Paula M. S. Fialho, Bernardo N. B. de Lima, Aldo Procacci
We establish a connection between the entropy compression method and the Moser-Tardos algorithmic version of the Lovász local lemma through the cluster expansion of the subset gas. We also show that the Moser-Tardos resampling algorithm and the entropy compression bactracking algorithm produce identical bounds.
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A recent theorem of Bissacot, et al. proved using results about the cluster expansion in statistical mechanics extends the Lovász Local Lemma by weakening the conditions under which its conclusions holds. In this note, we prove an algorithmic analog of this result, extending Moser and Tardos's recent algorithmic Local Lemma, and providing an alternative proof of the theorem of Bissacot, et al. applicable in the Moser-Tardos algorithmic framework.
The Moser-Tardos Framework with Partial Resampling
将Moser-Tardos重采样算法推广至部分重采样，适用于由随机变量之和决定的坏事件，改进调度、图横贯和分组路由等应用，渐近解决Szabó-Tardos猜想并提升近似比。
Moser-Tardos Algorithm with small number of random bits
本文研究了一种并行莫瑟-塔罗斯算法的变体，证明在依赖图具有亚指数增长的条件下，算法使用的随机位数期望为常数，且与变量数无关。该结论通过利用相同随机位重新采样远距离变量实现。
Lopsidependency in the Moser-Tardos framework: Beyond the Lopsided Lovasz Local Lemma
本文为Moser-Tardos算法提出一个强于偏倚Lovász局部引理的新收敛判据，改进k-SAT可满足性界，并引入新的并行算法，覆盖顺序算法适用的大多数场景。
A constructive algorithm for the LLL on permutations
针对随机排列语境下LLL算法缺失的问题，提出一种构造性随机多项式时间算法，应用于拉丁横贯等组合问题，并保留Moser-Tardos算法的多数性质。
Algorithmic and enumerative aspects of the Moser-Tardos distribution
本文研究Moser-Tardos算法中间及输出分布的随机性，证明其可加速坏事件查找、改进算法复杂度，并利用最小熵证明枚举问题解的数量更多，同时提出在LLL条件违反时仍有效的变体。
New bounds for the Moser-Tardos distribution
本文为Moser-Tardos分布建立了比LLL分布更强的概率界，在变量赋值和排列设置中分别改进了相关事件的概率估计，并应用于k-SAT、CNF公式最小蕴含子、独立横贯及拉丁横贯等问题。
The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Finding such an outcome by an efficient algorithm has been an active research topic for decades. Breakthrough work of Moser and Tardos (2009) presented an efficient algorithm for a general setting primarily characterized by a product structure on the probability space. In this work we present an efficient algorithm for a much more general setting. Our main assu…
The Lovász Local Lemma is a very powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos have shown that the LLL gives more than just pure existence results: there is an effective randomized algorithm that can be used to find a desired object. In order to analyze this algorithm Moser and Tardos developed the so-called entropy compression method. It turned out that one could obtain better combinatorial results by a direct application of the entropy compression method rather than simply appealin…
Parallel algorithms and concentration bounds for the Lovasz Local Lemma via witness DAGs
本文提出一种基于见证DAG的并行LLL算法，在相同条件下仅用一次MIS计算，将运行时间降至$O(\log^2 n)$，并给出重采样算法的更紧运行时间界。
Following the groundbreaking algorithm of Moser and Tardos for the Lovasz Local Lemma (LLL), there has been a plethora of results analyzing local search algorithms for various constraint satisfaction problems. The algorithms considered fall into two broad categories: resampling algorithms, analyzed via different algorithmic LLL conditions; and backtracking algorithms, analyzed via entropy compression arguments. This paper introduces a new convergence condition that seamlessly handles resampling, backtracking, and hybrid algorithms, i.e., algorithms that perform both resampling and backtracking…
The Lovász Local Lemma is a very powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos have shown that the LLL gives more than just pure existence results: there is an effective randomized algorithm that can be used to find a desired object. In order to analyze this algorithm, Moser and Tardos developed the so-called entropy compression method. It turned out that one could obtain better combinatorial results by a direct application of the entropy compression method rather than simply appeali…
Oblivious resampling oracles and parallel algorithms for the Lopsided Lovasz Local Lemma
本文提出重采样预言机的 oblivious 性质，据此实现统一的并行 LLLL 算法，首次给出彩虹完美匹配和彩虹哈密顿圈的 RNC 算法，并构建了相应的顺序与交换重采样预言机。
Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, the works of Grytczuk et al. on words, and Dujmović et al. on colorings, Esperet and Parreau developed a framework to prove upper bounds for several chromatic numbers (in particular acyclic chromatic index, star chromatic number and Thue chromatic number) using the so-called \emph{entropy compression method}. Inspired by this work, we propose a more general framework and a better analysis. This leads to improved upper bounds on chromatic numbers and indices. In particular, every graph with maximum degree $Δ$ has an a…
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively $k$-choosable if given lists of at least $k$ colours at each vertex, there is a nonrepetitive colouring such that each vertex is coloured from its own list. It is known that every graph with maximum degree $Δ$ is $cΔ^2$-choosable, for some constant $c$. We prove this result with $c=1$ (ignoring lower order terms). We then prove that every subdivision of a graph with sufficiently many division vertices per edge is nonrepe…
A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge colouring of a path is nonrepetitive if the sequence of colours of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colourings for arbitrarily long paths using only three colours. A recent generalization of this concept implies that we may obtain such colourings even if we are forced to choose edge colours from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these…
A subset $D \subseteq V $of a graph $G = (V, E)$ is a $(1, j)$-set if every vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$ vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as $γ_{(1,j)} (G)$, is called the $(1, j)$-domination number of $G$. Given a graph $G = (V, E)$ and an integer $k$, the decision version of the $(1, j)$-set problem is to decide whether $G$ has a $(1, j)$-set of cardinality at most $k$. In this paper, we first obtain an upper bound on $γ_{(1,j)} (G)$ using probabilistic methods, for bounded minimum and maximum degree gr…
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