Primal separation and approximation for the {0,1/2}-closure

The primal separation problem for $\{0,1/2\}$-cuts is: Given a vertex $\hat{x}$ of the integer hull of a polytope P and some fractional point $x^{⁎}\in P$, does there exist a $\{0,1/2\}$-cut that is tight at $\hat{x}$ and violated by $x^{⁎}$? We present two cases for which primal separation is solvable in polynomial time. Furthermore, we show that the optimization problem over the $\{0,1/2\}$-closure can be solved in polynomial time up to a factor $(1+\epsilon )$, for any fixed $\epsilon >0$.