Given a collection of events in a probability space it is often desirable to bound from below the probability of arbitrary intersections of them, in the presence of negative correlation, i.e., when the occurrence of each event may make the occurrence of certain other events less likely. A classic tool for modelling negative correlation is a "dependency" graph, whose vertices correspond to events and whose edges indicate dependence. When the dependency graph is sufficiently sparse, the famous Lovász Local Lemma gives an explicit, strictly positive lower bound for the probability of every intersection. Here we extend the dependency graph setting from probabilities of intersections of events to arbitrary supermodular functions and derive corresponding explicit, strictly positive lower bounds.