Hilbert Bases, Unimodular Triangulations, and Binary Covers of Rational Polyhedral Cones

We present a hierarchy of covering properties of rational convex cones with respect to the unimodular subcones spanned by the Hilbert basis. For two of the concepts from the hierarchy we derive characterizations: a description of partitions that leads to a natural integer programming formulation for the Hilbert Partition problem, and a characterization of \binary covers" that admits a linear algebra test over GF(2) for the existence of Binary Hilbert Covers. Implementation of our test leads to interesting new examples, among them: cones that have a Hilbert Partition but no Regular one; a 4-dimensional cone with unimodular facets that has no Hilbert Partition; and two 5-dimensional cones that do not have any Binary Hilbert Cover.