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# Trichotomy for Integer Linear Systems Based on Their Sign Patterns

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LIPIcs.STACS.2012.613.pdf
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## Cite As

Kei Kimura and Kazuhisa Makino. Trichotomy for Integer Linear Systems Based on Their Sign Patterns. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 613-623, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
https://doi.org/10.4230/LIPIcs.STACS.2012.613

## Abstract

In this paper, we consider solving the integer linear systems, i.e., given a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n 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 h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy: - ILS_=(r) is linearly solvable, if r < 1, - ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and - ILS_=(r) is strongly NP-hard, if r > 1. This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time.
##### Keywords
• Integer linear system
• Sign pattern
• Complexity index
• TVPI system
• Horn system

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