scholarly journals A Meeting Point of Probability, Graphs, and Algorithms: The Lovász Local Lemma and Related Results—A Survey

Algorithms ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 355
Author(s):  
András Faragó
Keyword(s):  
Probabilistic Method ◽  
Fundamental Result ◽  
Lovász Local Lemma ◽  
Survey Paper ◽  
Local Lemma ◽  
Versatile Tool ◽  
Lovasz Local Lemma ◽  
High Level ◽  
Core Message ◽  
Weakly Dependent

A classic and fundamental result, known as the Lovász Local Lemma, is a gem in the probabilistic method of combinatorics. At a high level, its core message can be described by the claim that weakly dependent events behave similarly to independent ones. A fascinating feature of this result is that even though it is a purely probabilistic statement, it provides a valuable and versatile tool for proving completely deterministic theorems. The Lovász Local Lemma has found many applications; despite being originally published in 1973, it still attracts active novel research. In this survey paper, we review various forms of the Lemma, as well as some related results and applications.

scholarly journals Lower Bounds on van der Waerden Numbers: Randomized- and Deterministic-Constructive

10.37236/551 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
William Gasarch ◽  
Bernhard Haeupler
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Efficient Algorithm ◽  
Probabilistic Method ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Constructive Proofs ◽  
Different Types ◽  
Lovasz Local Lemma

The van der Waerden number $W(k,2)$ is the smallest integer $n$ such that every $2$-coloring of 1 to $n$ has a monochromatic arithmetic progression of length $k$. The existence of such an $n$ for any $k$ is due to van der Waerden but known upper bounds on $W(k,2)$ are enormous. Much effort was put into developing lower bounds on $W(k,2)$. Most of these lower bound proofs employ the probabilistic method often in combination with the Lovász Local Lemma. While these proofs show the existence of a $2$-coloring that has no monochromatic arithmetic progression of length $k$ they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm. This paper clarifies these notions and gives definitions for deterministic- and randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on $W(k,2)$ in this light. We show how known nonconstructive lower bound proofs based on the Lovász Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms.

scholarly journals Can Colour-Blind Distinguish Colour Palettes?

10.37236/2319 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Large Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Comprehensive Information ◽  
Lovasz Local Lemma ◽  
Delta G ◽  
Irregular Graphs

Let $c:E(G)\rightarrow [k]$ be  a colouring, not necessarily proper, of edges of a graph $G$. For a vertex $v\in V$, let $\overline{c}(v)=(a_1,\ldots,a_k)$, where $ a_i =|\{u:uv\in E(G),\;c(uv)=i\}|$, for $i\in [k].$ If we re-order the sequence $\overline{c}(v)$ non-decreasingly, we obtain a sequence $c^*(v)=(d_1,\ldots,d_k)$, called a palette of a vertex $v$. This can be viewed as the most comprehensive information about colours incident with $v$ which can be delivered by a person who is unable to name colours but distinguishes one from another. The smallest $k$ such that $c^*$ is a proper colouring of vertices of $G$ is called the colour-blind index of a graph $G$, and is denoted by dal$(G)$. We conjecture that there is a constant $K$ such that dal$(G)\leq K$ for every graph $G$ for which the parameter is well defined. As our main result we prove that $K\leq 6$ for regular graphs of sufficiently large degree, and for irregular graphs with $\delta (G)$ and $\Delta(G)$ satisfying certain conditions. The proofs are based on the Lopsided Lovász Local Lemma. We also show that $K=3$ for all regular bipartite graphs, and for complete graphs of order $n\geq 8$.

An Alternate Proof of the Algorithmic Lovász Local Lemma

2017 ◽  
pp. 61-65
Author(s):  
Ioannis Giotis ◽  
Lefteris Kirousis ◽  
Kostas I. Psaromiligkos ◽  
Dimitrios M. Thilikos
Keyword(s):  
Lovász Local Lemma ◽  
Alternate Proof ◽  
Local Lemma ◽  
Lovasz Local Lemma

scholarly journals Uniform Sampling Through the Lovász Local Lemma

2019 ◽  
Vol 66 (3) ◽  
pp. 1-31 ◽  
Author(s):  
Heng Guo ◽  
Mark Jerrum ◽  
Jingcheng Liu
Keyword(s):  
Lovász Local Lemma ◽  
Uniform Sampling ◽  
Local Lemma ◽  
Lovasz Local Lemma

scholarly journals Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma

2017 ◽  
Vol 49 (1) ◽  
pp. 1-23
Author(s):  
Christoph Hofer-Temmel
Keyword(s):  
Point Process ◽  
Lower Bounds ◽  
Hard Sphere ◽  
Hard Core ◽  
Combinatorial Structure ◽  
Hard Sphere Model ◽  
Sphere Model ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Lovasz Local Lemma

AbstractA point process isR-dependent if it behaves independently beyond the minimum distanceR. In this paper we investigate uniform positive lower bounds on the avoidance functions ofR-dependent simple point processes with a common intensity. Intensities with such bounds are characterised by the existence of Shearer's point process, the uniqueR-dependent andR-hard-core point process with a given intensity. We also present several extensions of the Lovász local lemma, a sufficient condition on the intensity andRto guarantee the existence of Shearer's point process and exponential lower bounds. Shearer's point process shares a combinatorial structure with the hard-sphere model with radiusR, the uniqueR-hard-core Markov point process. Bounds from the Lovász local lemma convert into lower bounds on the radius of convergence of a high-temperature cluster expansion of the hard-sphere model. This recovers a classic result of Ruelle (1969) on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive approach of Dobrushin (1996).

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