Spectral Subspace Sparsification
Huan Li, Aaron Schild

TL;DR
This paper presents a novel spectral sparsification method that efficiently produces near-linear size sparsifiers by sampling spanning trees and contracting edges, improving computation of effective resistances and solving variants of the Steiner point removal problem.
Contribution
It introduces a new spectral sparsification technique that generalizes Schur complement sparsifiers, with faster algorithms and reweighted minors, advancing graph sparsification and effective resistance approximation.
Findings
Produces sparsifiers with near-linear edges in subspace dimension
Achieves almost-linear time sparsifier construction independent of error parameter
Provides a fast data structure for approximating effective resistance changes
Abstract
We introduce a new approach to spectral sparsification that approximates the quadratic form of the pseudoinverse of a graph Laplacian restricted to a subspace. We show that sparsifiers with a near-linear number of edges in the dimension of the subspace exist. Our setting generalizes that of Schur complement sparsifiers. Our approach produces sparsifiers by sampling a uniformly random spanning tree of the input graph and using that tree to guide an edge elimination procedure that contracts, deletes, and reweights edges. In the context of Schur complement sparsifiers, our approach has two benefits over prior work. First, it produces a sparsifier in almost-linear time with no runtime dependence on the desired error. We directly exploit this to compute approximate effective resistances for a small set of vertex pairs in faster time than prior work (Durfee-Kyng-Peebles-Rao-Sachdeva '17).…
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Taxonomy
TopicsGraphene research and applications · Complexity and Algorithms in Graphs · Graph theory and applications
