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• W4313203103 abstract "A seminal work of [Ahn-Guha-McGregor, PODS’12] showed that one can compute a cut sparsifier of an unweighted undirected graph by taking a near-linear number of linear measurements on the graph. Subsequent works also studied computing other graph sparsifiers using linear sketching, and obtained near-linear upper bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS’14] and first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA’21]. All these linear sketching algorithms, however, only work on unweighted graphs, and are extended to weighted graphs by weight grouping, a non-linear operation not implementable in, for instance, general turnstile streams.In this paper, we initiate the study of weighted graph sparsification by linear sketching by investigating a natural class of linear sketches that we call incidence sketches, in which each measurement is a linear combination of the weights of edges incident on a single vertex. This class captures all aforementioned linear sketches for unweighted sparsification. It also covers linear sketches implementable in the simultaneous communication model, where edges are distributed across n machines. Our results are:1)Weighted cut sparsification: We give an algorithm that computes a $(1+epsilon)$-cut sparsifier using $tilde{O}(nepsilon^{-3})$ linear measurements, which is nearly optimal. This also implies a turnstile streaming algorithm with $tilde{O}(nepsilon^{-3})$ space. Our algorithm is achieved by building a so-called “weighted edge sampler” for each vertex.2)Weighted spectral sparsification: We give an algorithm that computes a $(1+epsilon)$-spectral sparsifier using $tilde{O}(n^{6/5}epsilon^{-4})$ linear measurements. This also implies a turnstile streaming algorithm with $tilde{O}(n^{6/5}epsilon^{-4})$ space. Key to our algorithm is a novel analysis of how the effective resistances change under vertex sampling. Complementing our algorithm, we then prove a superlinear lower bound of $Omega(n^{21/20-o(1)})$ measurements for computing some O(1)-spectral sparsifier using incidence sketches.3)Weighted spanner computation: We first show that any $o(n^{2})$ linear measurements can only recover a spanner of stretch that in general depends linearly on $frac{w_{max}}{w_{min}}$. We thus focus on graphs with $frac{w_{max}}{w_{min}}=O(1)$ and study the stretch’s dependence on n. On such graphs, the algorithm in [FiltserKapralov-Nouri, SODA’21] can obtain a spanner of stretch $tilde{O}left(n^{frac{2}{3}left(1-alpharight)}right)$ using $tilde{O}(n^{1+alpha})$ measurements for any $alphain [0,1]$. We prove that, for incidence sketches, this tradeoff is optimal up to an $n^{o(1)}$ factor for all $alphalt 1/10$.We prove both our lower bounds by analyzing the “effective resistances” in certain matrix-weighted graphs, where we develop a number of new tools for reasoning about such graphs – most notably (i) a matrix-weighted analog of the widely used expander decomposition of ordinary graphs, and (ii) a proof that a random vertex-induced subgraph of a matrix-weighted expander is also an expander. We believe these tools are of independent interest." @default.
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• W4313203103 date "2022-10-01" @default.
• W4313203103 modified "2023-09-29" @default.
• W4313203103 title "On Weighted Graph Sparsification by Linear Sketching" @default.
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• W4313203103 doi "https://doi.org/10.1109/focs54457.2022.00052" @default.
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