  We present a Markov chain example where non-reversibility and an added edge
jointly improve mixing time: when a random edge is added to a cycle of $n$
vertices and a Markov chain with a drift is introduced, we get mixing time of
$O(n^{3/2})$ with probability bounded away from 0. If only one of the two
modifications were performed, the mixing time would stay $\Omega(n^2)$.
