  The constructive Lov\'{a}sz Local Lemma has become a central tool for
designing efficient distributed algorithms. While it has been extensively
studied in the classic LOCAL model that uses unlimited bandwidth, much less is
known in the bandwidth-restricted CONGEST model.
  In this paper, we present bandwidth- and time-efficient algorithms for
various subclasses of LLL problems, including a large class of subgraph
sampling problems that are naturally formulated as LLLs. Lastly, we use our
LLLs to design efficient CONGEST algorithms for coloring sparse and
triangle-free graphs with few colors. These coloring algorithms are
exponentially faster than previous LOCAL model algorithms.
