NP-complete problems for systems of divisibilities of values of linear polynomials

Abstract

We study the algorithmic complexity of the subproblems of simultaneous divisibility of values of linear polynomials with non-negative coefficients in positive integers. In general, it is not known whether it is in NP, but it was previously proved it to be in NEXPTIME. The NP-completeness is proved for such two series of this problem, that in the first of them a number divides a linear polynomial, and in the second case linear polynomial is a divisor of a number. Values for the parameters were determined, for which the problem is NP-complete. For the values less than specified, it was proved the problem to be in the class P. We prove the NP-hardness of the particular case of the general problem of simultaneous divisibility of values of linear polynomials in which the coefficients of the variables may only take values from the set {1, 2} and the constant terms of linear polynomials may only take values from the set {1, 5}. The obtained results may be useful for determining the algorithmic complexity of the general problem. In Section 1, it is proved that the problem ∃x1 . . . xn &mj =1(Lj (x1 . . . xn) | K) is NP-complete for K ≥ 4 and exactly three non-zero coefficients in each linear polynomial. It is from the class P for 0 < K < 4. In Section 2, we studied the problem of simultaneous divisibility in positive integers from the nontrivial interval [D,D′], of values of linear polynomials with non-negative integer coefficients by a positive integer K: ∃x1 . . . xn ( &n i=1xi ∈ [D,D′] & &mj =1K | Lj (x1 . . . xn)). It is proved to be NP-complete for K ≥ 3. If the parameter of the problem is the number of non-zero coefficients in linear polynomials, the problem is NP-complete for only two non-zero coefficients. In the case of exactly one non-zero coefficient, the problem is from the class P. In Section 3, it is proved NP-hardness of ∃x1 . . . xn &mj =1(Lj (x1 . . . xn) | L′ j (x1 . . . xn)), the problem of simultaneous divisibility of the values of linear polynomials on the values of the linear polynomials with the coefficients of the variables from {1, 2} and constant terms from {1, 5}. Refs 13.