Fully Dynamic Spectral Sparsification for Directed Hypergraphs

TL;DR

This work develops a fully dynamic spectral sparsification framework for directed hypergraphs, achieving a $(1\pm\varepsilon)$-spectral hypersparsifier of size $O\left(\frac{n^2}{\varepsilon^2}\log^7 m\right)$ with amortized update time $O\left(r^2\log^3 m\right)$, and extends naturally to batch-parallel updates with $O(kr^2\log^3 m)$ work and $O(\log^2 m)$ depth for batches of size $k$. Building on the static directed-hypergraph sparsification framework, the authors introduce a novel static variant that recurses on $S_{i-1}$ and aggregates coresets $C_i$ to the sparsifier, trading exact recourse for polylogarithmic overhead in size. The dynamic construction first develops a decremental sparsifier and then reduces the fully dynamic problem to a sequence of decremental subproblems via batching, using a decomposability property to combine level sparsifiers. The result is the first spectral-based sparsification algorithm for directed hypergraphs in the batch-parallel setting, with near-optimal size and polylogarithmic update guarantees, enabling scalable dynamic analysis of complex hypergraph-structured data.

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 \textit{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.

Figures (1)

• Figure 1: Comparison of (a) the algorithm of Oko:2023aa and (b) our static algorithm. In each iteration $i$, their algorithm recurses on $H_{i-1}\xspace = C_{i-1}\xspace \cup S_{i-1}\xspace$ and computes $H_{i}\xspace = C_{i}\xspace \cup S_{i}\xspace$ for the next iteration. In contrast, our algorithm recurses solely on $S_{i-1}$, adds the coreset $C_{i}$ to the sparsifier $\widetilde{H}$, and computes the sampled hypergraph $S_{i}$ for the next iteration. After $k= O(\log m)$ iterations, our algorithm terminates and returns $\widetilde{H}\xspace = C_{1}\xspace \cup \dots C_{k}\xspace \cup S_k$, whereas the algorithm of Oko:2023aa returns $\widetilde{H}\xspace = C_{k}\xspace \cup S_{k}\xspace$ (the shaded parts in the figures). The increase in the size of $\widetilde{H}$ in our algorithm allows us to maintain $\widetilde{H}$ efficiently in the dynamic setting, as detailed in \ref{['subsec:dynamic']}.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: Decomposability
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4