Near-optimal Hypergraph Sparsification in Insertion-only and Bounded-deletion Streams
Sanjeev Khanna, Aaron Putterman, Madhu Sudan
TL;DR
This work resolves the space complexity of constructing $(1 obreak ext{±} obreak ext{ε})$ hypergraph cut-sparsifiers in streaming models. It proves that insertion-only streams admit a sparsification algorithm with $ ilde{O}(n r/ ext{ε}^2)$ bits, matching the static setting, and establishes a smooth transition to dynamic/bounded-deletion streams with a $ ilde{O}(n r ext{log}(k)/ ext{ε}^2)$ space bound, alongside a matching lower bound of $ ilde{Ω}(n r ext{log}(k/n))$. The key technique is strength-based sampling combined with contracting strong components into super-vertices, leveraging laminar structure to limit the number of components and reduce the effective vertex count across multiple sampling levels. The results separate insertion-only and dynamic streaming in hypergraph sparsification and provide a coherent interpolation as deletions grow, with concurrent work highlighting related improvements in spectral sparsification. Overall, the paper advances practical streaming sparsification by showing near-optimal space usage in insertion-only streams and a principled, parameterized transition to bounded-deletion dynamics.
Abstract
We study the problem of constructing hypergraph cut sparsifiers in the streaming model where a hypergraph on $n$ vertices is revealed either via an arbitrary sequence of hyperedge insertions alone ({\em insertion-only} streaming model) or via an arbitrary sequence of hyperedge insertions and deletions ({\em dynamic} streaming model). For any $ε\in (0,1)$, a $(1 \pm ε)$ hypergraph cut-sparsifier of a hypergraph $H$ is a reweighted subgraph $H'$ whose cut values approximate those of $H$ to within a $(1 \pm ε)$ factor. Prior work shows that in the static setting, one can construct a $(1 \pm ε)$ hypergraph cut-sparsifier using $\tilde{O}(nr/ε^2)$ bits of space [Chen-Khanna-Nagda FOCS 2020], and in the setting of dynamic streams using $\tilde{O}(nr\log m/ε^2)$ bits of space [Khanna-Putterman-Sudan FOCS 2024]; here the $\tilde{O}$ notation hides terms that are polylogarithmic in $n$, and we use $m$ to denote the total number of hyperedges in the hypergraph. Up until now, the best known space complexity for insertion-only streams has been the same as that for the dynamic streams. This naturally poses the question of understanding the complexity of hypergraph sparsification in insertion-only streams. Perhaps surprisingly, in this work we show that in \emph{insertion-only} streams, a $(1 \pm ε)$ cut-sparsifier can be computed in $\tilde{O}(nr/ε^2)$ bits of space, \emph{matching the complexity} of the static setting. As a consequence, this also establishes an $Ω(\log m)$ factor separation between the space complexity of hypergraph cut sparsification in insertion-only streams and dynamic streams, as the latter is provably known to require $Ω(nr \log m)$ bits of space.