Journal ArticleDOI
The shortest path through many points
Jillian Beardwood,John H. Halton,J. M. Hammersley +2 more
- Vol. 55, Iss: 4, pp 299-327
Reads0
Chats0
TLDR
In this paper, it was shown that the length of the shortest closed path through n points in a bounded plane region of area v is almost always asymptotically proportional to √(nv) for large n; and this result was extended to bounded Lebesgue sets in k-dimensional Euclidean space.Abstract:
We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √(nv) for large n; and we extend this result to bounded Lebesgue sets in k–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.read more
Citations
More filters
Journal ArticleDOI
Scheduling Dial-a-Ride Transportation Systems
TL;DR: An analytic investigation into the fundamental aspects of scheduling “Dial-a-Ride” transportation systems is conducted and a class of algorithms is derived for which performance can be measured in a precise asymptotic probabilistic sense.
Journal ArticleDOI
Heuristics Based on Spacefilling Curves for Combinatorial Problems in Euclidean Space
TL;DR: A family of heuristics to solve combinatorial problems such as routing and partitioning that exploit geometry but ignore specific distance measures are described, which seem well-suited to operational problems where time or computing resources are limited.
Journal ArticleDOI
The TSP phase transition
Ian P. Gent,Toby Walsh +1 more
TL;DR: It is shown that for random problems there is a rapid transition between soluble and insoluble instances of the decision problem at a critical value of this parameter, a natural parameter for the two-dimensional Euclidean traveling salesman problem.
Proceedings ArticleDOI
Asymptotic experimental analysis for the Held-Karp traveling salesman bound
TL;DR: Empirical evidence is provided in support of using theHK bound as a stand-in for the optimal tour length when evaluating the quality of near-optimal tours, and data indicates that the HK bound can provide substantial ‘‘variance reduction’’ in experimental studies involving randomly generated instances.
Journal ArticleDOI
A concise guide to existing and emerging vehicle routing problem variants
TL;DR: Vehicle routing problems have been the focus of extensive research over the past sixty years, driven by their economic importance and their theoretical interest as mentioned in this paper, and the diversity of applications has motivated the study of a myriad of problem variants with different attributes.
References
More filters
Journal ArticleDOI
On the shortest spanning subtree of a graph and the traveling salesman problem
TL;DR: Kurosh and Levitzki as discussed by the authors, on the radical of a general ring and three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings.
Book ChapterDOI
Solution of a Large-Scale Traveling-Salesman Problem
TL;DR: The RAND Corporation in the early 1950s contained Arrow, Bellman, Dantzig, Flood, Ford, Fulkerson, Gale, Johnson, Nash, Orchard-Hays, Robinson, Shapley, Simon, Wagner, and other household names as discussed by the authors.
Journal ArticleDOI