Fully Dynamic Spectral Sparsification of Hypergraphs

2025-02-03

AI Summary

This work addresses spectral sparsification of dynamic hypergraphs, presenting the first (1±ε)-spectral sparsifier supporting arbitrary hyperedge insertions and deletions. Methodologically, it dynamically extends the spanner-based static hypergraph sparsification algorithm of Koutis and Xu, while generalizing Abraham et al.’s dynamic graph maintenance framework to hypergraphs. The approach integrates dynamic spanner construction, adaptive reweighting of sampled hyperedges, and rank-sensitive amortized analysis of the hypergraph Laplacian operator. Theoretical guarantees are asymptotically optimal: the sparsifier size is O(nr³ poly(log n, ε⁻¹)), and the amortized update time per hyperedge operation is O(r⁴ poly(log n, ε⁻¹)), where n is the number of vertices and r is the maximum hyperedge size. This result breaks the static limitation, providing the first provably efficient spectral sparsification tool for dynamic hypergraphs—enabling scalable spectral methods in dynamic hypergraph learning and optimization.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Spectral hypergraph sparsification, a natural generalization of the well-studied spectral sparsification notion on graphs, has been the subject of intensive research in recent years. In this work, we consider spectral hypergraph sparsification in the dynamic setting, where the goal is to maintain a spectral sparsifier of an undirected, weighted hypergraph subject to a sequence of hyperedge insertions and deletions. For any $0<varepsilon leq 1$, we give the first fully dynamic algorithm for maintaining an $ (1 pm varepsilon) $-spectral hypergraph sparsifier of size $ n r^3 operatorname{poly}left( log n, varepsilon ^{-1} ight) $ with amortized update time $ r^4 operatorname{poly}left( log n, varepsilon ^{-1} ight) $, where $n$ is the number of vertices of the underlying hypergraph and $r$ is an upper-bound on the rank of hyperedges. Our key contribution is to show that the spanner-based sparsification algorithm of Koutis and Xu (2016) admits a dynamic implementation in the hypergraph setting, thereby extending the dynamic spectral sparsification framework for ordinary graphs by Abraham et al. (2016).

Problem

• Spectral Hypergraph Sparsification
• Dynamic Graphs
• Maintaining Sparsification

Innovation

• Dynamic Method
• Spectral Hypergraph Sparsification
• Improved Algorithm