  A tight criterion under which the abstract version Lov\'asz Local Lemma
(abstract-LLL) holds was given by Shearer decades ago. However, little is known
about that of the variable version LLL (variable-LLL) where events are
generated by independent random variables, though this model of events is
applicable to almost all applications of LLL. We introduce a necessary and
sufficient criterion for variable-LLL, in terms of the probabilities of the
events and the event-variable graph specifying the dependency among the events.
Based on this new criterion, we obtain boundaries for two families of
event-variable graphs, namely, cyclic and treelike bigraphs. These are the
first two non-trivial cases where the variable-LLL boundary is fully
determined. As a byproduct, we also provide a universal constructive method to
find a set of events whose union has the maximum probability, given the
probability vector and the event-variable graph. Though it is #P-hard in
general to determine variable-LLL boundaries, we can to some extent decide
whether a gap exists between a variable-LLL boundary and the corresponding
abstract-LLL boundary. In particular, we show that the gap existence can be
decided without solving Shearer's conditions or checking our variable-LLL
criterion. Equipped with this powerful theorem, we show that there is no gap if
the base graph of the event-variable graph is a tree, while gap appears if the
base graph has an induced cycle of length at least 4. The problem is almost
completely solved except when the base graph has only 3-cliques, in which case
we also get partial solutions. A set of reduction rules are established that
facilitate to infer gap existence of an event-variable graph from known ones.
As an application, various event-variable graphs, in particular combinatorial
ones, are shown to be gapful/gapless.
