DeepAI AI Chat
Log In Sign Up

Variable Version Lovász Local Lemma: Beyond Shearer's Bound

by   Kun He, et al.

A tight criterion under which the abstract version Lovász 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.


page 1

page 2

page 3

page 4


Moser-Tardos Algorithm: Beyond Shearer's Bound

In a seminal paper (Moser and Tardos, JACM'10), Moser and Tardos develop...

A Sharp Threshold Phenomenon for the Distributed Complexity of the Lovasz Local Lemma

The Lovász Local Lemma (LLL) says that, given a set of bad events that d...

Quantum Lovász Local Lemma: Shearer's Bound is Tight

Lovász Local Lemma (LLL) is a very powerful tool in combinatorics and pr...

An interactive version of Lovász local lemma: Arthur and Merlin implement Moser's algorithm

Assume we are given (finitely many) mutually independent variables and (...

The Lovász Local Lemma is Not About Probability

Given a collection of independent events each of which has strictly posi...

An interactive version of the Lovász local lemma

Assume we are given (finitely many) mutually independent variables and (...

Solving 3SAT By Reduction To Testing For Odd Hole

An algorithm is given for finding the solutions to 3SAT problems. The al...