A $(1 \pm \epsilon)$-sparsifier of a hypergraph $G(V,E)$ is a (weighted) subgraph that preserves the value of every cut to within a $(1 \pm \epsilon)$-factor. It is known that every hypergraph with $n$ vertices admits a $(1 \pm \epsilon)$-sparsifier with $\tilde{O}(n/\epsilon^2)$ hyperedges. In this work, we explore the task of building such a sparsifier by using only linear measurements (a \emph{linear sketch}) over the hyperedges of $G$, and provide nearly-matching upper and lower bounds for this task. Specifically, we show that there is a randomized linear sketch of size $\widetilde{O}(n r \log(m) / \epsilon^2)$ bits which with high probability contains sufficient information to recover a $(1 \pm \epsilon)$ cut-sparsifier with $\tilde{O}(n/\epsilon^2)$ hyperedges for any hypergraph with at most $m$ edges each of which has arity bounded by $r$. This immediately gives a dynamic streaming algorithm for hypergraph cut sparsification with an identical space complexity, improving on the previous best known bound of $\widetilde{O}(n r^2 \log^4(m) / \epsilon^2)$ bits of space (Guha, McGregor, and Tench, PODS 2015). We complement our algorithmic result above with a nearly-matching lower bound. We show that for every $\epsilon \in (0,1)$, one needs $\Omega(nr \log(m/n) / \log(n))$ bits to construct a $(1 \pm \epsilon)$-sparsifier via linear sketching, thus showing that our linear sketch achieves an optimal dependence on both $r$ and $\log(m)$.

翻译：超图 $G(V,E)$ 的 $(1 \pm \epsilon)$-稀疏化器是一个（带权）子图，能在 $(1 \pm \epsilon)$ 因子内保持每个割的值。已知每个具有 $n$ 个顶点的超图都允许存在一个具有 $\tilde{O}(n/\epsilon^2)$ 条超边的 $(1 \pm \epsilon)$-稀疏化器。在本工作中，我们探索仅通过使用对 $G$ 的超边的线性测量（一种\emph{线性素描}）来构建此类稀疏化器的任务，并为此任务提供了近乎匹配的上界和下界。具体而言，我们证明存在一个大小为 $\widetilde{O}(n r \log(m) / \epsilon^2)$ 比特的随机化线性素描，它以高概率包含足够的信息，可以为任何最多具有 $m$ 条边且每条边的元数（arity）以 $r$ 为界的超图，恢复一个具有 $\tilde{O}(n/\epsilon^2)$ 条超边的 $(1 \pm \epsilon)$ 割稀疏化器。这立即给出了一种用于超图割稀疏化的动态流算法，其空间复杂度相同，改进了先前已知的最佳空间界限 $\widetilde{O}(n r^2 \log^4(m) / \epsilon^2)$ 比特（Guha, McGregor, and Tench, PODS 2015）。我们用一个近乎匹配的下界来补充上述算法结果。我们证明对于每个 $\epsilon \in (0,1)$，需要通过线性素描构建 $(1 \pm \epsilon)$-稀疏化器需要 $\Omega(nr \log(m/n) / \log(n))$ 比特，从而表明我们的线性素描在 $r$ 和 $\log(m)$ 上都达到了最优依赖。