The lefthanded local lemma characterizes chordal dependency graphs

Shearer gave a general theorem characterizing the family L of dependency graphs labeled with probabilities pv which have the property that for any family of events with a dependency graph from L (whose vertexlabels are upper bounds on the probabilities of the events), there is a positive probability that none of the events from the family occur. We show that, unlike the standard Lovász Local Lemma—which is less powerful than Shearer’s condition on every nonempty graph—a recently proved ‘Lefthanded’ version of the Local Lemma is equivalent to Shearer’s condition for all chordal graphs. This also leads to a simple and efficient algorithm to check whether a given labeled chordal graph is in L.