Linear-Sized Spectral Sparsifiers and the Kadison-Singer Problem

TL;DR

This paper proves the existence of linear-sized spectral sparsifiers for all undirected, weighted graphs by leveraging the Kadison-Singer theorem, providing a new theoretical foundation for graph sparsification.

Contribution

It formalizes the connection between the Kadison-Singer result and spectral sparsifiers, introducing a recursive algorithm to construct such sparsifiers with linear edge count.

Findings

Spectral sparsifiers with linear edges exist for all undirected, weighted graphs.

The proof combines leverage score partitioning with recursive graph sparsification.

The approach extends previous results by using Kadison-Singer to guarantee sparsifier existence.

Abstract

The Kadison-Singer Conjecture, as proved by Marcus, Spielman, and Srivastava (MSS) [Ann. Math. 182, 327-350 (2015)], has been informally thought of as a strengthening of Batson, Spielman, and Srivastava's theorem that every undirected graph has a linear-sized spectral sparsifier [SICOMP 41, 1704-1721 (2012)]. We formalize this intuition by using a corollary of the MSS result to derive the existence of spectral sparsifiers with a number of edges linear in their number of vertices for all undirected, weighted graphs. The proof consists of two steps. First, following a suggestion of Srivastava [Asia Pac. Math. Newsl. 3, 15-20 (2013)], we show the result in the special case of graphs with bounded leverage scores by repeatedly applying the MSS corollary to partition the graph, while maintaining an appropriate bound on the leverage scores of each subgraph. Then, we extend to the general case by constructing a recursive algorithm that repeatedly (i) divides edges with high leverage scores into multiple parallel edges and (ii) uses the bounded leverage score case to sparsify the resulting graph.