Trichotomy for integer linear systems based on their sign patterns

In this paper, we consider solving the integer linear systems, i.e., given a matrix A ∈ R^m×n, a vector b ∈ R^m, and a positive integer d, to compute an integer vector x ∈ Dⁿ such that Ax ≥ b, where m and n denote positive integers, R denotes the set of reals, and D = {0,1,⋯, d - 1}. The problem is one of the most fundamental NP-hard problems in computer science. For the problem, we propose a complexity index η which is based only on the sign pattern of A. For a real γ, let ILS₌(γ) denote the family of the problem instances I with η(I) = γ. We then show the following trichotomy: ILS ₌(γ) is linearly solvable, if γ < 1, ILS ₌(γ) is weakly NP-hard and pseudo-polynomially solvable, if γ = 1, and ILS₌(γ) is strongly NP-hard, if γ > 1. This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time.