论文标题
$ k $ cut问题的最佳界限
Optimal Bounds for the $k$-cut Problem
论文作者
论文摘要
在$ k $ cut的问题中，我们希望找到最低的边缘集，其删除将给定的（多）图将其分解为$ k $连接的组件。 Karger \＆Stein的算法可以在大约$ o（n^{2k}）$时间中解决此问题。另一方面，关于$ k $ clique问题的猜想的下限意味着可能需要$ω（n^{（1-o（1））k}）$时间。 Gupta，Lee \＆Li的最新结果给出了$ n^{1.98k + O（1）} $ time的一般$ k $ -cut的新算法，以及具有更好性能的专业算法（例如，小整数边缘重量）。 在这项工作中，我们解决了通用图的问题。我们表明，Karger的收缩算法输出任何固定的$ k $ - 重量$αλ_K$，概率$ω__k（n^{ - αk}）$，其中$λ_k$表示最小$ k $ -cut的重量。这也给出了最低$ k $ - cuts的$ o_k（n^k）$的极端界限，并用大约$ n^k \ mathrm {polylog}（n）$ runtime计算$λ_k$的算法。两者都是紧密的较低级因素，假设最大$ k $ clique的硬度的算法下限。 我们结果中的第一个主要成分是使用葵花籽引理的重量小于$2λ_k/k $的削减次数的极端结合。第二个成分是对图表在karger过程中的收缩方式以及平均程度如何发展的细粒度分析。
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.
