arXiv 2005.083012020-05cs.DS

$k$-割问题的最优界

Optimal Bounds for the $k$-cut Problem

Anupam Gupta, David G. Harris, Euiwoong Lee, Jason Li

备注：: Final version of arXiv:1911.09165 with new and more general proofs
发表信息：: Journal of the ACM 69(1), Article #2 (2022)

AI总结：

本文解决一般图的$k$-割问题，证明Karger收缩算法以概率$\Omega_k(n^{-\alpha k})$输出权重$\alpha\lambda_k$的$k$-割，并给出最小$k$-割数量的极值界及$n^k\mathrm{polylog}(n)$时间的计算算法，均达到最优。

摘要：

在$k$-割问题中，我们希望找到一组权重最小的边，删除这些边能将给定的（多重）图分割成$k$个连通分量。Karger和Stein的算法可以在大约$O(n^{2k})$时间内解决该问题。另一方面，关于$k$-团问题的猜想所蕴含的下界表明，$\Omega(n^{(1-o(1))k})$时间可能是必需的。Gupta、Lee和Li最近的结果为一般$k$-割问题提供了$n^{1.98k + O(1)}$时间的新算法，以及针对某些图类（例如小整数边权）具有更好性能的专门算法。\n在本工作中，我们解决了一般图上的该问题。我们证明Karger的收缩算法以概率$\Omega_k(n^{-\alpha k})$输出任意权重为$\alpha\lambda_k$的固定$k$-割，其中$\lambda_k$表示最小$k$-割权重。这也给出了最小$k$-割数量为$O_k(n^k)$的极值界，以及一个运行时间约为$n^k \mathrm{polylog}(n)$的计算$\lambda_k$的算法。两者在低阶因子意义下都是紧的，其中算法下界假设最大权重$k$-团问题的困难性。\n我们结果的第一个主要成分是利用向日葵引理得到的关于权重小于$2\lambda_k/k$的割数量的极值界。第二个成分是对Karger过程中图如何收缩——以及平均度如何演变——的细粒度分析。

英文摘要：

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 $Ω(n^{(1-o(1))k})$ 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 $αλ_k$ with probability $Ω_k(n^{-αk})$, where $λ_k$ 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 $λ_k$ 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 $2 λ_k/k$, 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.

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