A constructive proof of the general lovász local lemma

Moser, Robin A.; Tardos, Gábor

doi:10.1145/1667053.1667060

Cited by 425 publications

(670 citation statements)

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“…, λ(n + H )) and the graph G n,H need not be independent. To get around this obstacle we introduce a new argument which we call the 'entropy decrement argument' (in analogy with the 'density increment argument' and 'energy increment argument' that appear in the literature surrounding Szemerédi's theorem on arithmetic progressions (see for example [29]), and also reminiscent of the 'entropy compression argument' of Moser and Tardos [26]). This argument, which is a simple consequence of the Shannon entropy inequalities, can be viewed as a quantitative version of the standard subadditivity argument that establishes the existence of Kolmogorov-Sinai entropy in topological dynamical systems; it allows one to select a scale parameter H (in some suitable range [H − , H + ]) for which the sequence (λ(n + 1), .…”

Section: Introductionmentioning

confidence: 99%

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

Tao

2016

Forum of Mathematics, Pi

209

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Let λ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts thatas x → ∞, for any fixed natural numbers a 1 , a 2 and nonnegative integer b 1 , b 2 with a 1 b 2 −a 2 b 1 = 0. In this paper we establish the logarithmically averaged versionof the Chowla conjecture as x → ∞, where 1 ω(x) x is an arbitrary function of x that goes to infinity as x → ∞, thus breaking the 'parity barrier' for this problem. Our main tools are the multiplicativity of the Liouville function at small primes, a recent result of Matomäki, Radziwiłł, and the author on the averages of modulated multiplicative functions in short intervals, concentration of measure inequalities, the Hardy-Littlewood circle method combined with a restriction theorem for the primes, and a novel 'entropy decrement argument'. Most of these ingredients are also available (in principle, at least) for the higher order correlations, with the main missing ingredient being the need to control short sums of multiplicative functions modulated by local nilsequences. Our arguments also extend to more general bounded multiplicative functions than the Liouville function λ, leading to a logarithmically averaged version of the Elliott conjecture in the two-point case. In a subsequent paper we will use this version of the Elliott conjecture to affirmatively settle the Erdős discrepancy problem.

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Section: Introductionmentioning

confidence: 99%

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

Tao

2016

Forum of Mathematics, Pi

209

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“…The second proof we present here, finally, is a fully constructive version published by Moser and Tardos [12] which does not suffer from any gap to the existential version anymore. While it is general enough so as to apply to many applications of the Local Lemma, we will formulate it in terms of satisfiability here.…”

Section: Local Lemma In Terms Of Sat -Proof and Algorithmmentioning

confidence: 91%

The Lovász Local Lemma and Satisfiability

Gebauer

Moser

Scheder

et al. 2009

Lecture Notes in Computer Science

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Abstract. We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics.In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness.Several open problems remain, some of which we mention in the concluding section.

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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: 91%