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, 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

• Type: Journal
• Year: 2016
• Where: DAM
• Authors: Kei Kimura, Kazuhisa Makino
• Keywords: DAM 2016