  The $\{0,\frac{1}{2}\}$-closure of a rational polyhedron $\{ x \colon Ax \le
b \}$ is obtained by adding all Gomory-Chv\'atal cuts that can be derived from
the linear system $Ax \le b$ using multipliers in $\{0,\frac{1}{2}\}$. We show
that deciding whether the $\{0,\frac{1}{2}\}$-closure coincides with the
integer hull is strongly NP-hard. A direct consequence of our proof is that,
testing whether the linear description of the $\{0,\frac{1}{2}\}$-closure
derived from $Ax \le b$ is totally dual integral, is strongly NP-hard.
