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An Improvement of the Lovász Local Lemma via Cluster Expansion

Published online by Cambridge University Press:  20 June 2011

RODRIGO BISSACOT ,
ROBERTO FERNÁNDEZ ,
ALDO PROCACCI  and
BENEDETTO SCOPPOLA
RODRIGO BISSACOT
Affiliation:
Departamento de Matemática–ICEx, Universidade Federal de Minas Gerais, CP 702 Belo Horizonte, MG, 30161-970Brazil, and Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, 76801, France (e-mail: rodrigo.bissacot@gmail.com)
ROBERTO FERNÁNDEZ
Affiliation:
Department of Mathematics, Utrecht University, PO Box 80010, 3508 TA Utrecht, The Netherlands (e-mail: R.Fernandez1@uu.nl)
ALDO PROCACCI
Affiliation:
Departamento de Matemática–ICEx, Universidade Federal de Minas Gerais, CP 702 Belo Horizonte, MG, 30161-970Brazil (e-mail: aldo@mat.ufmg.br)
BENEDETTO SCOPPOLA
Affiliation:
Dipartimento di Matematica, Universita Tor Vergata di Roma, 00133 Roma, Italy (e-mail: scoppola@mat.uniroma2.it)
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Abstract

An old result by Shearer relates the Lovász local lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard-core lattice gas on graphs. We use this connection and a recent result on the analyticity of the logarithm of the partition function of the abstract polymer gas to get an improved version of the Lovász local lemma. As an application we obtain tighter bounds on conditions for the existence of Latin transversal matrices.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 709 - 719
Copyright
Copyright © Cambridge University Press 2011

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