The Lov\'{a}sz Local Lemma (LLL) states that the probability that none of a
set of "bad" events happens is nonzero if the probability of each event is
small compared to the number of bad events it depends on. A series of results
have provided algorithms to efficiently construct structures whose existence is
(non-constructively) guaranteed by the full asymmetric LLL, culminating in the
recent breakthrough of Moser & Tardos. We show that the output distribution of
the Moser-Tardos procedure has sufficient randomness, leading to two classes of
algorithmic applications. We first show that when an LLL application provides a
small amount of slack, the running time of the Moser-Tardos algorithm is
polynomial in the number of underlying independent variables (not events!), and
can thus be used to give efficient constructions in cases where the underlying
proof applies the LLL to super-polynomially many events (or where finding a bad
event that holds is computationally hard). We demonstrate our method on
applications including: the first constant-factor approximation algorithm for
the Santa Claus problem, as well as efficient algorithms for acyclic edge
coloring, non-repetitive graph colorings, and Ramsey-type graphs. Second, we
show applications to cases where a few of the bad events can hold, leading to
the first such algorithmic applications of the LLL: MAX $k$-SAT is an
illustrative example of this.