  We consider recent formulations of the algorithmic Lovasz Local Lemma by
Achlioptas-Iliopoulos-Kolmogorov [2] and by Achlioptas-Iliopoulos-Sinclair [3].
These papers analyze a random walk algorithm for finding objects that avoid
undesired "bad events" (or "flaws"), and prove that under certain conditions
the algorithm is guaranteed to find a "flawless" object quickly. We show that
conditions proposed in these papers are incomparable, and introduce a new
family of conditions that includes those in [2, 3] as special cases. We also
consider another condition that appeared in [3] in the context of sparse k-SAT
formulas. This condition imposes a constraint for each variable of the problem,
whereas traditional LLL formulations impose a constraint for each clause.
Achlioptas et al. handled the variable-based condition via a reduction to a
different condition and then applying a single-clause backtracking algorithm.
We propose a new condition that directly captures the sparse k-SAT application
considered in [3], and allows the use of the standard local search algorithm
(which offers important advantages such as parallelization). Finally, we extend
our previous notion of "commutativity" from [20] and prove several implications
of commutativity using some new tools that we develop. In particular, we
simplify the result of Iliopoulos [16] about approximating the LLL
distribution.
