TY - GEN

T1 - Trichotomy for integer linear systems based on their sign patterns

AU - Kimura, Kei

AU - Makino, Kazuhisa

PY - 2012

Y1 - 2012

N2 - 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 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.

AB - 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 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.

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U2 - 10.4230/LIPIcs.STACS.2012.613

DO - 10.4230/LIPIcs.STACS.2012.613

M3 - Conference contribution

AN - SCOPUS:84880281339

SN - 9783939897354

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 613

EP - 623

BT - 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012

T2 - 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012

Y2 - 29 February 2012 through 3 March 2012

ER -