Trichotomy for Integer Linear Systems Based on Their Sign Patterns
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.
Integer linear system
Sign pattern
Complexity index
TVPI system
Horn system
613-623
Regular Paper
Kei
Kimura
Kei Kimura
Kazuhisa
Makino
Kazuhisa Makino
10.4230/LIPIcs.STACS.2012.613
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