Towards Resistance Sparsifiers

TL;DR

This paper investigates resistance sparsification of graphs, showing dense regular expanders admit near-optimal sparse subgraphs that preserve effective resistances, and explores structural properties of expanders related to sparsification.

Contribution

It proves that dense regular expanders have resistance sparsifiers of size  O(n/) and introduces a structural question about sparse regular expanders within dense ones.

Findings

Dense regular expanders admit  O(n/) resistance sparsifiers.

A positive answer to a structural question about sparse regular expanders is provided.

The work relates resistance sparsification to spectral sparsification and graph structure.

Abstract

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a $(1+\epsilon)$-resistance sparsifier of size $\tilde O(n/\epsilon)$, and conjecture this bound holds for all graphs on $n$ nodes. In comparison, spectral sparsification is a strictly stronger notion and requires $\Omega(n/\epsilon^2)$ edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers.