Challenging computations of hilbert bases of cones associated with algebraic statistics

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.