Summary: 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 \(\text{polylog}(r)\) 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 minimal integer solutions of a monotone system of linear inequalities can be done in quasi-polynomial time.
For the entire collection see [Zbl 0967.00069].