Chaining, Group Leverage Score Overestimates, and Fast Spectral Hypergraph Sparsification

Abstract: We present an algorithm that given any $n$-vertex, $m$-edge, rank $r$ hypergraph constructs a spectral sparsifier with $O(n \varepsilon^{-2} \log n \log r)$ hyperedges in nearly-linear $\widetilde{O}(mr)$ time. This improves in both size and efficiency over a line of work (Bansal-Svensson-Trevisan 2019, Kapralov-Krauthgamer-Tardos-Yoshida 2021) for which the previous best size was $O(\min{n \varepsilon^{-4} \log^3 n,nr^3 \varepsilon^{-2} \log n})$ and runtime was $\widetilde{O}(mr + n^{O(1)})$. Independent Result: In an independent work, Lee (Lee 2022) also shows how to compute a spectral hypergraph sparsifier with $O(n \varepsilon^{-2} \log n \log r)$ hyperedges.