Spectral Subspace Sparsification

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). Secondly, it yields sparsifiers that are reweighted minors of the input graph. As a result, we give a near-optimal answer to a variant of the Steiner point removal problem. A key ingredient of our algorithm is a subroutine of independent interest: a near-linear time algorithm that, given a chosen set of vertices, builds a data structure from which we can query a multiplicative approximation to the decrease in the effective resistance between two vertices after identifying all vertices in the chosen set to a single vertex with inverse polynomial additional additive error in near-constant time.