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We develop tools for analyzing focused stochastic local search algorithms. These are algorithms which search a state space probabilistically by repeatedly selecting a constraint that is violated in the current state and moving to a random…

Discrete Mathematics · Computer Science 2015-08-18 Dimitris Achlioptas , Fotis Iliopoulos

The Lov\'asz Local Lemma (the LLL for short) is a powerful tool in probabilistic combinatorics that is used to verify the existence of combinatorial objects with desirable properties. Recent years saw the development of various…

Combinatorics · Mathematics 2026-05-29 Anton Bernshteyn , Jing Yu

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 Lovasz Local Lemma (LLL) is a probabilistic tool which has been used to show the existence of a variety of combinatorial structures with good "local" properties. The "LLL-distribution" can be used to show that the resulting structures…

Combinatorics · Mathematics 2023-10-13 David G. Harris

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

Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local Lemma (LLL), a series of works have exploited a key ingredient of the original analysis, the witness tree lemma, in order to: derive deterministic, parallel and…

Discrete Mathematics · Computer Science 2019-06-11 Fotis Iliopoulos

The Lov\'asz Local Lemma (LLL) is a very powerful tool in combinatorics and probability theory to show the possibility of avoiding all bad events under some weakly dependent conditions. In a seminal paper, Ambainis, Kempe, and Sattath (JACM…

Computational Complexity · Computer Science 2024-09-30 Kun He , Qian Li , Xiaoming Sun , Jiapeng Zhang

We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lov\'{a}sz Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al.~\cite{Ronitt} that have received a lot of…

Data Structures and Algorithms · Computer Science 2020-07-08 Dimitris Achlioptas , Themis Gouleakis , Fotis Iliopoulos

The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general…

Quantum Physics · Physics 2013-10-10 Andris Ambainis , Julia Kempe , Or Sattath

The famous Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. Kratochvil et al. applied this technique to prove that a k-CNF in which…

Data Structures and Algorithms · Computer Science 2008-09-15 Robin A. Moser

The resampling algorithm of Moser \& Tardos is a powerful approach to develop constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random…

Combinatorics · Mathematics 2023-10-13 David G. Harris , Aravind Srinivasan

The constructive Lov\'{a}sz Local Lemma has become a central tool for designing efficient distributed algorithms. While it has been extensively studied in the classic LOCAL model that uses unlimited bandwidth, much less is known in the…

Data Structures and Algorithms · Computer Science 2024-05-14 Magnús M. Halldórsson , Yannic Maus , Saku Peltonen

We consider recent formulations of the algorithmic Lovasz Local Lemma by Achlioptas-Iliopoulos-Kolmogorov [2] and by Achlioptas-Iliopoulos-Sinclair [3]. These papers analyze a random walk algorithm for finding objects that avoid undesired…

Data Structures and Algorithms · Computer Science 2020-08-17 Vladimir Kolmogorov

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 2017-05-08 Anton Bernshteyn

The Lov\'asz 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.…

Combinatorics · Mathematics 2017-11-21 Nicholas J. A. Harvey , Jan Vondrák

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

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…

Discrete Mathematics · Computer Science 2020-08-20 Dimitris Achlioptas , Fotis Iliopoulos , Alistair Sinclair

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

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

The lefthanded Lov\'asz local lemma (LLLL) is a generalization of the Lov\'asz local lemma (LLL), a powerful technique from the probabilistic method. We prove a computable version of the LLLL and use it to effectivize a collection of…

Logic · Mathematics 2024-06-19 Daniel Mourad