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
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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
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Quantizers and the worst-case Euclidean traveling salesman problem
TL;DR: By generalizing the classic “strip” algorithm used by Few, γk is the least possible expected value of 1k |z − Q(z)|2, over all functions Q from R onto an M-subset of R (Q is called a quantizer for R).
Task allocation and vehicle routing in dynamic environments
TL;DR: Task Allocation and Vehicle Routing in Dynamic Environments (TAVR) as discussed by the authors is a task that deals with task allocation and vehicle routing in dynamic environments, such as dynamic environments.
Journal ArticleDOI
Cube versus torus models and the euclidean minimum spanning tree constant
TL;DR: It is shown that the length of the minimum spanning tree through points drawn uniformly from the d-dimensional torus is almost surely asymptotically equivalent to the length in the traditional d-cube model, which implies that the analytical expression recently obtained for theminimum spanning tree (MST) constant is in fact valid for thetraditional d-Cube model.
Journal ArticleDOI
Geometry of the minimal spanning tree of a random 3-regular graph
TL;DR: The techniques of this paper can be used to establish the scaling limit of the MST in the setting of various different random graph models provided one additional technical condition is verified.
Journal ArticleDOI
Solution Approaches for the Stochastic Capacitated Traveling Salesmen Location Problem with Recourse
Luca Bertazzi,Francesca Maggioni +1 more
TL;DR: A heuristic proposed for the Capacitated Traveling Salesman Problem and the optimal solution of a stochastic second-order cone formulation with an approximate objective function are compared.
References
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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.
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