Theory of linear and integer programming
Theory of linear and integer programming
Non-negative integer basis algorithms for linear equations with integer coefficients
Journal of Automated Reasoning
Efficient solution of linear diophantine equations
Journal of Symbolic Computation
An efficient incremental algorithm for solving systems of linear Diophantine equations
Information and Computation
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
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
Basis of solutions for a system of linear inequalities in integers: computation and applications
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
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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 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 prove that, given a linear Diophantine system and a set of solutions, asking whether this set constitutes the Hilbert beisis of the system, is also coNP-complete in the strong sense, answering this way an open problem formulated by Henk and Weismantel in 1996. Our result also allows us to solve another open problem, formulated by Edmonds and Giles in 1982, where we prove that Eisking whether a given set of vectors constitutes the Hilbert b2usis of an unknown linear Diophantine system, is coNP-complete in the strong sense.