  We study the problem of computing approximate minimum edge cuts by
distributed algorithms. We use a standard synchronous message passing model
where in each round, $O(\log n)$ bits can be transmitted over each edge (a.k.a.
the CONGEST model). We present a distributed algorithm that, for any weighted
graph and any $\epsilon \in (0, 1)$, with high probability finds a cut of size
at most $O(\epsilon^{-1}\lambda)$ in $O(D) + \tilde{O}(n^{1/2 + \epsilon})$
rounds, where $\lambda$ is the size of the minimum cut. This algorithm is based
on a simple approach for analyzing random edge sampling, which we call the
random layering technique. In addition, we also present another distributed
algorithm, which is based on a centralized algorithm due to Matula [SODA '93],
that with high probability computes a cut of size at most $(2+\epsilon)\lambda$
in $\tilde{O}((D+\sqrt{n})/\epsilon^5)$ rounds for any $\epsilon>0$.
  The time complexities of both of these algorithms almost match the
$\tilde{\Omega}(D + \sqrt{n})$ lower bound of Das Sarma et al. [STOC '11], thus
leading to an answer to an open question raised by Elkin [SIGACT-News '04] and
Das Sarma et al. [STOC '11].
  Furthermore, we also strengthen the lower bound of Das Sarma et al. by
extending it to unweighted graphs. We show that the same lower bound also holds
for unweighted multigraphs (or equivalently for weighted graphs in which
$O(w\log n)$ bits can be transmitted in each round over an edge of weight $w$),
even if the diameter is $D=O(\log n)$. For unweighted simple graphs, we show
that even for networks of diameter $\tilde{O}(\frac{1}{\lambda}\cdot
\sqrt{\frac{n}{\alpha\lambda}})$, finding an $\alpha$-approximate minimum cut
in networks of edge connectivity $\lambda$ or computing an
$\alpha$-approximation of the edge connectivity requires $\tilde{\Omega}(D +
\sqrt{\frac{n}{\alpha\lambda}})$ rounds.
