Subgraph Sparsification and Nearly Optimal Ultrasparsifiers

TL;DR

This paper introduces a new approach to spectral subgraph sparsification, providing polynomial-time algorithms for constructing nearly optimal ultrasparsifiers that improve spectral properties with fewer edges.

Contribution

It presents a novel condition for the existence of spectral sparsifiers in subgraph settings and offers algorithms to efficiently find such sparsifiers, advancing spectral graph theory.

Findings

Existence of spectral sparsifiers under certain conditions

Construction of $(n-1+k)$-edge spectral sparsifiers with near-optimal bounds

Application to maximizing algebraic connectivity with limited edges

Abstract

We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs $G$ and $W$ and an integer $k$, we are asked to find a $k$-edge weighted graph $W_k$ such that $G+W_k$ is a good spectral sparsifer of $G+W$. We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on $G$ and $W$ such that a good sparsifier exists and give a polynomial time algorithm to find the sparsifer. %$O(\frac{n}{k})\log n \tilde{O}(\log \log n)$ As a significant application of our technique, we show that for each positive integer $k$, every $n$-vertex weighted graph has an $(n-1+k)$-edge spectral sparsifier with relative condition number at most $\frac{n}{k} \log n \tilde{O}(\log\log n)$ where $\tilde{O}()$ hides lower order terms. Our bound is within a factor of $\tilde{O}(\log \log n)$ from optimal. This nearly settles a question left open by Spielman and Teng about ultrasparsifiers, which is a key component in their nearly linear-time algorithms for solving diagonally dominant symmetric linear systems. We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges.