Variable-Version Lovász Local Lemma: Beyond Shearer's Bound

A tight criterion under which the abstract version Lovász Local Lemma (abstract-LLL) holds was given by Shearer [43] 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. ∗Part of the work has been published at FOCS2017 †Institute of Computing Technology, Chinese Academy of Sciences. University of Chinese Academy of Sciences. Beijing, China. Email:[email protected] ‡Department of Artificial Intelligence, Ant Financial Services Group, China. Email:[email protected]. §Correspondence author ¶Institute of Computing Technology, Chinese Academy of Sciences. University of Chinese Academy of Sciences. Beijing, China. Email:[email protected]. ‖Disco Group, ETH Zürich, Switzerland. Email:[email protected]. ∗∗State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences. University of Chinese Academy of Sciences. Beijing, China. Email:[email protected] 1 ar X iv :1 70 9. 05 14 3v 1 [ cs .D M ] 1 5 Se p 20 17