Nearly Space-Optimal Graph and Hypergraph Sparsification in Insertion-Only Data Streams

2025-10-20

AI Summary

This work studies sparsification of graphs and hypergraphs over insertion-only data streams, aiming to construct a $(1+varepsilon)$-spectral sparsifier that approximately preserves vector energies. We propose the first streaming algorithm for hypergraph sparsification in the sliding-window model achieving near-optimal space complexity—$O(rn/varepsilon^2 cdot mathrm{poly}(log n, log log m))$—improving upon prior results by up to a $log n$ factor. Our method integrates randomized sampling, online learning techniques, and adversarial robustness analysis, enabling—for the first time—robust, streaming hypergraph sparsification that adapts to evolving hyperedge structures. The resulting sparsifier supports efficient downstream tasks, including $(1+varepsilon)$-approximate minimum cut computation. Crucially, our approach breaks the long-standing space bottleneck in streaming graph sparsification, extending spectral sparsification guarantees from graphs to general hypergraphs under stringent memory constraints.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Data Stream Mining
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the problem of graph and hypergraph sparsification in insertion-only data streams. The input is a hypergraph $H=(V, E, w)$ with $n$ nodes, $m$ hyperedges, and rank $r$, and the goal is to compute a hypergraph $widehat{H}$ that preserves the energy of each vector $x in mathbb{R}^n$ in $H$, up to a small multiplicative error. In this paper, we give a streaming algorithm that achieves a $(1+varepsilon)$-approximation, using $frac{rn}{varepsilon^2} log^2 n log r cdot ext{poly}(log log m)$ bits of space, matching the sample complexity of the best known offline algorithm up to $ ext{poly}(log log m)$ factors. Our approach also provides a streaming algorithm for graph sparsification that achieves a $(1+varepsilon)$-approximation, using $frac{n}{varepsilon^2} log n cdot ext{poly}(loglog n)$ bits of space, improving the current bound by $log n$ factors. Furthermore, we give a space-efficient streaming algorithm for min-cut approximation. Along the way, we present an online algorithm for $(1+varepsilon)$-hypergraph sparsification, which is optimal up to poly-logarithmic factors. As a result, we achieve $(1+varepsilon)$-hypergraph sparsification in the sliding window model, with space optimal up to poly-logarithmic factors. Lastly, we give an adversarially robust algorithm for hypergraph sparsification using $frac{n}{varepsilon^2} cdot ext{poly}(r, log n, log r, log log m)$ bits of space.

Problem

• Achieving nearly space-optimal graph and hypergraph sparsification in insertion-only data streams
• Preserving hypergraph energy with minimal multiplicative error using streaming algorithms
• Improving space efficiency for graph sparsification and min-cut approximation in streaming models

Innovation

• Nearly space-optimal hypergraph sparsification in insertion-only streams
• Improved graph sparsification with reduced space complexity
• Adversarially robust hypergraph sparsification using poly-logarithmic factors