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∈^Rm×n, a vector b∈^Rm, and a positive integer d, to compute an integer vector x∈^Dn such that Ax≥b or to determine the infeasibility of the system, 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 depends 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 solvable in linear time, if γ<1,ILS(γ) is weakly NP-hard and pseudo-polynomially solvable, if γ=1,ILS(γ) is strongly NP-hard, if γ>1. This, for example, includes the previous results that Horn systems and two-variable-per-inequality (TVPI) systems can be solved in pseudo-polynomial time.