Alternative proofs of the asymmetric Lovász local lemma and Shearer's lemma

Ioannis Giotis 1 Lefteris Kirousis 1 John Livieratos 1 Kostas Psaromiligkos 1 Dimitrios M. Thilikos 1, 2
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : We provide new algorithmic proofs for two forms of the Lovász Local Lemma: the Asymmetric version and Shearer's Lemma. Our proofs directly compute an upper bound for the probability that the corresponding Moser-type algorithms last for at least n steps. These algorithms iteratively sample the probability space; when and if they halt, a correct sampling, i.e. one where all undesirable events are avoided, is obtained. Our computation shows that this probability is exponentially small in n. In contrast most extant proofs for the Lovász Local Lemma and its variants use counting arguments that give estimates of only the expectation that the algorithm lasts for at least n steps. For the asymmetric version, we use the results of Bender and Richmond on the multivariable Lagrange inversion. For Shearer's Lemma, we follow the work of Kolipaka and Szegedy, combined with Gelfand's formula for the spectral radius of a matrix.
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-01890510
Contributor : Dimitrios M. Thilikos <>
Submitted on : Monday, October 8, 2018 - 4:37:53 PM
Last modification on : Tuesday, October 9, 2018 - 1:14:47 AM
Long-term archiving on : Wednesday, January 9, 2019 - 4:32:17 PM

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Ioannis Giotis, Lefteris Kirousis, John Livieratos, Kostas Psaromiligkos, Dimitrios M. Thilikos. Alternative proofs of the asymmetric Lovász local lemma and Shearer's lemma. GASCom: Random and Exhaustive Generation of Combinatorial Structures, Jun 2018, Athens, Greece. pp.148-155. ⟨lirmm-01890510⟩

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