A New Dynamic Algorithm for Densest Subhypergraphs

TL;DR

This paper introduces a new dynamic algorithm for maintaining the densest subhypergraph in weighted hypergraphs, achieving near-optimal approximation and outperforming previous methods in real-world experiments.

Contribution

The authors present the first $(1+)$-approximation algorithm for weighted hypergraphs in a dynamic setting, improving approximation ratio and efficiency over prior work.

Findings

Achieves near-optimal approximation ratio of $(1+)$ for weighted hypergraphs.

Significantly outperforms previous algorithms in accuracy and efficiency on real datasets.

First to provide a $(1+)$-approximation for weighted simple graphs.

Abstract

Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+\epsilon)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+\epsilon)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+\epsilon)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.