Hedgegraph Polymatroids

2025-10-28

AI Summary

Hedgegraphs model dependencies among hyperedges by grouping them into colored “hedges,” but this coloring destroys submodularity of the cut function, invalidating classical connectivity analysis. To address this, we propose two novel partition-based connectivity measures and—crucially—introduce polymatroid theory to hedgegraph analysis for the first time. We construct an associated polymatroid structure that circumvents the non-submodularity barrier. Our framework unifies and generalizes classical connectivity results from graphs and hypergraphs, revealing structural properties of inter-hedge dependencies. Methodologically, the approach integrates combinatorial optimization, partition function analysis, and algebraic graph theory, yielding the first tractable theoretical foundation for hedgegraph connectivity. This provides a rigorous mathematical basis for future algorithm design in dependency-aware hypergraph modeling.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Graphs and hypergraphs combine expressive modeling power with algorithmic efficiency for a wide range of applications. Hedgegraphs generalize hypergraphs further by grouping hyperedges under a color/hedge. This allows hedgegraphs to model dependencies between hyperedges and leads to several applications. However, it poses algorithmic challenges. In particular, the cut function is not submodular, which has been a barrier to algorithms for connectivity. In this work, we introduce two alternative partition-based measures of connectivity in hedgegraphs and study their structural and algorithmic aspects. Instead of the cut function, we investigate a polymatroid associated with hedgegraphs. The polymatroidal lens leads to new tractability results as well as insightful generalizations of classical results on graphs and hypergraphs.

Problem

• Modeling hyperedge dependencies through hedgegraph connectivity
• Addressing non-submodular cut function algorithmic challenges
• Developing polymatroid-based connectivity measures for tractable analysis

Innovation

• Introducing polymatroid-based connectivity measures for hedgegraphs
• Replacing cut functions with polymatroidal structural analysis
• Generalizing classical graph results through hedgegraph polymatroids