Exact Algorithms for Minimum Dilation Triangulation

TL;DR

This paper introduces new practical algorithms and theoretical bounds for the Minimum Dilation Triangulation problem, significantly expanding the size of instances that can be solved exactly and improving known lower bounds on dilation.

Contribution

It presents scalable methods for computing and certifying optimal triangulations for large point sets and establishes a new lower bound on dilation for regular polygons.

Findings

Extended solvable instances from 200 to 30,000 points

Developed scalable shortest-path evaluation techniques

Improved lower bound on dilation of regular polygons

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 $n = 100$.