The Lovász Local Lemma and Its Applications to some Combinatorial Arrays

Authors:
D. Deng;D. R. Stinson;R. Wei
Affiliations:
School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada ddeng@cacr.math.uwaterloo.ca;School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada dstinson@cacr.math.uwaterloo.ca;Department of Computer Science, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada wei@ccc.cs.lakeheadu.ca
Venue:
Designs, Codes and Cryptography
Year:
2004

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Abstract

The Lovász Local Lemma is a useful tool in the “probabilistic method” that has found many applications in combinatorics. In this paper, we discuss applications of the Lovász Local Lemma to some combinatorial set systems and arrays, including perfect hash families, separating hash families, ▵-free systems, splitting systems, and generalized cover-free families. We obtain improved bounds for some of these set sytems. Also, we compare some of the bounds obtained from the local lemma to those using the basic probabilistic method as well as the well-known “expurgation” method. Finally, we briefly consider a “high probability” variation of the method, wherein a desired object is obtained with high probability in a suitable space.