  Following the groundbreaking algorithm of Moser and Tardos for the Lovasz
Local Lemma (LLL), there has been a plethora of results analyzing local search
algorithms for various constraint satisfaction problems. The algorithms
considered fall into two broad categories: resampling algorithms, analyzed via
different algorithmic LLL conditions; and backtracking algorithms, analyzed via
entropy compression arguments. This paper introduces a new convergence
condition that seamlessly handles resampling, backtracking, and hybrid
algorithms, i.e., algorithms that perform both resampling and backtracking
steps. Unlike all past LLL work, our condition replaces the notion of a
dependency or causality graph by quantifying point-to-set correlations between
bad events. As a result, our condition simultaneously: (i)~captures the most
general algorithmic LLL condition known as a special case; (ii)~significantly
simplifies the analysis of entropy compression applications; (iii)~relates
backtracking algorithms, which are conceptually very different from resampling
algorithms, to the LLL; and most importantly (iv)~allows for the analysis of
hybrid algorithms, which were outside the scope of previous techniques. We give
several applications of our condition, including a new hybrid vertex coloring
algorithm that extends the recent breakthrough result of Molloy for coloring
triangle-free graphs to arbitrary graphs.
