Algorithmic Shearer ’ s Lemma for general probability spaces

In the last two lectures we presented an algorithmic proof of the LLL (and in fact Shearer’s lemma) when the underlying probability space consists of independent random variables. In this lecture we present an “algorithmic proof” of Shearer’s Lemma for a general probability space. We note that the first abstract algorithmic framework for the LLL beyond [Moser and Tardos (2010)] was proposed by [Achlioptas and Iliopoulos (2014)]. Here we follow [Harvey and Vondrak (2015)]. Let Ω be a finite space with probability measure μ, and E1, . . . , En ⊂ Ω events in that probability space with dependency graph G. Similarly to the Moser-Tardos algorithm, we will choose an ω ∈ Ω at random and then repeatedly “resample” in a certain way until we find an ω ∈ ⋂n i=1 E i. To formalize this we define resampling oracles.