we can boost any constant approximation algorithm for CVP to a (1+\epsilon)-approximation algorithm with the improved run time, either using a straightforward sampling routine or using the deterministic algorithm of Dadush for the construction of an epsilon net.