  The maximal clique enumeration (MCE) problem has numerous applications in
biology, chemistry, sociology, and graph modeling. Though this problem is well
studied, most current research focuses on finding solutions in large sparse
graphs or very dense graphs, while sacrificing efficiency on the most difficult
medium-density benchmark instances that are representative of data sets often
encountered in practice. We show that techniques that have been successfully
applied to the maximum clique problem give significant speed gains over the
state-of-the-art MCE algorithms on these instances. Specifically, we show that
a simple greedy pivot selection based on a fixed maximum-degree first ordering
of vertices, when combined with bit-parallelism, performs consistently better
than the theoretical worst-case optimal pivoting of the state-of-the-art
algorithms of Tomita et al. [Theoretical Computer Science, 2006] and Naud\'e
[Theoretical Computer Science, 2016].
  Experiments show that our algorithm is faster than the worst-case optimal
algorithm of Tomita et al. on 60 out of 74 standard structured and random
benchmark instances: we solve 48 instances 1.2 to 2.2 times faster, and solve
the remaining 12 instances 3.6 to 47.6 times faster. We also see consistent
speed improvements over the algorithm of Naud\'e: solving 61 instances 1.2 to
2.4 times faster. To the best of our knowledge, we are the first to achieve
such speed-ups compared to these state-of-the-art algorithms on these standard
benchmarks.
