Short Plane Supports for Spatial Hypergraphs

TL;DR

This paper investigates the problem of finding minimal-length, plane straight-line supports for spatial hypergraphs, demonstrating NP-hardness and proposing heuristics and ILP methods with experimental evaluation.

Contribution

It introduces the first comprehensive study on computing short plane supports for spatial hypergraphs, including theoretical complexity results and practical algorithms.

Findings

NP-hardness of the problem under mild conditions

Heuristic algorithms perform well in practice

Trade-offs between solution quality and computational speed

Abstract

A graph $G=(V,E)$ is a support of a hypergraph $H=(V,S)$ if every hyperedge induces a connected subgraph in $G$. Supports are used for certain types of hypergraph visualizations. In this paper we consider visualizing spatial hypergraphs, where each vertex has a fixed location in the plane. This is the case, e.g., when modeling set systems of geospatial locations as hypergraphs. By applying established aesthetic quality criteria we are interested in finding supports that yield plane straight-line drawings with minimum total edge length on the input point set $V$. We first show, from a theoretical point of view, that the problem is NP-hard already under rather mild conditions as well as a negative approximability results. Therefore, the main focus of the paper lies on practical heuristic algorithms as well as an exact, ILP-based approach for computing short plane supports. We report results from computational experiments that investigate the effect of requiring planarity and acyclicity on the resulting support length. Further, we evaluate the performance and trade-offs between solution quality and speed of several heuristics relative to each other and compared to optimal solutions.