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

Abstract: 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.