Near-optimal Hypergraph Sparsification in Insertion-only and Bounded-deletion Streams

2025-04-22

AI Summary

This paper studies streaming hypergraph cut sparsification under two models: insertion-only streams (edges only added) and bounded-deletion streams (edges both added and deleted). Methodologically, it introduces an online sparsification framework integrating hierarchical edge-importance characterization, streaming hashing, and sampling-weight estimation. The main contribution is the first (1±ε)-approximate cut sparsifier for insertion-only streams achieving near-optimal space complexity Õ(nr/ε²) bits—matching the static lower bound tightly. It further establishes an intrinsic Ω(log m) space separation between insertion-only and dynamic (unbounded-deletion) streams, revealing a fundamental advantage of the former. This result constitutes the first hypergraph cut sparsifier in the insertion-only model attaining static-optimal space efficiency, substantially improving upon the known Ω(nr log m) lower bound for dynamic streams.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Machine Learning: Learning on the Edge & Model Compression
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $epsilon in (0,1)$, a $(1 pm epsilon)$ hypergraph cut-sparsifier of a hypergraph $H$ is a reweighted subgraph $H'$ whose cut values approximate those of $H$ to within a $(1 pm epsilon)$ factor. Prior work shows that in the static setting, one can construct a $(1 pm epsilon)$ hypergraph cut-sparsifier using $ ilde{O}(nr/epsilon^2)$ bits of space [Chen-Khanna-Nagda FOCS 2020], and in the setting of dynamic streams using $ ilde{O}(nrlog m/epsilon^2)$ bits of space [Khanna-Putterman-Sudan FOCS 2024]; here the $ ilde{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 $(1 pm epsilon)$ cut-sparsifier can be computed in $ ilde{O}(nr/epsilon^2)$ bits of space, emph{matching the complexity} of the static setting. As a consequence, this also establishes an $Omega(log m)$ 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 $Omega(nr log m)$ bits of space.

Problem

• Construct hypergraph cut sparsifiers in streaming models
• Optimize space complexity for insertion-only streams
• Compare insertion-only vs dynamic stream complexities

Innovation

• Insertion-only streams achieve static complexity
• Dynamic streams require more space
• Hypergraph cut-sparsifiers use reweighted subgraphs