Near-Optimal Size Linear Sketches for Hypergraph Cut Sparsifiers

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 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 $\tilde{O}(nr\log(m)/\epsilon^{2})$ 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 $\tilde{O}(nr^{2}\log^{4}({m})/\epsilon^{2})$ 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 $\log(m)$. The starting point for our improved algorithm is importance sampling of hyperedges based on the new notion of $k$ -cut strength introduced in the recent work of Quanrud (SODA 2024). The natural algorithm based on this concept leads to $\log m$ levels of sampling where errors can potentially accumulate, and this accounts for the polylog $(m)$ losses in the sketch size of the natural algorithm. We develop a more intricate analysis of the accumulation in error to show most levels do not contribute to the error and actual loss is only polylog $(n)$. Combining with careful preprocessing (and analysis) this enables us to get rid of all extraneous $\log m$ factors in the sketch size, but the quadratic dependence on $r$ remains. This dependence originates from use of correlated $\ell_{0}$ -samplers to recover a large number of low-strength edges in a hypergraph simultaneously by looking at neighborhoods of individual vertices. In graphs, this leads to discovery of $\Omega(n)$ edges in a single shot, whereas in hypergraphs, this may potentially only reveal $O$($n$/$r$) new edges, thus requiring $\Omega(r)$ rounds of recovery. To remedy this we introduce a new technique of random fingerprinting of hyperedges which effectively eliminates the correlations created by large arity hyperedges, and leads to a scheme for recovering hyperedges of low strength with an optimal dependence on $r$. Putting all these ingredients together yields our linear sketching algorithm. Our lower bound is established by a reduction from the universal relation problem in the one-way communication setting.