Theory of linear and integer programming
Theory of linear and integer programming
A Unification Algorithm for Associative-Commutative Functions
Journal of the ACM (JACM)
On the complexity of integer programming
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Hilbert Bases, Caratheodory's Theorem and Combinatorial Optimization
Proceedings of the 1st Integer Programming and Combinatorial Optimization Conference
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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.