Twice-Ramanujan Sparsifiers

We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every $d>1$ and every undirected, weighted graph $G=(V,E,w)$ on $n$ vertices, there exists a weighted graph $H=(V,F,\tilde{w})$ with at most $\lceil d(n-1) \rceil$ edges such that for every $x \in \mathbb{R}^{V}$, [ x^{T}L_{G}x \leq x^{T}L_{H}x \leq (\frac{d+1+2\sqrt{d}}{d+1-2\sqrt{d}})\cdot x^{T}L_{G}x ] where $L_{G}$ and $L_{H}$ are the Laplacian matrices of $G$ and $H$, respectively. Thus, $H$ approximates $G$ spectrally at least as well as a Ramanujan expander with $dn/2$ edges approximates the complete graph. We give an elementary deterministic polynomial time algorithm for constructing $H$.

💡 Research Summary

The paper addresses a long‑standing open problem in spectral graph sparsification: can every graph be approximated by a sparsifier whose number of edges is linear in the number of vertices while preserving the Laplacian quadratic form within a constant factor? The authors answer affirmatively and provide a constructive, deterministic polynomial‑time algorithm.

Main theorem. For any undirected weighted graph (G=(V,E,w)) with (|V|=n) and any real parameter (d>1), there exists a weighted subgraph (H=(V,F,\tilde w)) with at most (\lceil d(n-1)\rceil) edges such that for every vector (x\in\mathbb{R}^V)
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