Directed Lov\'asz Local Lemma and Shearer's Lemma

TL;DR

This paper introduces a new directed dependency notion for the Lovász Local Lemma, resulting in weaker conditions and improved bounds, with applications to Shearer's lemma and algorithmic analysis.

Contribution

It defines a novel directed dependency graph for the LLL, strengthening existing notions, and demonstrates its advantages over classical approaches including Shearer's lemma.

Findings

Stronger LLL conditions with sparser dependency graphs

Better bounds in specific examples compared to Shearer's lemma

Exponential bounds on algorithm success probability within fixed steps

Abstract

Moser and Tardos (2010) gave an algorithmic proof of the lopsided Lov\'asz local lemma (LLL) in the variable framework, where each of the undesirable events is assumed to depend on a subset of a collection of independent random variables. For the proof, they define a notion of a lopsided dependency between the events suitable for this framework. In this work, we strengthen this notion, defining a novel directed notion of dependency and prove LLL for the corresponding graph. We show that this graph can be strictly sparser (thus the sufficient condition for LLL weaker) compared with graphs that correspond to other extant lopsided versions of dependency. Thus, in a sense, we address the problem "find other simple local conditions for the constraints (in the variable framework) that advantageously translate to some abstract lopsided condition" posed by Szegedy (2013). We also give an example where our notion of dependency graph gives better results than the classical Shearer lemma. Finally, we prove Shearer's lemma for the dependency graph we define. For the proofs, we perform a direct probabilistic analysis that yields an exponentially small upper bound for the probability of the algorithm that searches for the desired assignment to the variables not to return a correct answer within $n$ steps. In contrast, the method of proof that became known as the entropic method, gives an estimate of only the expectation of the number of steps until the algorithm returns a correct answer, unless the probabilities are tinkered with.