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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Book ChapterDOI
Vehicle Routing and the Supply Chain
Shoshana Anily,Julien Bramel +1 more
TL;DR: In this article, the authors present models, analyses and approaches for solving a number of different distribution problems where retailers/customers are external, i.e., not part of the same company, or where retailers are internal to the firm.
Journal ArticleDOI
Total distance approximations for routing solutions
TL;DR: This paper presents regression-based estimation models that provide fast predictions for the travel distance in the traveling salesman problem, the capacitated vehicle routing problem with Time Windows, and the multi-region multi-depot pickup and delivery problem (MR-MDPDP).
Convergence rates of minimal graphs with random vertices
TL;DR: Almost sure (a.m.s.) and Lp-norm convergence rates are investigated and asymptotic lower bounds for the respective rates of convergence are obtained, using minimax techniques from nonparametric function estimation.
Journal ArticleDOI
A probabilistic analysis of tour partitioning heuristics for the capacitated vehicle routing problem with unsplit demands
TL;DR: This work analyzes the asymptotic performance of a class of heuristics called route first-cluster second, and in particular, the empirically well-studied, Sweep algorithm and Optimal Partitioning heuristic, for any distribution of the demands and when customers are independent and identically distributed in a given region.
Journal ArticleDOI
Minimization and maximization versions of the quadratic travelling salesman problem
Aichholzer Oswin,Anja Fischer,Frank Fischer,J. Fabian Meier,Ulrich Pferschy,Alexander Pilz,Rostislav Staněk +6 more
TL;DR: It is shown that the sum of inner turning angles in an optimal solution always equals .
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.
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Solution of a Large-Scale Traveling-Salesman Problem
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