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

On the Complexity of Recognizing Integrality and Total Dual Integrality of the $\{0,1/2\}$-Closure