A Distributed Minimum Cut Approximation Scheme

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.