We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph $G=(V,E)$. The buffered expansion of a set $S \subseteq V$ with a buffer $B \subseteq V \setminus S$ is the edge expansion of $S$ after removing all the edges from set $S$ to its buffer $B$. An $\varepsilon$-buffered $k$-partitioning is a partitioning of a graph into disjoint components $P_i$ and buffers $B_i$, in which the size of buffer $B_i$ for $P_i$ is small relative to the size of $P_i$: $|B_i| \le \varepsilon |P_i|$. The buffered expansion of a buffered partition is the maximum of buffered expansions of the $k$ sets $P_i$ with buffers $B_i$. Let $h^{k,\varepsilon}_G$ be the buffered expansion of the optimal $\varepsilon$-buffered $k$-partitioning, then for every $\delta>0$, $$h_G^{k,\varepsilon} \le O_\delta(1) \cdot \Big( \frac{\log k}{ \varepsilon}\Big) \cdot \lambda_{\lfloor (1+\delta) k\rfloor},$$ where $\lambda_{\lfloor (1+\delta)k\rfloor}$ is the $\lfloor (1+\delta)k\rfloor$-th smallest eigenvalue of the normalized Laplacian of $G$. Our inequality is constructive and avoids the ``square-root loss'' that is present in the standard Cheeger inequalities (even for $k=2$). We also provide a complementary lower bound, and a novel generalization to the setting with arbitrary vertex weights and edge costs. Moreover our result implies and generalizes the standard higher-order Cheeger inequalities and another recent Cheeger-type inequality by Kwok, Lau, and Lee (2017) involving robust vertex expansion.

翻译：我们证明了带缓冲分区的广义高阶Cheeger不等式。考虑图$G=(V,E)$。集合$S \subseteq V$关于缓冲区$B \subseteq V \setminus S$的缓冲扩张是移除从集合$S$到其缓冲区$B$的所有边后$S$的边扩张。一个$\varepsilon$-缓冲$k$-分区是将图划分为不相交的组件$P_i$和缓冲区$B_i$，其中$P_i$的缓冲区$B_i$的大小相对于$P_i$的大小较小：$|B_i| \le \varepsilon |P_i|$。缓冲分区的缓冲扩张是$k$个集合$P_i$及其缓冲区$B_i$的缓冲扩张的最大值。设$h^{k,\varepsilon}_G$为最优$\varepsilon$-缓冲$k$-分区的缓冲扩张，则对于任意$\delta>0$，有$$h_G^{k,\varepsilon} \le O_\delta(1) \cdot \Big( \frac{\log k}{ \varepsilon}\Big) \cdot \lambda_{\lfloor (1+\delta) k\rfloor},$$其中$\lambda_{\lfloor (1+\delta)k\rfloor}$是$G$的归一化拉普拉斯矩阵的第$\lfloor (1+\delta)k\rfloor$小特征值。我们的不等式是构造性的，避免了标准Cheeger不等式中存在的“平方根损失”（即使对于$k=2$的情况）。我们还提供了一个互补的下界，以及一个适用于任意顶点权重和边成本的广义推广。此外，我们的结果蕴含并推广了标准高阶Cheeger不等式以及Kwok、Lau和Lee（2017）近期提出的涉及鲁棒顶点扩张的另一类Cheeger型不等式。