A constructive proof of the Lovász local lemma

Moser, Robin A.

doi:10.1145/1536414.1536462

Cited by 117 publications

(145 citation statements)

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“…The symmetric LLL proves the existence of a satisfying color assignment but does not yield an efficient algorithm to find one. Beginning with Beck [7], a long line of research has sought to find efficient (and ideally deterministic) algorithms for computing satisfying assignments [7,1,19,9,32,22,23,24,8,12,14,25]. Most of these results required a major weakening of the standard symmetric LLL constraint ep(d + 1) < 1.…”

Section: Introductionmentioning

confidence: 99%

Distributed algorithms for the Lovász local lemma and graph coloring

Chung

Pettie

2014

Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing

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The Lovász Local Lemma (LLL), introduced by Erdős and Lovász in 1975, is a powerful tool of the probabilistic method that allows one to prove that a set of n "bad" events do not happen with non-zero probability, provided that the events have limited dependence. However, the LLL itself does not suggest how to find a point avoiding all bad events. Since the work of Beck (1991) there has been a sustained effort to find a constructive proof (i.e. an algorithm) for the LLL or weaker versions of it. In a major breakthrough Moser and Tardos (2010) showed that a point avoiding all bad events can be found efficiently. They also proposed a distributed/parallel version of their algorithm that requires O(log 2 n) rounds of communication in a distributed network.In this paper we provide two new distributed algorithms for the LLL that improve on both the efficiency and simplicity of the Moser-Tardos algorithm. For clarity we express our results in terms of the symmetric LLL though both algorithms deal with the asymmetric version as well. Let p bound the probability of any bad event and d be the maximum degree in the dependency graph of the bad events. When epd 2 < 1 we give a truly simple LLL algorithm running in O(log 1/epd 2 n) rounds. Under the tighter condition ep(d + 1) < 1, we give a slightly slower algorithm running in O(log 2 d · log 1/ep(d+1) n) rounds. Furthermore, we give an algorithm that runs in sublogarithmic rounds under the condition p · f (d) < 1, where f (d) is an exponential function of d. Although the conditions of the LLL are locally verifiable, we prove that any distributed LLL algorithm requires Ω(log * n) rounds. In many graph coloring problems the existence of a valid coloring is established by one or more applications of the LLL. Using our LLL algorithms, we give logarithmic-time *

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Section: Introductionmentioning

confidence: 99%

Distributed algorithms for the Lovász local lemma and graph coloring

Chung

Pettie

2014

Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing

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“…2. A long line of research [Bec91, Alo91, MR98, CS00, Sri08, Mos08] has culminated in a very recent breakthrough result by Moser [Mos09] (see also [MT09]), who gave an algorithmic proof of the LLL that allows to efficiently construct the desired satisfying assignment (and more generally the object whose existence is asserted by the LLL [MT09]). Moser's algorithm itself is a rather simple random walk on assignments; an innovative information theoretic argument proves its correctness (see also [For09]).…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

A quantum Lovász local lemma

Ambainis

Kempe

Sattath

2012

J. ACM

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The Lovász 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 geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions.Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most 2 k /(e · k) of them, has a joint satisfiable state.We then apply our results to the recently studied model of random k-QSAT. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. [LLM + 09] we greatly extend the known satisfiable region for random k-QSAT to a density of Ω(2 k /k 2 ). Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.

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“…The basic technique we use dates back to Beck's seminal paper [4] in which he showed how to convert some applications of the Local Lemma into efficient algorithms. We remark that the recent work by Moser [21] and Moser and Tardos [22] would also apply to the first two phases of our procedure (and, in fact, would yield much simpler algorithms with no loss in the constants) but it does not seem to apply to the third stage.…”

Section: Algorithmsmentioning

confidence: 92%