In the $k$-cut problem, we want to find the smallest set of edges whose deletion breaks a given (multi)graph into $k$ connected components. Algorithms of Karger & Stein and Thorup showed how to find such a minimum $k$-cut in time approximately $O(n^{2k})$. The best lower bounds come from conjectures about the solvability of the $k$-clique problem, and show that solving $k$-cut is likely to require time $\Omega(n^k)$. Recent results of Gupta, Lee & Li have given special-purpose algorithms that solve the problem in time $n^{1.98k + O(1)}$, and ones that have better performance for special classes of graphs (e.g., for small integer weights). In this work, we resolve the problem for general graphs, by showing that the Contraction Algorithm of Karger outputs any fixed $k$-cut of weight $\alpha \lambda_k$ with probability $\Omega_k(n^{-\alpha k})$, where $\lambda_k$ denotes the minimum $k$-cut size. This also gives an extremal bound of $O_k(n^k)$ on the number of minimum $k$-cuts and an algorithm to compute a minimum $k$-cut in similar runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight $k$-clique. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks -- and how the average degree evolves -- in the Karger process. The second ingredient is an extremal bound on the number of cuts of size less than $2 \lambda_k/k$, using the Sunflower lemma.

翻译：在 $k$ cut 问题中, 我们想要找到最微小的边缘, 其删除将给定的( 倍数) 折成 $ k$ 连接的元件。 Karger & Stein 和 Thorup 的 Algorithms 显示如何在时间上找到这样一个最低的美元- cut, 大约为 $O (n ⁇ 2k) 美元。 最低的界限来自关于 $k美元 clock 问题的猜测。 并显示, 解决 $k 的削减可能需要时间 $( 倍数) 。 最近Gupta, Lee & Li 给出了在时间上解决问题的最小值为 $1. 98k + O(1) 美元, 而那些对特殊图表类别( 例如, 小整数) 的性能更好。 在这项工作中, 我们通过显示 Karger 的 Algorialthm 输出任何固定的 美元- 美元- 美元- 美元- talfa 美元- lax 美元- test $ $ data_ k modeal_ modeal_ rial_ modeal_ modeal_ modeal_ mox a rial_ modeal_ rial_ modeal_ rial_ modeal_ rial_ rial_ modeal_ rial_ modeal_ rial_ modeal_ mox) a.