Exact Algorithms for Minimum Dilation Triangulation

2025-02-25

AI Summary

This paper addresses the Minimum-Dilation Triangulation (MDT) problem for point sets in the plane: computing a triangulation that minimizes the maximum dilation—the ratio of the shortest Euclidean path distance between any two vertices in the triangulation to their straight-line Euclidean distance. Despite its significance, the problem has long lacked a polynomial-time algorithm, and its objective—comprising numerous nested square roots—defies exact evaluation. We propose a geometry-driven exact framework integrating structured search pruning, symmetry analysis of regular polygons, batch shortest-path queries, and high-precision arithmetic. Our method scales exact MDT computation from ≤200 points to up to 30,000 points—the first such breakthrough. We rigorously establish a lower bound of 1.44116 on the dilation of the regular 84-gon, substantially narrowing the theoretical gap. Moreover, we fully determine the optimal MDT for all regular *n*-gons with *n* ≤ 100.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We provide a spectrum of new theoretical insights and practical results for finding a Minimum Dilation Triangulation (MDT), a natural geometric optimization problem of considerable previous attention: Given a set $P$ of $n$ points in the plane, find a triangulation $T$, such that a shortest Euclidean path in $T$ between any pair of points increases by the smallest possible factor compared to their straight-line distance. No polynomial-time algorithm is known for the problem; moreover, evaluating the objective function involves computing the sum of (possibly many) square roots. On the other hand, the problem is not known to be NP-hard. (1) We provide practically robust methods and implementations for computing an MDT for benchmark sets with up to 30,000 points in reasonable time on commodity hardware, based on new geometric insights into the structure of optimal edge sets. Previous methods only achieved results for up to $200$ points, so we extend the range of optimally solvable instances by a factor of $150$. (2) We develop scalable techniques for accurately evaluating many shortest-path queries that arise as large-scale sums of square roots, allowing us to certify exact optimal solutions, with previous work relying on (possibly inaccurate) floating-point computations. (3) We resolve an open problem by establishing a lower bound of $1.44116$ on the dilation of the regular $84$-gon (and thus for arbitrary point sets), improving the previous worst-case lower bound of $1.4308$ and greatly reducing the remaining gap to the upper bound of $1.4482$ from the literature. In the process, we provide optimal solutions for regular $n$-gons up to $n = 100$.

Problem

• Develop exact algorithms for Minimum Dilation Triangulation
• Scalable techniques for shortest-path queries evaluation
• Establish lower bounds on dilation for geometric shapes

Innovation

• Enhances MDT computation with new geometric insights.
• Develops scalable techniques for accurate path evaluation.
• Establishes improved lower bounds for dilation in polygons.