Challenging Computations of Hilbert Bases of Cones Associated with Algebraic Statistics

Abstract

In this paper we present two independent computational proofs that the monoid derived from 5×5×3 contingency tables is normal, completing the classification by Hibi and Ohsugi. We show that Vlach's vector disproving normality for the monoid derived from 6×4×3 contingency tables is the unique minimal such vector up to symmetry. Finally, we compute the full Hilbert basis of the cone associated with the nonnormal monoid of the semigraphoid for |N|=5. The computations are based on extensions of the packages LattE-4ti2 and Normaliz.

2000 AMS Subject Classification:

• 13P99
• 14M25
• 52B20

Keywords:

• Affine monoid
• normalization
• rational cone
• Hilbert basis
• contingency table

Acknowledgments

B. Ichim was partially supported by CNCSIS grant RP-1 no. 7/01.07.2009 during the preparation of this work. M. Köppe was partially supported by grant DMS-0914873 of the National Science Foundation.

Notes

¹Available at http://www.math.uos.de/normaliz.

²Available at http://www.4ti2.de.

³Available at http://www.latte-4ti2.de.

⁴Available at http://www.math.uos.de/normaliz.

⁵Available at http://cs.anu.edu.au/bdm/nauty/.