integer valued outer normal vectors

Question

Suppose a bounded polyhedra $C$ is given by

$$x\in \mathbb R^n: Ax\leq b$$

The matrix $A\in\mathbb R^{m\times n}$ contains only elements from $\{-1,0,1\}$, which implies the outer normal vectors of the facets of $C$ are integers?

This property seems very special for certain classes of polyhedra? Is there any example or implication for this property?

Thanks.

Answer 1

It's very unclear as to what you're asking as there's really nothing special about matrices containing only $\{0,1,-1\}$ entries. The normals to such matrices can point in many possible directions.

You might be interested in the concept of totally unimodular matrices, which are square matrices with determinant equal to $\pm 1$. These matrices resemble what you're describing - many (but not all) matrices with $\{0,1,-1\}$ entries are totally unimodular. The ramifications of this property is that all extreme points of the polyhedron described by the matrix are integers, which implies that integer programs with this constraint matrix have the same solution set as a linear program with the same constraint matrix, i.e. the linear relaxation has the same solution as the integer problem.