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

TL;DR

This paper proves that determining whether the $rac{1}{2}$-closure of a rational polyhedron matches its integer hull or is totally dual integral is strongly NP-hard, highlighting computational complexity challenges in this area.

Contribution

It establishes the NP-hardness of recognizing integrality and total dual integrality of the $rac{1}{2}$-closure, a previously unresolved complexity question.

Findings

Deciding if the $rac{1}{2}$-closure equals the integer hull is strongly NP-hard.

Testing total dual integrality of the $rac{1}{2}$-closure is strongly NP-hard.

The results highlight computational difficulty in verifying certain polyhedral properties.

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.