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Fully Dynamic Spectral Sparsification of Hypergraphs

Gramoz Goranci, Ali Momeni

TL;DR

This work addresses maintaining a $(1 \pm \varepsilon)$-spectral hypersparsifier of a dynamic hypergraph under insertions and deletions. The authors deliver the first fully dynamic algorithm with sparsifier size $O(n r^3 \varepsilon^{-2} \log^2 m \log^5 n \log W)$ and amortized update time $O(r^4 \varepsilon^{-2} \log^2 m \log^5 n \log r)$, parameterized by the hyperedge rank $r$ and weight ratio bound $W$. The approach weaves a sampling scheme guided by effective resistance with a dynamic, monotone $t$-bundle hyperspanner structure, enabling a constant-probability sampling outside a bordered set and a peeling-based sparsification that preserves spectral properties. By extending the Koutis–Xu spanner framework to hypergraphs and leveraging nested dynamic sparsifiers, the paper achieves nearly optimal sparsifier size and polylogarithmic update times, enabling scalable dynamic hypergraph processing for downstream tasks such as clustering, embedding, and graph-based learning.

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} \right) $ with amortized update time $ r^4 \operatorname{poly}\left( \log n, \varepsilon ^{-1} \right) $, 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).

Fully Dynamic Spectral Sparsification of Hypergraphs

TL;DR

This work addresses maintaining a -spectral hypersparsifier of a dynamic hypergraph under insertions and deletions. The authors deliver the first fully dynamic algorithm with sparsifier size and amortized update time , parameterized by the hyperedge rank and weight ratio bound . The approach weaves a sampling scheme guided by effective resistance with a dynamic, monotone -bundle hyperspanner structure, enabling a constant-probability sampling outside a bordered set and a peeling-based sparsification that preserves spectral properties. By extending the Koutis–Xu spanner framework to hypergraphs and leveraging nested dynamic sparsifiers, the paper achieves nearly optimal sparsifier size and polylogarithmic update times, enabling scalable dynamic hypergraph processing for downstream tasks such as clustering, embedding, and graph-based learning.

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 , we give the first fully dynamic algorithm for maintaining an -spectral hypergraph sparsifier of size with amortized update time , where is the number of vertices of the underlying hypergraph and 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).

Paper Structure

This paper contains 20 sections, 23 theorems, 36 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Related Work
  4. Preliminaries
  5. Spectral Hypergraph Sparsification.
  6. Effective Resistance
  7. Spanners and Hyperspanners.
  8. The Associated Graph.
  9. $t$-Bundle $\alpha$-spanners and $t$-Bundle $\alpha$-Hyperspanners.
  10. Chernoff Bound Chernoff:1952aa.
  11. The Algorithm
  12. Static $(1 \pm \varepsilon)$-Spectral Hypersparsifier
  13. Dynamic $(1 \pm \varepsilon)$-Spectral Hypersparsifier
  14. Decremental Monotone $t$-Bundle $\mathcal{O}\mathopen{}\mathclose{\left( \log n \right)}\xspace$-Hyperspanners
  15. Decremental Implementation of \ref{['alg:light_spectral_sparsify']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $0 < \varepsilon \leq 1$, $\gamma \geq 1$ be a constant, and ${ H\xspace} = (V, E, \boldsymbol{w})$ be an initially empty $n$-vertex hypergraph guaranteed to have at most $m$ hyperedges of rank at most $r$ and weight ratio $\max _{i,j} w_i / w_j \leq W$ throughout any sequence of hyperedge inser and an expected amortized update time of

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • Lemma 3.2: Decomposability
  • proof
  • proof : Proof of \ref{['lem:r/2']}
  • Lemma 3.3: Adapted from Oko:2023aa
  • proof : Proof sketch
  • ...and 39 more