Extensions of the Algorithmic Lovasz Local Lemma

TL;DR

This paper advances the algorithmic Lovasz Local Lemma by comparing recent conditions, introducing a new family of conditions for better applicability, especially in sparse k-SAT problems, and extending the concept of commutativity.

Contribution

It introduces a new family of conditions that unify and extend previous formulations, enabling more efficient algorithms for finding flaw-free objects, particularly in sparse k-SAT contexts.

Findings

Conditions are incomparable and can be unified under a new family.

A new condition directly applicable to sparse k-SAT problems is proposed.

Extended the notion of commutativity, simplifying related approximation results.

Abstract

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.