Derandomizing the Lovasz Local Lemma via logspace statistical tests
Abstract
The Lovász 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 badevent has probability $p$ and affects at most $d$ other badevents, 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 decisiontree 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 logspace statistical tests. The only requirement here is that badevents 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 badevents, including any conditional probability calculations, are required. Additionally, the proof is a straightforward combination of general derandomization results and highlevel analysis of the MoserTardos algorithm. We illustrate with applications to defective vertex coloring, domatic partition, and independent transversals.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.06672
 Bibcode:
 2018arXiv180706672H
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 This is superseded by arXiv:1909.08065