  In this paper, we study the problem of approximating the minimum cut in a
distributed message-passing model, the CONGEST model. The minimum cut problem
has been well-studied in the context of centralized algorithms. However, there
were no known non-trivial algorithms in the distributed model until the recent
work of Ghaffari and Kuhn. They gave algorithms for finding cuts of size
$O(\epsilon^{-1}\lambda)$ and $(2+\epsilon)\lambda$ in
$O(D)+\tilde{O}(n^{1/2+\epsilon})$ rounds and $\tilde{O}(D+\sqrt{n})$ rounds
respectively, where $\lambda$ is the size of the minimum cut. This matches the
lower bound they provided up to a polylogarithmic factor. Yet, no scheme that
achieves $(1+\epsilon)$-approximation ratio is known. We give a distributed
algorithm that finds a cut of size $(1+\epsilon)\lambda$ in
$\tilde{O}(D+\sqrt{n})$ time, which is optimal up to polylogarithmic factors.
