发表机构
National University of Singapore; University of Copenhagen（新加坡国立大学; 哥本哈根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出内向聚类图分解原语，生成分层网络分解，并应用于LOCAL模型，实现列表边着色和局部平衡割的确定性算法，时间复杂度为$\widetilde O(\log^2 n)$。
AI 中文摘要
我们引入了一种称为内向聚类的图分解原语，它通过保证每个被聚类的顶点在其自身聚类中保留至少$\left(\frac12-\varepsilon\right)$比例的相关邻居，从而强化了标准的低直径聚类。重复应用该原语可产生具有$O(\log n)$层和弱直径$O(\log n)$的分层内向网络分解。我们在$\mathsf{LOCAL}$模型中给出了两个应用。对于每个常数$\varepsilon>0$，我们获得了一个$\widetilde O(\log^2 n)$轮的确定性算法，用于最大度$\Delta\geq\Delta_0(\varepsilon)$的图上的列表$\left(\frac32+\varepsilon\right)\Delta$-边着色；对于二部图，该结果对所有$\Delta$成立。对于每个常数$0<\varepsilon<1/4$，我们还获得了一个$\widetilde O(\log^2 n)$轮的确定性算法，用于$\left(\frac14-\varepsilon\right)$-局部平衡割，其中每个顶点至少有$\left(\frac14-\varepsilon\right)$比例的邻居在对侧。所得算法非常简单：边着色按逆序处理各层并对每个聚类着色，而局部平衡割按正序处理各层并在每个聚类内计算局部最大割。内向保证使得这些过程超越了通常的网络分解贪心机制。我们使用Miller--Peng--Xu低直径聚类和简单的修剪过程在$O(\log^2 n)$随机轮内构造分解，并通过Ghaffari和Grunau [FOCS 2024]的递归网络分解算法的白盒适配在$\widetilde O(\log^2 n)$轮内确定性构造。
英文摘要
We introduce a graph decomposition primitive called introvert clustering, which strengthens standard low-diameter clustering by guaranteeing that every clustered vertex keeps at least a $\left(\frac12-\varepsilon\right)$-fraction of its relevant neighbors in its own cluster. Repeatedly applying this primitive yields a layered introvert network decomposition with $O(\log n)$ layers and weak diameter $O(\log n)$. We give two applications in the $\mathsf{LOCAL}$ model. For every constant $\varepsilon>0$, we obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for list $\left(\frac32+\varepsilon\right)Δ$-edge coloring on graphs of maximum degree $Δ\geqΔ_0(\varepsilon)$; for bipartite graphs, the result holds for all $Δ$. For every constant $0<\varepsilon<1/4$, we also obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for a $\left(\frac14-\varepsilon\right)$-locally balanced cut, where every vertex has at least a $\left(\frac14-\varepsilon\right)$-fraction of its neighbors on the opposite side. The resulting algorithms are remarkably simple: edge coloring processes the layers in reverse order and colors each cluster, while locally balanced cut processes them forward and computes a locally maximum cut within each cluster. The introvert guarantee enables these procedures beyond the usual greedy regime of network decomposition. We construct the decomposition in $O(\log^2 n)$ randomized rounds using Miller--Peng--Xu low-diameter clustering and a simple trimming procedure, and deterministically in $\widetilde O(\log^2 n)$ rounds via a white-box adaptation of the recursive network decomposition algorithm of Ghaffari and Grunau [FOCS 2024].
