7 research outputs found
Complexity of Minimum Corridor Guarding Problems
In this paper, the complexity of minimum corridor guarding problems is discussed. These problem can be described as: given a connected orthogo-nal arrangement of vertical and horizontal line segments and a guard with unlimited visibility along a line segment, find a tree or a closed tour with minimum total length along edges of the arrangement, such that if the guard runs on the tree or on the closed tour, all line segments are visited by the guard. These problems are proved to be NP-complete. Keywords: computational complexity, computational geometry, corridor guarding, NP-complet
Online Searching with an Autonomous Robot
We discuss online strategies for visibility-based searching for an object
hidden behind a corner, using Kurt3D, a real autonomous mobile robot. This task
is closely related to a number of well-studied problems. Our robot uses a
three-dimensional laser scanner in a stop, scan, plan, go fashion for building
a virtual three-dimensional environment. Besides planning trajectories and
avoiding obstacles, Kurt3D is capable of identifying objects like a chair. We
derive a practically useful and asymptotically optimal strategy that guarantees
a competitive ratio of 2, which differs remarkably from the well-studied
scenario without the need of stopping for surveying the environment. Our
strategy is used by Kurt3D, documented in a separate video.Comment: 16 pages, 8 figures, 12 photographs, 1 table, Latex, submitted for
publicatio
Polygon Exploration with Time-Discrete Vision
With the advent of autonomous robots with two- and three-dimensional scanning
capabilities, classical visibility-based exploration methods from computational
geometry have gained in practical importance. However, real-life laser scanning
of useful accuracy does not allow the robot to scan continuously while in
motion; instead, it has to stop each time it surveys its environment. This
requirement was studied by Fekete, Klein and Nuechter for the subproblem of
looking around a corner, but until now has not been considered in an online
setting for whole polygonal regions.
We give the first algorithmic results for this important algorithmic problem
that combines stationary art gallery-type aspects with watchman-type issues in
an online scenario: We demonstrate that even for orthoconvex polygons, a
competitive strategy can be achieved only for limited aspect ratio A (the ratio
of the maximum and minimum edge length of the polygon), i.e., for a given lower
bound on the size of an edge; we give a matching upper bound by providing an
O(log A)-competitive strategy for simple rectilinear polygons, using the
assumption that each edge of the polygon has to be fully visible from some scan
point.Comment: 28 pages, 17 figures, 2 photographs, 3 tables, Latex. Updated some
details (title, figures and text) for final journal revision, including
explicit assumption of full edge visibilit
A PTAS for Euclidean TSP with Hyperplane Neighborhoods
In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given
a collection of geometric regions in some space. The goal is to output a tour
of minimum length that visits at least one point in each region. Even in the
Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying
more tractable special cases of the problem. In this paper, we focus on the
fundamental special case of regions that are hyperplanes in the -dimensional
Euclidean space. This case contrasts the much-better understood case of
so-called fat regions.
While for an exact algorithm with running time is known,
settling the exact approximability of the problem for has been repeatedly
posed as an open question. To date, only an approximation algorithm with
guarantee exponential in is known, and NP-hardness remains open.
For arbitrary fixed , we develop a Polynomial Time Approximation Scheme
(PTAS) that works for both the tour and path version of the problem. Our
algorithm is based on approximating the convex hull of the optimal tour by a
convex polytope of bounded complexity. Such polytopes are represented as
solutions of a sophisticated LP formulation, which we combine with the
enumeration of crucial properties of the tour. As the approximation guarantee
approaches , our scheme adjusts the complexity of the considered polytopes
accordingly.
In the analysis of our approximation scheme, we show that our search space
includes a sufficiently good approximation of the optimum. To do so, we develop
a novel and general sparsification technique to transform an arbitrary convex
polytope into one with a constant number of vertices and, in turn, into one of
bounded complexity in the above sense. Hereby, we maintain important properties
of the polytope
The traveling salesman problem for lines, balls and planes
We revisit the traveling salesman problem with neighborhoods (TSPN) and
propose several new approximation algorithms. These constitute either first
approximations (for hyperplanes, lines, and balls in , for ) or improvements over previous approximations achievable in comparable times
(for unit disks in the plane).
\smallskip (I) Given a set of hyperplanes in , a TSP tour
whose length is at most times the optimal can be computed in
time, when is constant.
\smallskip (II) Given a set of lines in , a TSP tour whose
length is at most times the optimal can be computed in polynomial
time for all .
\smallskip (III) Given a set of unit balls in , a TSP tour
whose length is at most times the optimal can be computed in polynomial
time, when is constant.Comment: 30 pages, 9 figures; final version to appear in ACM Transactions on
Algorithm