On Weighted Graph Sparsification by Linear Sketching
Yu Chen, Sanjeev Khanna, Huan Li

TL;DR
This paper extends linear sketching techniques to weighted graphs, providing near-optimal algorithms for weighted cut and spectral sparsification, and establishing bounds for weighted spanner computation.
Contribution
It introduces incidence sketches for weighted graphs and develops algorithms with near-optimal measurements for weighted sparsification and bounds for weighted spanners.
Findings
Weighted cut sparsifier computed with $ ilde{O}(n ext{ poly}(1/\e))$ measurements.
Weighted spectral sparsifier achieved with $ ilde{O}(n^{6/5} ext{ poly}(1/\e))$ measurements.
Lower bounds established for spectral sparsification using incidence sketches.
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. 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. Our results are: 1. Weighted cut sparsification: We give an…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Sparse and Compressive Sensing Techniques · Machine Learning and Algorithms
