Asymptotic constant-factor approximation algorithm for the Traveling Salesperson Problem for Dubins' vehicle

This article proposes the first known algorithm that achieves a constant-factor approximation of the minimum length tour for a Dubins' vehicle through $n$ points on the plane. By Dubins' vehicle, we mean a vehicle constrained to move at constant speed along paths with bounded curvature without reversing direction. For this version of the classic Traveling Salesperson Problem, our algorithm closes the gap between previously established lower and upper bounds; the achievable performance is of order $n^{2/3}$

A Pseudopolynomial Algorithm for Alexandrov's Theorem

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution leads to the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron to arbitrary precision given the metric, and prove a pseudopolynomial bound on its running time. Along the way, we develop pseudopolynomial algorithms for computing shortest paths and weighted Delaunay triangulations on a polyhedral surface, even when the surface edges are not shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes; an abbreviated v2 was at WADS 200

Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published $3$-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page

09111 Abstracts Collection -- Computational Geometry

From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 ``Computational Geometry \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available