The Lefthanded Local Lemma characterizes chordal dependency graphs

TL;DR

This paper proves that the Lefthanded Local Lemma exactly characterizes chordal dependency graphs in Shearer's family, providing a simple algorithm to verify membership and highlighting the lemma's power in this class.

Contribution

It establishes the equivalence between the Lefthanded Local Lemma and Shearer's condition specifically for chordal graphs, enhancing understanding of dependency graph properties.

Findings

Lefthanded Local Lemma is equivalent to Shearer's condition for chordal graphs

Provides an efficient algorithm to check if a chordal graph belongs to f

Clarifies the power of the Lefthanded Local Lemma in characterizing dependency graphs

Abstract

Shearer gave a general theorem characterizing the family $\LLL$ of dependency graphs labeled with probabilities $p_v$ which have the property that for any family of events with a dependency graph from $\LLL$ (whose vertex-labels 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 Lov\'asz 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 $\LLL$.