Practical PTAS Implementation in Hyperbolic Geometry via Quadtrees

Develop and implement polynomial-time approximation schemes in hyperbolic geometry based on quadtree decompositions, specifically targeting the construction of hyperbolic Steiner minimal trees, to enable user-tunable approximation–time trade-offs and bridge the empirical gap to optimal performance.

Background

The authors report a remaining empirical gap between their heuristic results and the theoretical upper bound for reductions in hyperbolic Steiner minimal trees. They propose PTAS-based approaches, inspired by Euclidean geometric PTAS methods, as a promising route to systematically control the performance–time trade-off.

Although the existence of quadtrees in hyperbolic space has been established theoretically, the paper notes that a practical implementation of such PTAS algorithms for hyperbolic problems—crucially including SMT construction—has not yet been realized, and resolving this would advance the state of the art.