Derandomizing the Lovasz Local Lemma via log-space statistical tests

TL;DR

This paper introduces a deterministic parallel algorithm for the Lovász Local Lemma that relies on log-space statistical tests, broadening applicability to many graph theory and combinatorics problems.

Contribution

It presents a new NC algorithm for the LLL based on log-space derandomization, requiring only that bad-events be automaton-computable, unlike previous methods needing low decision-tree complexity.

Findings

The algorithm applies to most LLL applications in graph theory and combinatorics.

It does not require auxiliary information or conditional probability calculations.

Demonstrated on defective vertex coloring, domatic partition, and independent transversals.

Abstract

The Lov\'{a}sz Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its simplest form, it asserts that whenever a bad-event has probability $p$ and affects at most $d$ other bad-events, and $e p (d+1) < 1$, then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of Moser & Tardos (2010) gives randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. Notably, prior deterministic LLL algorithms have required stringent conditions on $\mathcal B$; for example, they have required that events in $\mathcal B$ have low decision-tree complexity or depend on a small number of variables. For this reason, they can only be applied to small fraction of the numerous LLL applications in practice. We describe an alternate deterministic parallel (NC) algorithm for the LLL, based on a general derandomization method of Sivakumar (2002) using log-space statistical tests. The only requirement here is that bad-events should be computable via a finite automaton with $\text{poly}(d)$ states. This covers most LLL applications to graph theory and combinatorics. No auxiliary information about the bad-events, including any conditional probability calculations, are required. Additionally, the proof is a straightforward combination of general derandomization results and high-level analysis of the Moser-Tardos algorithm. We illustrate with applications to defective vertex coloring, domatic partition, and independent transversals.