Constant Arboricity Spectral Sparsifiers

TL;DR

This paper demonstrates that any graph can be approximated spectrally by a union of a constant number of forests, and that existing sparsifiers are unions of logarithmically many forests, enabling efficient boundary estimation.

Contribution

It introduces a novel spectral approximation of graphs using a constant number of forests and improves understanding of Spielman-Srivastava sparsifiers as unions of O(log n) forests.

Findings

Graphs are spectrally similar to unions of constant forests

Spielman-Srivastava sparsifiers are unions of O(log n) forests

Enables nearly optimal query time for boundary estimation

Abstract

We show that every graph is spectrally similar to the union of a constant number of forests. Moreover, we show that Spielman-Srivastava sparsifiers are the union of O(logn) forests. This result can be used to estimate boundaries of small subsets of vertices in nearly optimal query time.