Near-optimal Size Linear Sketches for Hypergraph Cut Sparsifiers

TL;DR

This paper develops nearly optimal linear sketching methods for hypergraph cut sparsifiers, achieving improved space complexity and matching lower bounds, advancing dynamic streaming algorithms for hypergraph sparsification.

Contribution

Introduces a nearly-matching upper and lower bound for linear sketching of hypergraph cut sparsifiers, improving space complexity bounds and providing a dynamic streaming algorithm.

Findings

Linear sketch of size $ ilde{O}(n r ext{log}(m) / ext{epsilon}^2)$ bits suffices for hypergraph sparsification.

Improves previous space bounds from $ ilde{O}(n r^2 ext{log}^4(m) / ext{epsilon}^2)$ bits.

Lower bound of $ ilde{ ext{Omega}}(n r ext{log}(m/n) / ext{log}(n))$ bits matches the upper bound dependence on $r$ and $ ext{log}(m)$.

Abstract

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 $\log(m)$.