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It is known that any rational convex polytope expressed as $\{ x\in\mathbb{R}^d : Ax \ge b \}$, where $A\in\mathbb{Z}^{k\times d}$ and $b\in\mathbb{Z}^k$, can be written as the convex hull of finitely many points.

My question is, given the above representation in terms of hyperplanes, one can determine (easily) whether the polytope is integral ---that is, whether the polytope can be written as the convex hull of points in the integer lattice.

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Are you looking for a theoretical solution for a class of examples, or a computational solution for specific instances? – Gordon Royle Oct 24 '12 at 5:28
I am currently interested in figuring out whether, under some reasonable conditions, certain polytopes I am studying are integral. The class of polytopes I am looking at must satisfy some specific symmetry conditions. If for certain parameters I could argue that the polytope must be totally unimodular (as someone pointed out below), then that would be nice ... – John Doe Oct 27 '12 at 18:46
up vote 1 down vote accepted

An incomplete solution:

There is a polynomial-time test for total unimodularity, and if $A$ is totally unimodular, then the polytope is integral.

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What if $A$ is not unimodular? E.g., $A=\begin{pmatrix}1 & 0\\ 0 & 1\\ 2 & 1 \end{pmatrix}$ and $b=\begin{pmatrix}0 \\ 0 \\ 2\end{pmatrix}$? – J.C. Ottem Oct 24 '12 at 4:52
Obviously he doesn't (yet) have an answer for that case. $\:$ – Ricky Demer Oct 24 '12 at 4:53
(the example should have been $A=\begin{pmatrix}1 & 0\\ 0 &1\\ 2 & -1\end{pmatrix}$ and $b=\begin{pmatrix}0 \\ 0 \\ -2\end{pmatrix}$.) – J.C. Ottem Oct 24 '12 at 4:56
Right, if $A$ is not totally unimodular then this test is inconclusive. I'm not aware of an efficient test that works for all $A,b$, though I'd certainly be interested to see one! – Michael Biro Oct 24 '12 at 5:29

This paper shows that testing integrality is coNP-complete.

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