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The Lov\'asz Local Lemma is a versatile result in probability theory, characterizing circumstances in which a collection of $n$ `bad events', each occurring with probability at most $p$ and dependent on a set of underlying random variables,…

Data Structures and Algorithms · Computer Science 2025-02-18 Peter Davies-Peck

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…

Data Structures and Algorithms · Computer Science 2023-10-13 David G. Harris

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.…

Data Structures and Algorithms · Computer Science 2015-11-19 Nicholas Harvey , Jan Vondrak

The Lov\'{a}sz Local Lemma is a central tool in probabilistic combinatorics, providing a sufficient condition under which a finite collection of undesirable events with limited dependencies can be simultaneously avoided with positive…

Combinatorics · Mathematics 2026-04-30 Igal Sason

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…

Data Structures and Algorithms · Computer Science 2019-09-20 David G. Harris

The Lov\'{a}sz Local Lemma (LLL) states that the probability that none of a set of "bad" events happens is nonzero if the probability of each event is small compared to the number of bad events it depends on. A series of results have…

Data Structures and Algorithms · Computer Science 2011-10-04 Bernhard Haeupler , Barna Saha , Aravind Srinivasan

Locally Checkable Labeling (LCL) problems include essentially all the classic problems of $\mathsf{LOCAL}$ distributed algorithms. In a recent enlightening revelation, Chang and Pettie [arXiv 1704.06297] showed that any LCL (on bounded…

Data Structures and Algorithms · Computer Science 2017-05-17 Manuela Fischer , Mohsen Ghaffari

We show that any randomised Monte Carlo distributed algorithm for the Lov\'asz local lemma requires $\Omega(\log \log n)$ communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the…

Distributed, Parallel, and Cluster Computing · Computer Science 2015-11-04 Sebastian Brandt , Orr Fischer , Juho Hirvonen , Barbara Keller , Tuomo Lempiäinen , Joel Rybicki , Jukka Suomela , Jara Uitto

The Lovasz Local Lemma (LLL) is a powerful result in probability theory that states that the probability that none of a set of bad events happens is nonzero if the probability of each event is small compared to the number of events that…

Data Structures and Algorithms · Computer Science 2019-08-07 Karthekeyan Chandrasekaran , Navin Goyal , Bernhard Haeupler

The Lovasz Local Lemma due to Erdos and Lovasz is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events,…

Data Structures and Algorithms · Computer Science 2007-05-23 Aravind Srinivasan

Moser & Tardos have developed a powerful algorithmic approach (henceforth "MT") to the Lovasz Local Lemma (LLL); the basic operation done in MT and its variants is a search for "bad" events in a current configuration. In the initial stage…

Discrete Mathematics · Computer Science 2023-10-13 David G. Harris , Aravind Srinivasan

Recently, Brandt, Maus and Uitto [PODC'19] showed that, in a restricted setting, the dependency of the complexity of the distributed Lov\'asz Local Lemma (LLL) on the chosen LLL criterion exhibits a sharp threshold phenomenon: They proved…

Data Structures and Algorithms · Computer Science 2020-06-09 Sebastian Brandt , Christoph Grunau , Václav Rozhoň

The Lov\'{a}sz Local Lemma (LLL) says that, given a set of bad events that depend on the values of some random variables and where each event happens with probability at most $p$ and depends on at most $d$ other events, there is an…

Data Structures and Algorithms · Computer Science 2019-08-21 Sebastian Brandt , Yannic Maus , Jara Uitto

The complexity of distributed edge coloring depends heavily on the palette size as a function of the maximum degree $\Delta$. In this paper we explore the complexity of edge coloring in the LOCAL model in different palette size regimes. 1.…

Distributed, Parallel, and Cluster Computing · Computer Science 2018-04-20 Yi-Jun Chang , Qizheng He , Wenzheng Li , Seth Pettie , Jara Uitto

The Lov\'{a}sz 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…

Combinatorics · Mathematics 2019-09-13 Anton Bernshteyn

The probabilistic method is a technique for proving combinatorial existence results by means of showing that a randomly chosen object has the desired properties with positive probability. A particularly powerful probabilistic tool is the…

Combinatorics · Mathematics 2022-02-08 Anton Bernshteyn

The Lov\'{a}sz Local Lemma (LLL) is a probabilistic tool which shows that, if a collection of "bad" events $\mathcal B$ in a probability space are not too likely and not too interdependent, then there is a positive probability that no…

Data Structures and Algorithms · Computer Science 2023-10-13 David G. Harris

The Lov\'asz Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum degree. This paper presents a general…

Combinatorics · Mathematics 2021-04-14 Ian M. Wanless , David R. Wood

We consider the recent formulation of the Algorithmic Lov\'asz Local Lemma [10,2,3] for finding objects that avoid `bad features', or `flaws'. It extends the Moser-Tardos resampling algorithm [17] to more general discrete spaces. At each…

Data Structures and Algorithms · Computer Science 2018-09-05 Vladimir Kolmogorov

The Local Computation Algorithm (LCA) model is a popular model in the field of sublinear-time algorithms that measures the complexity of an algorithm by the number of probes the algorithm makes in the neighborhood of one node to determine…

Data Structures and Algorithms · Computer Science 2021-12-06 Sebastian Brandt , Christoph Grunau , Václav Rozhoň
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