NP-completeness conditions for some types of systems of linear diophantine dis-equations consistency checking

Abstract

Three series of number-theoretic problems concerning systems of Diophantine linear dis-equations with explicitly pointed out parameters are proposed in this part of the paper. Conditions upon the parameters implying that every problem of a series is an NP-complete one are proved. It is proved that for every m and m (m < m ) the consistency problem for a system of Diophantine linear dis-equations every of which contains exactly 3 variables (even if every coefficient belongs to {−1, 1}) is NP-complete. This problem also admits a simple geometrical interpretation concerning NP-completeness of the checking whether inside a many-dimensional cube there exists an integer-valued point which does not belong to any of the given hyperplanes which cut off equal segments of three axes and are parallel to the other ones. If every dis-equation of a system of Diophantine linear dis-equations contains exactly 2 variables then the problem remains an NP-complete one under the condition that m −m > 2. It is also proved that if a solution of a system of Diophantine linear dis-equations every of which contains exactly 3 variables must belong to a domain, which is defined by a system of polynomial inequalities, contains an n-dimentional cube and is contained in an n-dimentional parallelogramm, then it is an NP-complete one. Refs 15.