Near-optimal Linear Sketches and Fully-Dynamic Algorithms for Hypergraph Spectral Sparsification

TL;DR

The paper introduces a unified vertex-sampling framework that reduces hypergraph spectral sparsification to effective-resistance sampling on ordinary graphs derived from vertex-sampled subhypergraphs, thereby bypassing the need for expensive weight-assignments. This framework yields near-linear-time, nearly-linear-size spectral sparsifiers and extends to linear sketches, fully dynamic, and online computation models, with provably near-optimal space and update-time guarantees. Central to the approach are concepts of collective energy and minimax resistance, which underpin a certificate-based recovery of important hyperedges via vertex sampling and recursive recovery. The results substantially close gaps in efficiently maintaining spectral sparsifiers for evolving hypergraphs, enabling faster clustering and semi-supervised learning in dynamic settings while matching lower bounds in key models. Practically, the work provides concrete algorithms and data-structures for dynamic/hypersparse environments, with implications for scalable graph signal processing and real-time hypergraph analytics.

Abstract

A hypergraph spectral sparsifier of a hypergraph $G$ is a weighted subgraph $H$ that approximates the Laplacian of $G$ to a specified precision. Recent work has shown that similar to ordinary graphs, there exist $\widetilde{O}(n)$-size hypergraph spectral sparsifiers. However, the task of computing such sparsifiers turns out to be much more involved, and all known algorithms rely on the notion of balanced weight assignments, whose computation inherently relies on repeated, complete access to the underlying hypergraph. We introduce a significantly simpler framework for hypergraph spectral sparsification which bypasses the need to compute such weight assignments, essentially reducing hypergraph sparsification to repeated effective resistance sampling in \textit{ordinary graphs}, which are obtained by \textit{oblivious vertex-sampling} of the original hypergraph. Our framework immediately yields a simple, new nearly-linear time algorithm for nearly-linear size spectral hypergraph sparsification. Furthermore, as a direct consequence of our framework, we obtain the first nearly-optimal algorithms in several other models of computation, namely the linear sketching, fully dynamic, and online settings.

Theorems & Definitions (149)

• Theorem 1.1: Informal
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Key Technical Theorem
• Lemma 2.2
• Lemma 2.3
• Definition 2.1: Vertex Potentials and Spanning Hyperedges
• Lemma 2.3: Collective Energy Lemma