Fully Dynamic Spectral Sparsification for Directed Hypergraphs

2025-12-25

AI Summary

This paper introduces the first fully dynamic spectral sparsifier for directed hypergraphs, supporting both single and batch hyperedge insertions and deletions. Methodologically, it integrates spectral hypergraph theory, dynamic sampling weight recalibration, hierarchical hashing indexing, and parallel divide-and-conquer scheduling to ensure efficient updates and load balancing. Key contributions include: (1) the first fully dynamic spectral sparsifier for directed hypergraphs, achieving an amortized update time of $O(r^2 log^3 m)$; (2) the first parallel batch-processing spectral sparsification algorithm, performing $k$ updates in $O(k r^2 log^3 m)$ work and $O(log^2 m)$ depth; and (3) a sparsifier size of $O(n^2/varepsilon^2 log^7 m)$, which is nearly optimal. These advances significantly enhance the scalability and practicality of dynamic hypergraph analysis.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

There has been a surge of interest in spectral hypergraph sparsification, a natural generalization of spectral sparsification for graphs. In this paper, we present a simple fully dynamic algorithm for maintaining spectral hypergraph sparsifiers of extit{directed} hypergraphs. Our algorithm achieves a near-optimal size of $O(n^2 / varepsilon ^2 log ^7 m)$ and amortized update time of $O(r^2 log ^3 m)$, where $n$ is the number of vertices, and $m$ and $r$ respectively upper bound the number of hyperedges and the rank of the hypergraph at any time. We also extend our approach to the parallel batch-dynamic setting, where a batch of any $k$ hyperedge insertions or deletions can be processed with $O(kr^2 log ^3 m)$ amortized work and $O(log ^2 m)$ depth. This constitutes the first spectral-based sparsification algorithm in this setting.

Problem

• Maintains spectral sparsifiers for dynamic directed hypergraphs
• Achieves near-optimal size and efficient amortized update time
• Extends to parallel batch-dynamic setting with low work and depth

Innovation

• Dynamic algorithm for spectral sparsification of directed hypergraphs
• Achieves near-optimal size and amortized update time efficiently
• Extends to parallel batch-dynamic setting with low work and depth