I&C Journal 2026 Journal Article
Variable version Lovász local lemma: A tale of two boundaries
- Kun He
- Liang Li
- Xingwu Liu
- Yuyi Wang
- Mingji Xia
Shearer gave a tight criterion for the abstract version of the Lovász Local Lemma (abstract-LLL), but the corresponding picture for the variable version (variable-LLL), where events are generated from independent random variables, has remained largely open. We establish a necessary and sufficient criterion for variable-LLL expressed purely in terms of the event probabilities and the event-variable bigraph. This allows us to determine exactly the probability boundary for two fundamental families of event-variable graphs: cyclic and treelike bigraphs, giving the first nontrivial cases where the variable-LLL boundary is fully characterized. As a byproduct, we obtain a general constructive procedure that, for any given probability vector and event-variable graph, produces a set of events whose union has maximum possible probability; the method also applies when any two events are either independent or disjoint. We further show that computing the variable-LLL boundary is #P-hard in general, and focus on deciding whether there is a gap between the variable-LLL boundary and the corresponding abstract-LLL (Shearer) boundary. We prove that gap existence can be decided without evaluating Shearer's condition or our criterion. Using this theorem, we show that there is no gap when the base graph of the event-variable graph is a tree, whereas any induced cycle of length at least four forces a gap. Finally, we develop reduction rules that propagate gapful/gapless property and apply them to several combinatorial event-variable graphs.