发表机构
Carnegie Mellon University（卡内基梅隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出加权图 Minimum k-Cut 的随机化和确定性算法，时间复杂度为 k^{O(k^2)} n^{k-1}，匹配条件下界，其中 k=3 时达到 O(n^2 log^2 n) 随机时间和 O~(n^2) 确定时间。
AI 中文摘要
Minimum $k$-Cut 问题要求找到一组最小权重的边，其移除后使得一个无向加权图至少具有 $k$ 个连通分量。我们仅考虑 $k \ge 3$ 的情况。在 Max-Weight Clique 猜想下，对于每个固定的 $k$，加权 Minimum $k$-Cut 需要 $n^{k-1-o(1)}$ 的时间。之前最快的加权图算法运行时间为 $n^{k-2}(m+n)(\log n)^{O(k^2)}$ 随机化时间~\cite{LV26}；对于 $k=3$，这给出了一个 $\widetilde O(nm)$ 时间的算法。我们给出了随机化和确定性算法，在指数上匹配条件下界。在一个具有 $n$ 个顶点、$m$ 条边的加权图上，我们的随机化算法以高概率在 $k^{O(k^2)}n^{k-1}\log^2n$ 时间内运行。我们的确定性算法在 $k^{O(k^2)}n^{k-1}\log^{O(1)}n$ 时间内运行。特别地，加权 Minimum $3$-Cut 可以在 $O(n^2 \log^2 n)$ 随机化时间和 $\widetilde O(n^2)$ 确定性时间内解决。算法有两个主要组成部分。首先，我们给出了一个更快的加权 Minimum $3$-Cut 算法。在处理具有非常小的一侧的最优解和具有两个轻侧的最优解之后，剩余的最优解具有一个唯一的结构化侧。树打包将其完成问题简化为一批 $2$-respecting 割问题。其次，我们通过枚举一个有界的光割候选族，并递归地完成每个候选的任一侧，将 Minimum $k$-Cut 归约为 Minimum $3$-Cut。如果枚举产生太多割，那么我们可以直接生成一个最优 $k$-割。我们使用一个确定性的近最小割骨架对 $3$-割算法进行去随机化，并使用该骨架、构造性光割界以及 \cite{BSS12} 的确定性谱稀疏化器，使用一个专门的 $4$-割算法对归约进行去随机化。
英文摘要
The Minimum $k$-Cut problem asks for a minimum-weight set of edges whose removal leaves an undirected weighted graph with at least $k$ connected components. We consider only $k \ge 3$. Under the Max-Weight Clique conjecture, weighted Minimum $k$-Cut requires $n^{k-1-o(1)}$ time for every fixed $k$. The fastest previous algorithm for weighted graphs ran in $n^{k-2}(m+n)(\log n)^{O(k^2)}$ randomized time~\cite{LV26}; for $k=3$, this gave an $\widetilde O(nm)$-time algorithm. We give randomized and deterministic algorithms matching the conditional lower bound in the exponent. On an $n$-vertex, $m$-edge weighted graph, our randomized algorithm runs with high probability in \begin{equation*} k^{O(k^2)}n^{k-1}\log^2n \end{equation*} time. Our deterministic algorithm runs in \begin{equation*} k^{O(k^2)}n^{k-1}\log^{O(1)}n \end{equation*} time. In particular, weighted Minimum $3$-Cut can be solved in $O(n^2 \log^2 n)$ randomized time and in $\widetilde O(n^2)$ deterministic time. The algorithms have two main components. First, we give a faster algorithm for weighted Minimum $3$-Cut. After handling optima with a very small side and optima with two light sides, the remaining optimum has a unique structured side. Tree packing reduces its completion to a batched collection of $2$-respecting cut problems. Second, we reduce Minimum $k$-Cut to Minimum $3$-Cut by enumerating a bounded family of light-cut candidates and recursively completing either side of each candidate. If the enumeration produces too many cuts, then we can instead produce an optimum $k$-cut directly. We derandomize the $3$-cut algorithm using a deterministic near-minimum-cut skeleton, and derandomize the reduction using a specialized $4$-cut algorithm using the skeleton, the constructive light-cut bounds, and the deterministic spectral sparsifier of \cite{BSS12}.
