A new notion of commutativity for the algorithmic Lov\'{a}sz Local Lemma

TL;DR

This paper introduces a new, more general notion of commutativity for the algorithmic Lovász Local Lemma, leading to refined properties and simpler proofs for the behavior of LLL-inspired algorithms.

Contribution

It proposes a novel matrix-based commutativity concept that enhances understanding of LLL algorithms' properties and simplifies their analysis.

Findings

New notion of matrix commutativity introduced

Refined properties of LLL algorithms established

Simplified proofs for algorithmic properties obtained

Abstract

The Lov\'{a}sz Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs.