Generating all minimal integral solutions to AND–OR systems of monotone inequalities: Conjunctions are simpler than disjunctions

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 : Z n ↦ 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.