We present a deterministic (global) mincut algorithm for weighted, undirected graphs that runs in $m^{1+o(1)}$ time, answering an open question of Karger from the 1990s. To obtain our result, we de-randomize the construction of the \emph{skeleton} graph in Karger's near-linear time mincut algorithm, which is its only randomized component. In particular, we partially de-randomize the well-known Benczur-Karger graph sparsification technique by random sampling, which we accomplish by the method of pessimistic estimators. Our main technical component is designing an efficient pessimistic estimator to capture the cuts of a graph, which involves harnessing the expander decomposition framework introduced in recent work by Goranci et al. (SODA 2021). As a side-effect, we obtain a structural representation of all approximate mincuts in a graph, which may have future applications.

中文翻译：

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几乎线性时间内的确定性 Mincut

我们提出了一种用于加权无向图的确定性（全局）mincut 算法，该算法在 $m^{1+o(1)}$ 时间内运行，回答了 1990 年代 Karger 的一个悬而未决的问题。为了获得我们的结果，我们在 Karger 的近线性时间 mincut 算法中对 \emph {skeleton} 图的构造进行去随机化，这是其唯一的随机组件。特别是，我们通过随机抽样对著名的 Benczur-Karger 图稀疏技术进行了部分去随机化，这是我们通过悲观估计量的方法完成的。我们的主要技术组件是设计一个有效的悲观估计器来捕获图的切割，这涉及利用 Goranci 等人最近的工作中引入的扩展器分解框架。（苏打水 2021）。作为副作用，我们获得了图中所有近似最小切割的结构表示，