A. Czumaj et C. Scheideler, Coloring nonunifoum hypergraphs: A new algorithmic approach to the generalLovasz local lemma, RAND STR AL, 17(3-4), 2000, pp. 213-237
The Lovasz local lemma is a sieve method to prove the existence of certain
structures with certain prescribed properties. In most of its applications
the Lovasz local lemma does not supply a polynomial-time algorithm for find
ing these structures. Beck was the first who gave a method of converting so
me of these existence proofs into efficient algorithmic procedures, at the
cost of losing a little in the estimates. He applied his technique to the s
ymmetric form of the Lovasz local lemma and, in particular, to the problem
of 2-coloring uniform hypergraphs. In this paper we investigate the general
form of the Lovasz local lemma. Our main result is a randomized algorithm
for 2-coloring nonuniform hypergraphs that runs in expected lineal time. Ev
en for uniform hypergraphs no algorithm with such a runtime bound was previ
ously known, and no polynomial-time algorithm was known at all for the clas
s of nonuniform hypergraphs we will consider in this paper. Our algorithm a
nd its analysis provide a novel approach to the general Lovasz local lemma
that may be of independent interest. We also show how to extend our result
to the c-coloring problem. (C) 2000 John Wiley & Sons, Inc.