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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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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Remarks on d-Dimensional TSP Optimal Tour Length Behaviour
TL;DR: The well-known $O(n^{1-1/d})$ behaviour of the optimal tour length for TSP in d-dimensional Cartesian space causes breaches of the triangle inequality, which is attempted to remedy.
Continuous approximation models for the fleet replacement and composition problem
O Jabali,Güneş Erdoğan +1 more
TL;DR: In this article, the authors consider a finite planning horizon, throughout which they optimize the fleet replacement and composition problem while explicitly accounting for vehicle routing costs, including vehicle purchasing cost, maintenance cost, salvage revenue and routing costs.
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Review of Length Approximations for Tours with Few Stops
Youngmin Choi,Paul Schonfeld +1 more
TL;DR: In this article, the shortest tour distance for visiting all points exactly once and returning to the origin is computed by solving the well-known traveling salesman problem (TSP), which is a special case of the TSP.
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Stacks, queues, and deques with order-statistic operations
Dana Richards,Jeffrey S. Salowe +1 more
TL;DR: This work shows how to implement an order-statistic deque so that inject(x, D), eject, eject, push, andpop(D) take O(logk) amortized time andfind(k, D) takes worst-case constant time; the time bounds can be made worst case using a technique of Gajewska and Tarjan.
Optimal Cost Prediction in the Vehicle Routing Problem Through Supervised Learning
TL;DR: The goal of this thesis is to apply supervised learning to predict the optimal cost of single-vehicle pick-up and delivery problem, leading to a simpler implementation compared with combinatorial neural networks.
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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