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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

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

We reveal a connection between the incompressibility method and the Lovasz local lemma in the context of Ramsey theory. We obtain bounds by repeatedly encoding objects of interest and thereby compressing strings. The method is demonstrated…

Combinatorics · Mathematics 2008-04-07 Pascal Schweitzer

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 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 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 Lovasz Local Lemma [EL75] is a powerful tool to prove the existence of combinatorial objects meeting a prescribed collection of criteria. The technique can directly be applied to the satisfiability problem, yielding that a k-CNF formula…

Data Structures and Algorithms · Computer Science 2008-10-29 Robin A. Moser

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 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

The Quantum Lov\'asz Local Lemma (QLLL) [AKS12] establishes non-constructively that any quantum system constrained by a local Hamiltonian has a zero-energy ground state, if the local Hamiltonian terms overlap only in a certain restricted…

Quantum Physics · Physics 2013-11-27 Martin Schwarz , Toby S. Cubitt , Frank Verstraete

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) 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\'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

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 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

In this paper we investigate the extent to which the Lov\'asz Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lov\'asz Local Lemma is used to produce a…

Combinatorics · Mathematics 2019-08-29 Anton Bernshteyn

We illustrate the use of probability theory in existential proofs, focusing on the Lov\'asz Local Lemma. This result gives a lower bound for the probability of avoiding a suitable finite collection of events. We describe some applications…

Combinatorics · Mathematics 2019-09-25 Irfan Alam

The Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive…

Data Structures and Algorithms · Computer Science 2009-05-21 Robin A. Moser , Gábor Tardos

In a seminal paper (Moser and Tardos, JACM'10), Moser and Tardos developed a simple and powerful algorithm to find solutions to combinatorial problems in the variable Lov{\'a}sz Local Lemma (LLL) setting. Kolipaka and Szegedy (STOC'11)…

Data Structures and Algorithms · Computer Science 2021-11-15 Kun He , Qian Li , Xiaoming Sun

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