Abstract
We consider monotone ∨, ∧-formulae φ of m atoms, each of which is a monotone inequality of the form fi(x)≥ ti over the integers, where for i = 1,...,m, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$f_i : \mathbb{Z}^n \mapsto \mathbb{R}$\end{document} is a given monotone function and ti is a given threshold. We show that if the ∨-degree of φ is bounded by a constant, then for linear, transversal and polymatroid monotone inequalities all minimal integer vectors satisfying φ can be generated in incremental quasi-polynomial time. In contrast, the enumeration problem for the disjunction of m inequalities is NP-hard when m is part of the input. We also discuss some applications of the above results in disjunctive programming, data mining, matroid and reliability theory.