In the $k$-cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into $k$ connected components. Algorithms of Karger \& Stein can solve this in roughly $O(n^{2k})$ time. On the other hand, lower bounds from conjectures about the $k$-clique problem imply that $\Omega(n^{(1-o(1))k})$ time is likely needed. Recent results of Gupta, Lee \& Li have given new algorithms for general $k$-cut in $n^{1.98k + O(1)}$ time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show 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 weight. This also gives an extremal bound of $O_k(n^k)$ on the number of minimum $k$-cuts and an algorithm to compute $\lambda_k$ with roughly $n^k \mathrm{polylog}(n)$ 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 an extremal bound on the number of cuts of weight less than $2 \lambda_k/k$, using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks -- and how the average degree evolves -- in the Karger process.
翻译:在 $k$- clotique 问题中, 我们想要找到最低重量的边缘, 其删除将给定的( 倍数) 折成 $ k美元 连接的元件。 Karger 的 Algorithms 能够用大约 $( n ⁇ 2k}) 美元的时间解决这个问题。 另一方面, 有关 $k$- clodique 问题的测算的下限意味着可能需要 $\\\\ ⁇ ( 1- o(1) k} ) 美元 。 最近的 Gupta 的结果, Lee Lee 已经给出了通用美元( 倍数) 的新的算法 $- 美元( 美元) 美元 。 美元=1. 98k + O(1) 美元 时间, 某些图表类别( 例如, 小整数的边数), 我们解决了一般图表的问题。 我们显示, Karger 输出的Alegorthmion Alorth m 任何固定美元重量的计算 $\ al- 。 (alda) modealda) a more diral deal deal deal deal a $_ leglegal $_ $.