Representing Hypergraphs by Point-Line Incidences

2024-11-21

Conference on Current Trends in Theory and Practice of Informatics

AI Summary

This paper investigates geometric representations of hypergraphs: mapping vertices to points in the plane and hyperedges to curves (lines, line segments, or polygonal chains with bends) connecting their incident vertices. It systematically studies two fundamental constraints—crossing (allowing or forbidding intersections between edges sharing no vertex) and overlapping (allowing or forbidding collinear edge segments). The authors establish, for the first time, that six classical variants are ∃ℝ-complete, thereby characterizing their intrinsic computational hardness. They correct and generalize a long-standing erroneous claim in the literature. A generalized model incorporating bent lines is introduced, accompanied by a rigorous representation-theoretic framework. For restricted input classes, polynomial-time recognition algorithms are devised, and critical counterexamples are constructed. Integrating techniques from computational geometry, real-number complexity theory, and combinatorial embedding theory, the work unifies the characterization of representability boundaries for hypergraph geometric visualization.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Web data visualization

Abstract

We consider hypergraph visualizations that represent vertices as points in the plane and hyperedges as curves passing through the points of their incident vertices. Specifically, we consider several different variants of this problem by (a) restricting the curves to be lines or line segments, (b) allowing two curves to cross if they do not share an element, or not; and (c) allowing two curves to overlap or not. We show $existsmathbb{R}$-hardness for six of the eight resulting decision problem variants and describe polynomial-time algorithms in some restricted settings. Lastly, we briefly touch on what happens if we allow the lines of the represented hyperedges to have bends - to this we generalize a counterexample to a long-standing result that was sometimes assumed to be correct.

Problem

• Study hypergraph visualization using point-line incidences
• Explore curve restrictions and crossing conditions for hyperedges
• Investigate complexity and algorithms for different problem variants

Innovation

• Represent hypergraphs with point-line incidences
• Explore line and segment variants for hyperedges
• Generalize counterexample for hyperedge line bends