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

Abstract

Three series of number-theoretic problems concerning systems of modulo m comparisons and systems of Diophantine linear 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 m1 and m2 (m1 < m2 and m1 is not a divisor of m2) the consistency problem for a system of simultaneously modulo m1 and m2 comparisons every of which contains exactly 3 variables is NP-complete. It is also proved that for every m >2 the consistency on the subset containing at least two elements of the set {0, . . .,m − 1} problem for a system of modulo m comparisons every of which contains exactly 3 variables is NP-complete. If P =NP then the statement of the proved theorem can not be changed by means of replacing the term «3-comparison» with the term «2-comparison». For a system of Diophantine linear equations every of which contains exactly 3 variables it is proved the NP-completeness of the problem of its consistency on the interval of integers. If P =NP then the statement of the proved theorem can not be changed by means of replacing the term «3-equation» with the term «2-equation». This problem also admits a simple geometrical interpretation concerning NP-completeness of the checking whether there exists inside a many-dimensional cube an integer-valued point of intersection of hyperplanes which cut off equal segments of three axes and are parallel to the other ones. Among the problems of the series there are practically useful problems. If P =NP then, as the set of all values for a variable of a computer type integer may be considered as the set of integer values from a segment then theorem proves that there does not exist a polynomial algorithm solving such a system in the set of all numbers of the type integer. Refs 6.