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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Book ChapterDOI
Effective Covering of Supplied Nanostores in Emerging Cities
TL;DR: This chapter deals with a real-life application problem where a Moroccan company has to build an effective strategy to supply nanostores in a major city like Casablanca, and adopts a multi-objective approach to a multiple traveling salesman problem.
Convergence rate for geometric statistics of point processes with fast decay dependence
Tianshu Cong,Aihua Xia +1 more
TL;DR: In this paper , Błaszczyszyn, Yogeshwaran and Yukich considered the rates of a normal approximation in terms of the Wasserstein distance for statistics of point processes on R d satisfying fast decay dependence, and demonstrated the use of the central limit theorems for statistics arising from two families of point process: the rarified Gibbs point processes and the determinantal point processes with fast decay kernels.
Computing the Moments of Costs over the Solution Space of the TSP in Polynomial Time
TL;DR: Polynomial time algorithms are given to compute the third and fourth moments about the mean of tour costs over the solution space of the general symmetric Travelling Salesman Problem to provide a tractable method to computed the skewness and kurtosis of the probability distribution of tour Costs.
Journal ArticleDOI
Minimum Spanning Trees Across Well-Connected Cities and with Location-Dependent Weights
TL;DR: In this paper, it was shown that if the cities are well connected in a certain sense, then the convergence of the minimum spanning tree for nodes distributed throughout the unit square S with location dependent edge weights converges to zero in probability.
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Separating subadditive Euclidean functionals
Alan Frieze,Wesley Pegden +1 more
TL;DR: It is proved that the TSP on random points in Euclidean space is indeed asymptotically distinct from these and other natural lower bounds, and it is shown that this separation implies that branch‐and‐bound algorithms based on these naturalLower bounds must take nearly exponential time to solve the T SP to optimality, even in average case.
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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