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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Multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost
TL;DR: In this article, the problem is formulated as a multi-objective nonlinear integer programming problem with quadratic cost under three different situations, i.e. complete, partial or null information about the population.
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Approximating the Expected Values for Combinatorial Optimization Problems over Stochastic Points
Lingxiao Huang,Jian Li +1 more
TL;DR: In this article, the authors considered the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each vertex is uncertain.
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Probabilistic analysis of a network design problem heuristic
TL;DR: An efficient heuristic procedure for solving budget network design problems embedded in a circle on the euclidean plane where the nodes are independently and randomly distributed over the circle.
Journal ArticleDOI
Randomized near-neighbor graphs, giant components and applications in data science.
TL;DR: It is proved that it suffices to connect every point to c d,1 log log n points chosen randomly among its cd,2 log n-nearest neighbors to ensure a giant component of size n - o(n) with high probability.
Journal ArticleDOI
Heuristics and bounds for the travelling salesman location problem on the plane
David Simchi-Levi,Oded Berman +1 more
TL;DR: This paper presents heuristics for the travelling salesman location problem on the plane with rectilinear or Euclidean distances and shows a polynomial heuristic which produces solutions that are at most 50% worse than the optimal solution.
References
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On the shortest spanning subtree of a graph and the traveling salesman problem
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