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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 Lopsided Lov\'{a}sz Local Lemma (LLLL) is a powerful probabilistic principle which has been used in a variety of combinatorial constructions. While originally a general statement about probability spaces, it has recently been…

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

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

Lov\'asz Local Lemma (LLL) is a probabilistic tool that allows us to prove the existence of combinatorial objects in the cases when standard probabilistic argument does not work (there are many partly independent conditions). LLL can be…

Data Structures and Algorithms · Computer Science 2010-12-03 Andrey Rumyantsev

The Lov\'{a}sz Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works…

Data Structures and Algorithms · Computer Science 2025-09-09 David G. Harris , Fotis Iliopoulos , Vladimir Kolmogorov

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

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

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 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

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

Assume we are given (finitely many) mutually independent variables and (finitely many) "undesirable" events, each depending on a subset of the variables of at most $k$ elements, called the scope of the event. Assume that the probability of…

Discrete Mathematics · Computer Science 2018-08-08 Lefteris Kirousis , John Livieratos , Kostas I. Psaromiligkos

We present a poly $\log \log n$ time randomized CONGEST algorithm for a natural class of Lovasz Local Lemma (LLL) instances on constant degree graphs. This implies, among other things, that there are no LCL problems with randomized…

Distributed, Parallel, and Cluster Computing · Computer Science 2021-08-06 Yannic Maus , Jara Uitto

We introduce a variant of the $k$-nearest neighbor classifier in which $k$ is chosen adaptively for each query, rather than supplied as a parameter. The choice of $k$ depends on properties of each neighborhood, and therefore may…

Machine Learning · Computer Science 2019-05-31 Akshay Balsubramani , Sanjoy Dasgupta , Yoav Freund , Shay Moran

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

Our previous paper applied a lopsided version of the Lov\'asz Local Lemma that allows negative dependency graphs to the space of random injections from an $m$-element set to an $n$-element set. Equivalently, the same story can be told about…

Combinatorics · Mathematics 2014-02-25 Linyuan Lu , Laszlo A. Szekely

We give a fast algorithm for sampling uniform solutions of general constraint satisfaction problems (CSPs) in a local lemma regime. Suppose that the CSP has $n$ variables with domain size at most q, each constraint contains at most k…

Data Structures and Algorithms · Computer Science 2023-03-10 Kun He , Chunyang Wang , Yitong Yin

The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our paper is to…

Combinatorics · Mathematics 2016-04-21 Heidi Gebauer , Tibor Szabo , Gabor Tardos

One of the cornerstones of the distributed complexity theory is the derandomization result by Chang, Kopelowitz, and Pettie [FOCS 2016]: any randomized LOCAL algorithm that solves a locally checkable labeling problem (LCL) can be…

Distributed, Parallel, and Cluster Computing · Computer Science 2023-05-16 Sameep Dahal , Francesco d'Amore , Henrik Lievonen , Timothé Picavet , Jukka Suomela

We develop a framework for the rigorous analysis of focused stochastic local search algorithms. These are algorithms that search a state space by repeatedly selecting some constraint that is violated in the current state and moving to a…

Discrete Mathematics · Computer Science 2018-09-06 Dimitris Achlioptas , Fotis Iliopoulos , 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