  Over the last years the vertex enumeration problem of polyhedra has seen a
revival in the study of metabolic networks, which increased the demand for
efficient vertex enumeration algorithms for high-dimensional polyhedra given by
inequalities. It is a famous and long standing open question in polyhedral
theory and computational geometry whether the vertices of a polytope (bounded
polyhedron), described by a set of linear constraints, can be enumerated in
total polynomial time. In this paper we apply the concept of
branch-decomposition to the vertex enumeration problem of polyhedra $P = \{x :
Ax = b, x \geq 0\}$. For this purpose, we introduce the concept of $k$-module
and show how it relates to the separators of the linear matroid generated by
the columns of $A$. We then use this to present a total polynomial time
algorithm for polytopes $P$ for which the branch-width of the linear matroid
generated by $A$ is bounded by a constant $k$.
