We study the spectral implications of re-weighting a graph by the $\ell_\infty$-Lewis weights of its edges. Our main motivation is the ER-Minimization problem (Saberi et al., SIAM'08): Given an undirected graph $G$, the goal is to find positive normalized edge-weights $w\in \mathbb{R}_+^m$ which minimize the sum of pairwise \emph{effective-resistances} of $G_w$ (Kirchhoff's index). By contrast, $\ell_\infty$-Lewis weights minimize the \emph{maximum} effective-resistance of \emph{edges}, but are much cheaper to approximate, especially for Laplacians. With this algorithmic motivation, we study the ER-approximation ratio obtained by Lewis weights. Our first main result is that $\ell_\infty$-Lewis weights provide a constant ($\approx 3.12$) approximation for ER-minimization on \emph{trees}. The proof introduces a new technique, a local polarization process for effective-resistances ($\ell_2$-congestion) on trees, which is of independent interest in electrical network analysis. For general graphs, we prove an upper bound $\alpha(G)$ on the approximation ratio obtained by Lewis weights, which is always $\leq \min\{ \text{diam}(G), \kappa(L_{w_\infty})\}$, where $\kappa$ is the condition number of the weighted Laplacian. All our approximation algorithms run in \emph{input-sparsity} time $\tilde{O}(m)$, a major improvement over Saberi et al.'s $O(m^{3.5})$ SDP for exact ER-minimization. Finally, we demonstrate the favorable effects of $\ell_\infty$-LW reweighting on the \emph{spectral-gap} of graphs and on their \emph{spectral-thinness} (Anari and Gharan, 2015). En-route to our results, we prove a weighted analogue of Mohar's classical bound on $\lambda_2(G)$, and provide a new characterization of leverage-scores of a matrix, as the gradient (w.r.t weights) of the volume of the enclosing ellipsoid.

翻译：我们研究通过边的ℓ∞-路易斯权重对图进行重新赋权的谱学含义。主要动机源于ER-最小化问题（Saberi等, SIAM'08）：给定无向图$G$，目标是寻找正则化的正边权重$w\in \mathbb{R}_+^m$，使得$G_w$的成对有效电阻之和（基尔霍夫指数）最小化。相比之下，ℓ∞-路易斯权重最小化边的最大有效电阻，但其近似计算成本远低于前者，尤其在拉普拉斯矩阵场景中。基于这一算法动机，我们研究路易斯权重获得的ER-近似比。首要结果是：在树上，ℓ∞-路易斯权重能为ER-最小化问题提供常数近似比（≈3.12）。证明引入了一种新方法——树上有效电阻（ℓ2-拥塞）的局部极化过程，这一技术对电网络分析具有独立研究价值。对于一般图，我们证明路易斯权重获得的近似比存在上界α(G)，且始终满足α(G) ≤ min{ diam(G), κ(L_{w∞}) }，其中κ为加权拉普拉斯矩阵的条件数。所有近似算法均运行于输入稀疏时间$\tilde{O}(m)$内，较Saberi等人精确ER-最小化的$O(m^{3.5})$半定规划方法有重大改进。最后，我们展示了ℓ∞-路易斯权重重赋权对图谱间隙及谱细度（Anari & Gharan, 2015）的积极影响。在推导结论过程中，我们证明了莫哈尔经典上界λ2(G)的加权版本，并给出了矩阵杠杆得分的全新刻画——将其定义为包络椭球体积关于权重的梯度。