## DIMACS TR: 2001-12

## Dual-Bounded Generating Problems: All Minimal Integer Solutions for a Monotone System of Linear Inequalities

### Authors: Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Leonid Khachiyan and Kazuhisa Makino

**
ABSTRACT
**

We consider the problem of enumerating all minimal integer
solutions of a monotone system of linear inequalities. We first show
that for any monotone system of $r$ linear inequalities in $n$
variables, the number of maximal infeasible integer vectors is at
most $rn$ times the number of minimal integer solutions to the
system. This bound is accurate up to a $polylog(r)$ factor and
leads to a polynomial-time reduction of the enumeration problem to
a natural generalization of the well-known dualization problem for
hypergraphs, in which dual pairs of hypergraphs are replaced by
dual collections of integer vectors in a box. We provide a
quasi-polynomial algorithm for the latter dualization problem.
These results imply, in particular, that the problem of
incrementally generating
all minimal integer solutions to a monotone system of linear
inequalities can be done in quasi-polynomial time.

Paper Available at:
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