# Deterministic algorithms for the Lovasz Local Lemma: simpler, more general, and more parallel

@article{Harris2022DeterministicAF,
title={Deterministic algorithms for the Lovasz Local Lemma: simpler, more general, and more parallel},
author={David G. Harris},
journal={ArXiv},
year={2022},
volume={abs/1909.08065}
}
• David G. Harris
• Published 17 September 2019
• Computer Science, Mathematics
• ArXiv
The Lovasz 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 "symmetric" form, it asserts that whenever a bad-event has probability $p$ and affects at most $d$ bad-events, and $e p d < 1$, then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of Moser & Tardos (2010) gives nearly-automatic…
7 Citations
Oblivious Resampling Oracles and Parallel Algorithms for the Lopsided Lovász Local Lemma
A new structural property that holds for most known resampling oracles, which is called “obliviousness,” means that the interaction between two bad-events B, B′ depends only on the randomness used to resample B and not the precise state within B itself, which allows to build LLLL probability spaces from simpler “atomic” events.
New bounds for the Moser-Tardos distribution
New bounds on the MT-distribution are shown which are significantly stronger than those known to hold for the LLL-dist distribution for the variable-assignment setting and a tighter bound on the probability of a disjunctive event or singleton event is shown.
Moser-Tardos Algorithm: Beyond Shearer's Bound
• Computer Science
ArXiv
• 2021
It is shown that the e cient region of the Moser-Tardos algorithm goes beyond the Shearer’s bound of the underlying dependency graph, if the graph is not chordal.
Algorithms for weighted independent transversals and strong colouring
• Computer Science, Mathematics
SODA
• 2021
A randomized version of this algorithm that is much more widely applicable is developed, and its use is demonstrated by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.
Conflict-Free Coloring of Star-Free Graphs on Open Neighborhoods
• Mathematics
• 2020
Given a graph, the conflict-free coloring problem on open neighborhoods (CFON) asks to color the vertices of the graph so that all the vertices have a uniquely colored vertex in its open
Nonrepetitive graph colouring
• D. Wood
• Computer Science, Mathematics
The Electronic Journal of Combinatorics
• 2021
The goal is to give a unified and comprehensive presentation of the major results and proof methods, as well as to highlight numerous open problems about nonrepetitive colourings of graphs.
Pliable Index Coding via Conflict-Free Colorings of Hypergraphs
• Computer Science, Mathematics
2021 IEEE International Symposium on Information Theory (ISIT)
• 2021
This work presents achievable PICOD schemes using conflict-free colorings of the PIC OD hypergraph, and shows the existence of an achievable scheme which has length O(log Γ), where Γ refers to a parameter of the hypergraph that captures the maximum ‘incidence’ number of other edges on any edge.

## References

SHOWING 1-10 OF 41 REFERENCES
Parallel algorithms for the Lopsided Lovász Local Lemma
• David G. Harris
• Computer Science, Mathematics
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms
• 2019
New parallel algorithms for most forms of LLLL are given, which are simpler, faster, and more general than the algorithms of Harris and Harris & Srinivasan, and generalizes an algorithm given by Blelloch, Fineman, Shun for undirected graphs.
Deterministic algorithms for the Lovász Local Lemma
• Mathematics, Computer Science
SODA '10
• 2010
This work addresses the main problem left open by Moser and Tardos of derandomizing these algorithms efficiently and improves upon the deterministic algorithms ofMoser and of Moser-Tardos with running times.
New Constructive Aspects of the Lovasz Local Lemma
• Mathematics, Computer Science
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
• 2010
It is shown that the output distribution of the Moser-Tardos algorithm well-approximates the conditional LLL-distribution – the distribution obtained by conditioning on all bad events being avoided, and how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and non-constructive results.
Parallel Algorithms and Concentration Bounds for the Lovász Local Lemma via Witness DAGs
• Mathematics, Computer Science
SODA
• 2017
A new parallel algorithm is given that works under essentially the same conditions as the original algorithm of Moser and Tardos but uses only a single MIS computation, thus running in O(log2 n) time on an EREW PRAM.
Deterministic Parallel Algorithms for Fooling Polylogarithmic Juntas and the Lovász Local Lemma
This work gives a new algorithm to generate a probability space which can fool a given set of neighborhoods, and uses this for an NC2 algorithm for defective vertex coloring, which works for arbitrary degree graphs.
Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution
• Computer Science, Mathematics
SODA
• 2016
It is shown that, in certain conditions when the LLL condition is violated, a variant of the MT algorithm can still produce a distribution that avoids most of the bad events, and in some cases that this MT variant can run faster than the original MT algorithm itself and develop the first-known criterion for the case of the asymmetric LLL.
An Algorithmic Proof of the Lopsided Lovasz Local Lemma
• Mathematics, Computer Science
ArXiv
• 2015
This work develops efficient resampling oracles for the known uses of the Lopsided Lovasz Local Lemma, unifying previous algorithmic applications and presenting new results for packings of Latin transversals, rainbow matchings and rainbow spanning trees.
Sublogarithmic Distributed Algorithms for Lovász Local lemma, and the Complexity Hierarchy
• Mathematics, Computer Science
DISC
• 2017
It is proved that any o(\log n)-round randomized distributed algorithm for any LCL problem on bounded degree graphs can be automatically sped up to run in 2^{O(\sqrt{\log\log n})}\$ rounds.
New bounds for the Moser-Tardos distribution
New bounds on the MT-distribution are shown which are significantly stronger than those known to hold for the LLL-dist distribution for the variable-assignment setting and a tighter bound on the probability of a disjunctive event or singleton event is shown.
Moser and tardos meet Lovász
• Mathematics, Computer Science
STOC '11
• 2011
An alternative proof for the Shearer's bound not only highlights the connection between the variable and general versions of LLL, but also illustrates that variants of the Moser-Tardos algorithm can be useful in existence proofs.