We present a simple and faster algorithm for computing fair cuts on undirected graphs, a concept introduced in recent work of Li et al. (SODA 2023). Informally, for any parameter $\epsilon>0$, a $(1+\epsilon)$-fair $(s,t)$-cut is an $(s,t)$-cut such that there exists an $(s,t)$-flow that uses $1/(1+\epsilon)$ fraction of the capacity of every edge in the cut. Our algorithm computes a $(1+\epsilon)$-fair cut in $\tilde O(m/\epsilon)$ time, improving on the $\tilde O(m/\epsilon^3)$ time algorithm of Li et al. and matching the $\tilde O(m/\epsilon)$ time algorithm of Sherman (STOC 2017) for standard $(1+\epsilon)$-approximate min-cut. Our main idea is to run Sherman's approximate max-flow/min-cut algorithm iteratively on a (directed) residual graph. While Sherman's algorithm is originally stated for undirected graphs, we show that it provides guarantees for directed graphs that are good enough for our purposes.

翻译：我们提出了一种更简单、更快速的算法，用于计算无向图上的公平割，这一概念由 Li 等人（SODA 2023）在近期工作中引入。非正式地说，对于任意参数 $\epsilon>0$，一个 $(1+\epsilon)$-公平 $(s,t)$-割是指这样一个 $(s,t)$-割：存在一个 $(s,t)$-流，该流使用了割中每条边容量的 $1/(1+\epsilon)$ 部分。我们的算法在 $\tilde O(m/\epsilon)$ 时间内计算出一个 $(1+\epsilon)$-公平割，改进了 Li 等人的 $\tilde O(m/\epsilon^3)$ 时间算法，并与 Sherman（STOC 2017）针对标准 $(1+\epsilon)$-近似最小割的 $\tilde O(m/\epsilon)$ 时间算法相匹配。我们的核心思想是在一个（有向）残差图上迭代运行 Sherman 的近似最大流/最小割算法。虽然 Sherman 的算法最初是针对无向图提出的，但我们证明它能为有向图提供足够好的保证，以满足我们的需求。