更快的最小 k 割 I：简单且稀疏的加权图

中文摘要 AI 辅助

最小 $k$ 割问题要求找到最少的边，使得移除这些边后输入图至少具有 $k$ 个连通分量。此前，针对简单图的最佳算法运行时间为 $O_k(n^{(1-\varepsilon)k+O(1)})$（见文献~\cite{HL22}），这表明 $n^k$ 的障碍可以被打破至多项式开销范围内。我们首次给出了简单图上最小 $k$ 割问题的 $\widetilde O_k(n^{ck})$-时间算法，其中 $c<1$ 为绝对常数。更精确地，对于 $k=3$，运行时间为 $\widetilde O(n^2)$；对于 $k=4$，运行时间为 $\widetilde O(n^{55/19})$；对于 $k=5$，运行时间为 $\widetilde O(n^{4.112007})$；对于每个 $k\ge6$，运行时间为 \\[ k^{O(k^2)}n^{1+(6k-6)\frac{k-1.749614}{7k-10}}(\log n)^{O(k^2)}, \\] 其指数为 $\frac67k-0.132\ldots+O(1/k)$。该算法结合了三个组成部分。首先，对于加权最小 $k$ 割问题，我们给出一个随机化 \\[ k^{O(k^2)}n^{k-2}(m+n)\log^3(n) \\] -时间算法：它对边权重进行扰动，使得任何最小 $k$ 割的一侧具有严格小于平均值的边界。然后，这些边以高概率从树打包的对数大小样本中切割少量边。在枚举这些边之后，我们递归计算 $(k-1)$-割以将它们补全为它们所属的 $k$-割。扰动的一种变体以及处理整个打包支撑集给出一个确定性 $k^{O(k^2)}n^{k+O(1)}$-时间变体。其次，对于割大小 $s$，我们给出一个改进的 FPT 算法，使用近线性时间构造具有 $O(s\log^2 n\log\log n)$ 粘附度的 $(O(s\log^2 n\log\log n),s)$ 边不可破坏树分解；这也给出了最小 $k$ 割问题的近线性时间近似算法。第三，我们细化了文献~\cite{HL22} 的边界/岛屿框架，使用矩形矩阵乘法来恢复单例岛屿，并将其与改进的 FPT 算法进行平衡。

英文摘要

The minimum $k$-cut problem asks for the fewest edges whose removal leaves an input graph with at least $k$ connected components. Previously, the best algorithm for simple graphs ran in $O_k(n^{(1-\varepsilon)k+O(1)})$ time~\cite{HL22}, showing that the \(n^k\) barrier can be broken up to a polynomial overhead. We give the first $\widetilde O_k(n^{ck})$-time algorithm for Minimum $k$-Cut on simple graphs for an absolute constant $c<1$. More precisely, the running times are $\widetilde O(n^2)$ for $k=3$, $\widetilde O(n^{55/19})$ for $k=4$, and $\widetilde O(n^{4.112007})$ for $k=5$; for every $k\ge6$, the running time is \[ k^{O(k^2)}n^{1+(6k-6)\frac{k-1.749614}{7k-10}}(\log n)^{O(k^2)}, \] whose exponent is $\frac67k-0.132\ldots+O(1/k)$. The algorithm combines three ingredients. First, for weighted Minimum $k$-Cut we give a randomized \[ k^{O(k^2)}n^{k-2}(m+n)\log^3(n) \] -time algorithm: it perturbs the edge weights so that any minimum $k$-cut has a side with boundary strictly smaller than average. These then cut few edges of some tree in a logarithmic-size sample from a tree packing with high probability. After we enumerate them, we recursively compute $(k-1)$-cuts to complete them to the $k$-cuts of which they were a part. A variant of the perturbation and processing the entire packing support give a deterministic $k^{O(k^2)}n^{k+O(1)}$-time variant. Second, for cut size $s$, we give an improved FPT algorithm using a near-linear-time construction of an $(O(s\log^2 n\log\log n),s)$ edge-unbreakable tree decomposition with $O(s\log^2 n\log\log n)$ adhesion; this also gives a near-linear-time approximation algorithm for Minimum $k$-Cut. Third, we refine the border/island framework of~\cite{HL22}, using rectangular matrix multiplication to recover singleton islands and balancing it against the improved FPT algorithm.

发表机构

• Carnegie Mellon University（卡内基梅隆大学）

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