Infinite Lewis Weights in Spectral Graph Theory
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We study the spectral implications of re-weighting a graph by the ℓ_∞-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∈ℝ_+^m which minimize the sum of pairwise effective-resistances of G_w (Kirchhoff's index). By contrast, ℓ_∞-Lewis weights minimize the maximum effective-resistance of 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 ℓ_∞-Lewis weights provide a constant (≈ 3.12) approximation for ER-minimization on trees. The proof introduces a new technique, a local polarization process for effective-resistances (ℓ_2-congestion) on trees, which is of independent interest in electrical network analysis. For general graphs, we prove an upper bound α(G) on the approximation ratio obtained by Lewis weights, which is always ≤min{diam(G), κ(L_w_∞)}, where κ is the condition number of the weighted Laplacian. All our approximation algorithms run in input-sparsity time Õ(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 ℓ_∞-LW reweighting on the spectral-gap of graphs and on their spectral-thinness (Anari and Gharan, 2015). En-route to our results, we prove a weighted analogue of Mohar's classical bound on λ_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.
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