Abstract.
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 four-dimensional cone with unimodular facets that has no HILBERT PARTITION; and two five-dimensional cones that do not have any BINARY HILBERT COVER.
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Received September 5, 1997, and in revised form October 10, 1997.
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Firla, R., Ziegler, G. Hilbert Bases, Unimodular Triangulations, and Binary Covers of Rational Polyhedral Cones . Discrete Comput Geom 21, 205–216 (1999). https://doi.org/10.1007/PL00009416
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DOI: https://doi.org/10.1007/PL00009416