The shortest path through many points
01 Oct 1959-Vol. 55, Iss: 4, pp 299-327
TL;DR: 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.
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TL;DR: There is a deep and useful connection between statistical mechanics and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters), and a detailed analogy with annealing in solids provides a framework for optimization of very large and complex systems.
Abstract: There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and provides an unfamiliar perspective on traditional optimization problems and methods.
41,772 citations
01 Jan 1989
TL;DR: In this paper, the authors present a short abstract, which is a summary of the paper.Short abstract, isn't it? But it is short abstracts, not abstracts.
Abstract: Short abstract, isn't it?
1,666 citations
Cites background from "The shortest path through many poin..."
...In addition, for a random uniform distribution of N cities over a rectangular area of R units, an asymptotic expected length formula for the optimal tour has been derived [23]....
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TL;DR: An implementation of the Lin–Kernighan heuristic, one of the most successful methods for generating optimal or near-optimal solutions for the symmetric traveling salesman problem (TSP), is described.
1,462 citations
TL;DR: The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.
Abstract: We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c > 1 and given any n nodes in ℛ2, a randomized version of the scheme finds a (1 + 1/c)-approximation to the optimum traveling salesman tour in O(n(log n)O(c)) time. When the nodes are in ℛd, the running time increases to O(n(log n)(O(√c))d-1). For every fixed c, d the running time is n · poly(logn), that is nearly linear in n. The algorithmm can be derandomized, but this increases the running time by a factor O(nd). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time.All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as lp for p ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
1,113 citations
Cites background from "The shortest path through many poin..."
...The TSP Algorithm This section describes our approximation scheme for the TSP. Section 2.1 describes the algorithm for Euclidean TSP in W2....
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...In this paper, we show that Euclidean TSP has a PTAS....
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...Polynomial-time approximation schemes for Euclidean TSP and other geometric problem....
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...Lemma 3 is implicit in prior work on Euclidean TSP [Beardwood et al. 1959; Karp 1977] and can safely be called a folk theorem....
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...Unfortunately, even Euclidean TSP is NP-hard (Papadimitriou [1977]; Garey et al. [1976])....
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References
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01 Feb 1956
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.
Abstract: 7 A Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull Acad Sei URSS, Ser Math vol 5 (1941) pp 233-240 8 J Levitzki, On the radical of a general ring, Bull Amer Math Soc vol 49 (1943) pp 462^66 9 -, On 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, Trans Amer Math Soc vol 74 (1953) pp 384-409
5,104 citations
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.
Abstract: The RAND Corporation in the early 1950s contained “what may have been the most remarkable group of mathematicians working on optimization ever assembled” [6]: Arrow, Bellman, Dantzig, Flood, Ford, Fulkerson, Gale, Johnson, Nash, Orchard-Hays, Robinson, Shapley, Simon, Wagner, and other household names. Groups like this need their challenges. One of them appears to have been the traveling salesman problem (TSP) and particularly its instance of finding a shortest route through Washington, DC, and the 48 states [4, 7].
1,461 citations