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 $5\times 5\times 3$ contingency tables is normal, completing the classification by Hibi and Ohsugi. We show that Vlach's vector disproving normality for the monoid derived from $6\times 4\times 3$ contingency tables is the unique minimal such vector up to symmetry. Finally, we compute the full Hilbert basis of the cone associated with the non-normal monoid of the semi-graphoid for $|N|=5$. The computations are based on extensions of the packages LattE-4ti2 and Normaliz.Comment: 10 page

On the Complexity of Recognizing the Hilbert Basis of a Linear Diophantine System

. The problem of computing the Hilbert basis of a linear Diophantine system over nonnegative integers is often considered in automated deduction and integer programming. In automated deduction, the Hilbert basis of a corresponding system serves to compute the minimal complete set of associative-commutative uniers, whereas in integer programming the Hilbert bases are tightly connected to integer polyhedra and to the notion of total dual integrality. In this paper, we sharpen the previously known result that the problem, asking whether a given solution belongs to the Hilbert basis of a given system, is coNP-complete. We show that the problem has a pseudopolynomial algorithm if the number of equations in the system is xed, but it is coNP-complete in the strong sense if the given system is unbounded. This result is important in the scope of automated deduction, where the input is given in unary and therefore the previously known coNP-completeness result was unusable. Moreover, we prove t..

On the Complexity of Recognizing the Hilbert Basis of a Linear Diophantine System

Article dans revue scientifique avec comité de lecture.The problem of computing the Hilbert basis of a homogeneous linear Diophantine system over nonnegative integers is often considered in automated deduction and integer programming. In automated deduction, the Hilbert basis of a corresponding system serves to compute the minimal complete set of associative-commutative unifiers, whereas in integer programming the Hilbert bases are tightly connected to integer polyhedra and to the notion of total dual integrality. In this paper, we sharpen the previously known result that the problem, asking whether a given solution belongs to the Hilbert basis of a given system, is coNP-complete. We show that the problem has a pseudopolynomial algorithm if the number of equations in the system is fixed, but it is coNP-complete in the strong sense if the given system is unbounded. This result is important in the scope of automated deduction, where the input is given in unary and therefore the previously known coNP-completeness result was unusable. Moreover, we show that, from the complexity standpoint, it is not important to know the underlying homogeneous linear Diophantine system when we ask whether a given set of vectors constitutes a Hilbert basis