We give a deterministic algorithm for finding the minimum (weight) cut of an undirected graph on $n$ vertices and $m$ edges using $\text{polylog}(n)$ calls to any maximum flow subroutine. Using the current best deterministic maximum flow algorithms, this yields an overall running time of $\tilde O(m \cdot \min(\sqrt{m}, n^{2/3}))$ for weighted graphs, and $m^{4/3+o(1)}$ for unweighted (multi)-graphs. This marks the first improvement for this problem since a running time bound of $\tilde O(mn)$ was established by several papers in the early 1990s. To obtain this result, we introduce a new tool for finding minimum cuts of an undirected graph: *isolating cuts*. Given a set of vertices $R$, this entails finding cuts of minimum weight that separate (or isolate) each individual vertex $v\in R$ from the rest of the vertices $R\setminus \{v\}$. Na\"ively, this can be done using $|R|$ maxflow calls, but we show that just $O(\log |R|)$ suffice for finding isolating cuts for any set of vertices $R$. We call this the *isolating cut lemma*.
翻译:我们给出一个确定性算法, 以找到一个未调整的图表的最小值( 重量), 以美元为顶值, 以美元为底值, 以美元为底值, 以美元为底值, 以美元为底值, 以美元为底值, 以美元为底值, 以美元为底值。 这是自1990年代初期若干论文设定美元为底值的 O( m) 最大流算法以来, 问题的第一个改进。 为了获得这个结果, 我们引入了一个新的工具, 以找到未调整的图表的最小切分值 : * 孤立的切分 * 。 鉴于一组顶值 $R, 这意味着要找到最小的重量, 将每个未加权( 或分解) 美元为底值的( R) 美元 。