# Commutative Algorithms Approximate the LLL-distribution

## TL;DR

This paper proves that commutative algorithms, a broad class of algorithms for the Lovasz Local Lemma, approximate the LLL-distribution, enabling new applications like list coloring in sparse graphs.

## Contribution

It establishes that the witness tree lemma holds for commutative algorithms, extending the analysis of LLL algorithms to a wider class and enabling approximation of the LLL-distribution.

## Key findings

- Witness tree lemma holds for commutative algorithms
- Algorithms approximate the LLL-distribution effectively
- Application to list coloring in sparse graphs

## Abstract

Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local Lemma (LLL), a series of works have exploited a key ingredient of the original analysis, the witness tree lemma, in order to: derive deterministic, parallel and distributed algorithms for the LLL, to estimate the entropy of the output distribution, to partially avoid bad events, to deal with super-polynomially many bad events, and even to devise new algorithmic frameworks. Meanwhile, a parallel line of work, has established tools for analyzing stochastic local search algorithms motivated by the LLL that do not fall within the Moser-Tardos framework. Unfortunately, the aforementioned results do not transfer to these more general settings. Mainly, this is because the witness tree lemma, provably, no longer holds. Here we prove that for commutative algorithms, a class recently introduced by Kolmogorov and which captures the vast majority of LLL applications, the witness tree lemma does hold. Armed with this fact, we extend the main result of Haeupler, Saha, and Srinivasan to commutative algorithms, establishing that the output of such algorithms well-approximates the LLL-distribution, i.e., the distribution obtained by conditioning on all bad events being avoided, and give several new applications. For example, we show that the recent algorithm of Molloy for list coloring number of sparse, triangle-free graphs can output exponential many list colorings of the input graph.