We give an algorithm to find a minimum cut in an edge-weighted directed graph with $n$ vertices and $m$ edges in $\tilde O(n\cdot \max(m^{2/3}, n))$ time. This improves on the 30 year old bound of $\tilde O(nm)$ obtained by Hao and Orlin for this problem. Our main technique is to reduce the directed mincut problem to $\tilde O(\min(n/m^{1/3}, \sqrt{n}))$ calls of {\em any} maxflow subroutine. Using state-of-the-art maxflow algorithms, this yields the above running time. Our techniques also yield fast {\em approximation} algorithms for finding minimum cuts in directed graphs. For both edge and vertex weighted graphs, we give $(1+\epsilon)$-approximation algorithms that run in $\tilde O(n^2 / \epsilon^2)$ time.
翻译:我们给出了一种算法, 以 $n (n/ m ⁇ 1/3},\ sqrt{n}) 时间在 $\ tilde O (n\cdot\ max (m ⁇ 2/3}, n) 中找到最起码的剪切值。 这在Hao 和 Orlin 为此问题获得的 $\ tilde O (n) 30 年的捆绑 $\ tilde O (n) 和 Orlin (n) 上下。 我们的主要技术是将直接剪切问题降低到 $\ tilde O (n/m ⁇ 1/3},\ sqrt{n} $ ($em) 任何 最大流流流的调用量 。 使用最先进的最大流算法, 得出以上运行时间 。 我们的技术也产生快速的 em impress} 算法, 在定向图形中找到最小的剪切值 。 对于边缘和 加权图形, 我们给出了 $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 美元 美元。