  The Lov\'{a}sz Local Lemma is a very powerful tool in probabilistic
combinatorics, that is often used to prove existence of combinatorial objects
satisfying certain constraints. Moser and Tardos have shown that the LLL gives
more than just pure existence results: there is an effective randomized
algorithm that can be used to find a desired object. In order to analyze this
algorithm, Moser and Tardos developed the so-called entropy compression method.
It turned out that one could obtain better combinatorial results by a direct
application of the entropy compression method rather than simply appealing to
the LLL. The aim of this paper is to provide a generalization of the LLL which
implies these new combinatorial results. This generalization, which we call the
Local Cut Lemma, concerns a random cut in a directed graph with certain
properties. Note that our result has a short probabilistic proof that does not
use entropy compression. As a consequence, it not only shows that a certain
probability is positive, but also gives an explicit lower bound for this
probability. As an illustration, we present a new application (an improved
lower bound on the number of edges in color-critical hypergraphs) as well as
explain how to use the Local Cut Lemma to derive some of the results obtained
previously using the entropy compression method.
