Exact Algorithms for Minimum Dilation Triangulation

TL;DR

This work tackles the Minimum Dilation Triangulation problem, seeking a triangulation of a point set that minimizes the worst-case ratio between a shortest path in the triangulation and the direct Euclidean distance. It advances both theory and practice by (i) constructing a practical MDT edge supergraph through an ellipse-property-based enumeration with dead sectors and a quadtree, enabling exact solutions for up to $n=30{,}000$ points, (ii) developing scalable, exact evaluation of many shortest-path sums of square roots to certify optimality, and (iii) proving a new lower bound of $1.44116$ on the dilation of the regular $84$-gon and providing optimal solutions for regular $n$-gons up to $n=100$, thereby narrowing the gap to the best known upper bounds. The paper also presents two exact solvers, IncMDT and BinMDT, based on a triangulation supergraph and a SAT formulation with dilation-path separation, supplemented by dilation sampling to reduce expensive computations. Together, these contributions push practical MDT computation to large-scale instances and deepen understanding of worst-case dilations in planar triangulations, with implications for Euclidean spanners and related geometric optimization problems.

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 $n = 100$.

Figures (6)

• Figure 1: MDT solutions for four instances. The red edges indicate a dilation-defining path.
• Figure 2: Any path connecting $\ell$ and $r$ with length below $\kappa = \rho d(\ell,r)$ must lie within the ellipse (dashed black lines). The edge $st$ does not have the ellipse property as neither $s$ nor $t$ lie inside the ellipse; therefore, inserting $st$ makes connecting $\ell$ and $r$ by a sufficiently short path impossible.
• Figure 3: A dead sector $\mathcal{DS}_{\rho}(p, \ell, r)$, shaded in gray and red with the ellipse $E(\ell, r, \rho)$. We approximate the ellipse by a disk centered at $p$ with radius $\tilde{A}_{\rho}(p,\ell,r)$ (thereby ignoring the red area), which is the distance from $p$ to the farthest point $q$ of the rectangle $B(\ell, r, \rho)$ in $\mathcal{DS}_{\rho}(p, \ell, r)$.
• Figure 4: If $st$ (black) is present, the pairs $\ell, r$ (resp. $\ell',r'$) can only be connected by paths that have at least the length of the solid blue (resp. orange) path. Hence, the ratio between the solid and dashed blue (resp. orange) paths is a lower bound on the dilation of any triangulation containing $st$. The maximum of such lower bounds for $st$ is the dilation threshold $\vartheta(st)$ of $st$.
• Figure 5: Progress of the incremental algorithm on an instance with $n = 50$ points. Green edges indicate changes in the triangulation, red edges indicate a dilation-defining path.

Theorems & Definitions (3)

• Definition 1
• Definition 3
• Lemma 5