Trichotomy for Integer Linear Systems Based on Their Sign Patterns

Kei Kimura , Kazuhisa Makino

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.

Mots clés: Integer linear system; Sign pattern; Complexity index; TVPI system; Horn system

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Kei Kimura, Kazuhisa Makino. Trichotomy for Integer Linear Systems Based on Their Sign Patterns. STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), Feb 2012, Paris, France. pp.613-623. ⟨hal-00678207⟩