Nearly Space-Optimal Graph and Hypergraph Sparsification in Insertion-Only Data Streams

We study the problem of graph and hypergraph sparsification in insertion-only data streams. The input is a hypergraph $H=(V, E, w)$ with $n$ nodes, $m$ hyperedges, and rank $r$, and the goal is to compute a hypergraph $\widehat{H}$ that preserves the energy of each vector $x \in \mathbb{R}^n$ in $H$, up to a small multiplicative error. In this paper, we give a streaming algorithm that achieves a $(1+\varepsilon)$-approximation, using $\mathcal{O}\left(\frac{rn}{\varepsilon^2} \log^2 n \log r\right) \cdot$ poly $(\log \log m)$ bits of space, matching the sample complexity of the best known offline algorithm up to poly $(\log \log m)$ factors. Our approach also provides a streaming algorithm for graph sparsification that achieves a $(1+\varepsilon)$-approximation, using $\mathcal{O}\left(\frac{n}{\varepsilon^2} \log n\right)\cdot\text{poly}(\log\log n)$ bits of space, improving the current bound by $\log n$ factors. Furthermore, we give a space-efficient streaming algorithm for min-cut approximation. Along the way, we present an online algorithm for $(1+\varepsilon)$-hypergraph sparsification, which is optimal up to poly-logarithmic factors. Hence, we achieve $(1+\varepsilon)$-hypergraph sparsification in the sliding window model, with space optimal up to poly-logarithmic factors. Lastly, we give an adversarially robust algorithm for hypergraph sparsification using $\frac{n}{\varepsilon^2} \cdot$ poly $(r, \log n, \log r, \log \log m)$ bits of space.