# Trichotomy for integer linear systems based on their sign patterns

Kei Kimura, Kazuhisa Makino

4 被引用数 (Scopus)

## 抄録

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.

本文言語 英語 67-78 12 Discrete Applied Mathematics 200 https://doi.org/10.1016/j.dam.2015.07.004 出版済み - 2 19 2016 はい

• 離散数学と組合せ数学
• 応用数学

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