In 1996, Karger [Kar96] gave a startling randomized algorithm that finds a minimum-cut in a (weighted) graph in time $O(m\log^3n)$ which he termed near-linear time meaning linear (in the size of the input) times a polylogarthmic factor. In this paper, we give the first deterministic algorithm which runs in near-linear time for weighted graphs. Previously, the breakthrough results of Kawarabayashi and Thorup [KT19] gave a near-linear time algorithm for simple graphs. The main technique here is a clustering procedure that perfectly preserves minimum cuts. Recently, Li [Li21] gave an $m^{1+o(1)}$ deterministic minimum-cut algorithm for weighted graphs; this form of running time has been termed "almost-linear''. Li uses almost-linear time deterministic expander decompositions which do not perfectly preserve minimum cuts, but he can use these clusterings to, in a sense, "derandomize'' the methods of Karger. In terms of techniques, we provide a structural theorem that says there exists a sparse clustering that preserves minimum cuts in a weighted graph with $o(1)$ error. In addition, we construct it deterministically in near linear time. This was done exactly for simple graphs in [KT19, HRW20] and with polylogarithmic error for weighted graphs in [Li21]. Extending the techniques in [KT19, HRW20] to weighted graphs presents significant challenges, and moreover, the algorithm can only polylogarithmically approximately preserve minimum cuts. A remaining challenge is to reduce the polylogarithmic-approximate clusterings to $1+o(1/\log n)$-approximate so that they can be applied recursively as in [Li21] over $O(\log n)$ many levels. This is an additional challenge that requires building on properties of tree-packings in the presence of a wide range of edge weights to, for example, find sources for local flow computations which identify minimum cuts that cross clusters.

翻译：1996年，Karger [Kar96] 提出了一种令人瞩目的随机算法，能在 $O(m\log^3n)$ 时间内找到（加权）图的最小割，他将此称为近线性时间，即输入规模乘以多对数因子。本文首次给出了加权图上的确定性近线性时间算法。此前，Kawarabayashi 和 Thorup [KT19] 的突破性成果为简单图提供了近线性时间算法。本文的核心技术是一种完美保持最小割的聚类过程。最近，Li [Li21] 提出了加权图的 $m^{1+o(1)}$ 确定性最小割算法，这种运行时间被称为“近似线性”。Li 使用了近似线性时间的确定性扩展器分解，这种分解虽不能完美保持最小割，但可在某种意义上“去随机化”Karger 的方法。在技术方面，我们提供了一个结构定理，表明存在一种稀疏聚类，能以 $o(1)$ 误差保持加权图的最小割。此外，我们以近线性时间确定性构建了该聚类。这在 [KT19, HRW20] 中针对简单图已精确实现，而 [Li21] 针对加权图实现了多对数误差。将 [KT19, HRW20] 的技术扩展到加权图面临重大挑战，且该算法仅能以多对数近似保持最小割。一个遗留挑战是将多对数近似聚类降低到 $1+o(1/\log n)$ 近似，以便如 [Li21] 所述在 $O(\log n)$ 层上递归应用。这需要利用树打包在宽范围边权重下的性质，例如为识别跨越聚类的局部流计算找到源点，从而进一步增加难度。