Scalable Approximate Algorithm for Dynamic Densest Subhypergraphs with Solution-Guided Maintenance

2026-09-30

AI Summary

This work addresses the prohibitive maintenance overhead of densest subgraphs in dynamic hypergraphs by proposing the CAP algorithm. CAP introduces an endpoint assignment mechanism to establish a tight upper bound on optimal density and designs a solution-guided local repair strategy that triggers local search only when necessary, thereby reusing valid solutions to efficiently maintain a (1+ε)-approximate densest subhypergraph. Experimental results demonstrate that CAP achieves nearly four orders of magnitude speedup over existing baseline methods for hypergraphs, while remaining approximately 64 times faster than mainstream dynamic graph algorithms even on standard graphs. These findings confirm that CAP simultaneously optimizes both theoretical complexity and query efficiency for densest subgraph maintenance in dynamic settings.

Abstract

Hypergraphs model interactions involving groups of entities, and finding highly connected groups is a fundamental task in analyzing these data. When interactions arrive and expire, maintaining a dense group can require frequent and expensive changes to the underlying representation. We propose CAP, a scalable algorithm that explicitly maintains a $(1+ε)$-approximate densest subhypergraph under hyperedge insertions and deletions. The algorithm keeps a solution candidate together with an endpoint allocation that bounds the optimum density. The candidate's density guides maintenance: an update triggers repair only when this bound is too large to establish the required approximation. Local searches redistribute load or find a denser candidate, allowing CAP to retain useful solutions across many updates. We give an analysis that relates maintenance work to the size of the regions searched and the density lost by the maintained solution. It identifies conditions under which local repair remains inexpensive and provides a theoretical explanation for the observed efficiency. Experiments on real-world hypergraphs show speedups of up to nearly four orders of magnitude over the evaluated dynamic baselines, while supporting frequent queries and high accuracy. CAP also remains competitive with specialized dynamic graph methods on ordinary graphs, with speedups of up to approximately $64\times$ on the tested workloads.

Problem

• dynamic hypergraphs
• densest subhypergraph
• hyperedge insertions and deletions
• approximate algorithm

Innovation

• dynamic densest subhypergraph
• approximate algorithm
• solution-guided maintenance
• local search
• scalability