Trichotomy for Integer Linear Systems Based on Their Sign Patterns

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.