Generating All Minimal Integral Solutions to Monotone ∧,∨-Systems of Linear, Transversal and Polymatroid Inequalities

Abstract

We consider monotone ∨, ∧-formulae φ of m atoms, each of which is a monotone inequality of the form f i (x)≥ t i over the integers, where for i = 1,...,m, \(f_i : \mathbb{Z}^n \mapsto \mathbb{R}\) is a given monotone function and t i 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.